Spring 2015 Colloquia

Variable exponent dynamical boundary value problems on "bad" domains 
Alejandro Velez-Santiago (University of California, Riverside) 
March 3, 2015

Abstract: We are concerned with the solvability of a class quasi-linear parabolic boundary value problems involving the p(x)-Laplace operator and dynamical boundary conditions on a large class of bounded non-smooth and fractal domains. We will discuss well-posedness results, and will provide examples of domains and situations when the problem is uniquely solvable. A regularity result for the corresponding solutions of the problem is also discussed.

Geometric analysis on singular and non-compact spaces 
Jesse Gell-Redman (John Hopkins University) 
February 26, 2015

Abstract: The geometric objects which arise naturally in mathematics are frequently and in important cases not smooth. Instead, many have structured ends which look for example like cones, horns, or families of these; a paradigmatic example is the Riemann moduli space of surfaces of genus $g > 1$ with the Weil-Petersson metric, which near its singular locus is approximately Riemannian products of families of horns. This talk concerns geometric analysis on such spaces, especially the analysis of naturally arising differential operators like the Hodge-Laplacian, the Dirac operator, and the D'Alembertian (wave operator). The analysis of such operators is substantially more complex in the non-smooth setting, but we will present tools from microlocal analysis which provide both a clear picture of and a resolution for many problems. We will focus in particular on recent progress in index theory and spectral theory.

Potential methods for higher-order boundary-value problems 
Ariel Barton (University of Missouri)
February 25, 2015

Abstract: The theory of boundary-value problems for the Laplacian in Lipschitz domains is by now very well developed. Furthermore, many of the existing tools and known results for the Laplacian have been extended to the case of second-order linear equations of the form $\nabla\cdot A\nabla u=0$, where $A is a matrix of variable coefficients.
However, at present there are many open questions in the theory of higher-order elliptic differential equations. In this talk I will describe a generalization of layer potentials to the case of higher-order operators $\nabla^m\cdot A\nabla^m$; layer potentials are a common and very useful tool in the theory of second-order equations. I will then describe some applications of layer potentials to the theory of boundary-value problems.
This is joint work with Steve Hofmann and Svitlana Mayboroda.

Spatio-temporal Bayesian nonparametric variable selection models of functional MRI data
Linlin Zhang (Rice University)
January 13, 2015

Abstract: Functional magnetic resonance imaging (fMRI) is a common tool to detect changes in neuronal activity. It measures blood oxygenation level-dependent (BOLD) contrast that depends on changes in regional cerebral blood flow. Statistical methods play a crucial role in the analysis of fMRI data, due to their complex spatial and temporal correlation structure. Common modeling approaches rely on general linear models (GLMs). In this talk, I will present novel wavelet-based Bayesian nonparametric models for the analysis of fMRI data in the case of a single subject and multiple subjects. The proposed methods take into account both the temporal and spatial correlation structures in the fMRI data. The goal of the single-subject fMRI modeling is to provide a joint analytical framework to detect brain regions showing neuronal activity in response to a given stimulus, and simultaneously infer the association of spatially remote voxels. Furthermore, the multiple-subject modeling approach ties the strength of activation in response to an external stimulus within and across subjects via a spiked Bayesian nonparametric prior. Due to the complex spatio-temporal structure and high dimensionality of the fMRI data, the implementation of inference for the proposed method is computationally challenging using fully Markov chain Monte Carlo (MCMC). In addition to MCMC algorithm, we use variational Bayes (VB) method for the inference. The results on simulated data with VB method show good performance at a reduced computational cost.

Learning Bayesian network with mixed variables and its application to cancer systems biology 
Quingyang Zhang (Northwestern University) 
January 8, 2015

Abstract: A Bayesian network (BN) is a directed acyclic graph that encodes the joint distribution of a set of random variables. Compared to other graphical models, BN features the capability to rigorously model the causal relationships among variables. In the existing literature, most BN models either assume that all nodes follow a Gaussian distribution or a multinomial distribution. These assumptions impose limitations to BN, both methodologically and computationally when applied to complex problems that involve both continuous and discrete variables. In the motivating example of TCGA cancer data, each subject is represented by various molecular profiles, including gene expression level, promoter methylation level, copy number and somatic mutation. We aim to develop graphical models that can incorporate different profiles to infer the genetic/epigenetic pathways underlying cancer phenotypes. In this talk, I will discuss two new approaches for modeling mixed variables in BN, namely, Logistic BN using discretized variables and Gaussian-Probit BN without discretization of continuous variables, together with a Blockwise Coordinate Descent algorithm for network estimation. In addition, a stepwise correlation-based feature selector (SCBS) is proposed to select features from high-dimensional data. The effectiveness of these approaches will be demonstrated by both simulated data and the TCGA ovarian cancer data.

 

Fall 2014 Colloquia

Recovering a variety from its cohomology 
Sofia Tirabassi (University of Utah) 
December 18, 2014

Abstract: One of the possible paths in the study of projective varieties is to compute their cohomological invariants. It is then natural to ask to which extent these invariants carry information about geometry. I will present a joint work with Z. Jiang and M. Lahoz, in which we were able to recover, under certain hypotheses, a variety from some of its cohomological data.

Functional Time Series Analysis 
Greg Rice (University of Utah) 
December 16, 2014

Abstract: Abstract: Functional Data Analysis (FDA) is concerned with observations that are viewed as functions defined over some set. For example, the pollution level in a city on a given day can be thought of as a function defined on the time of day. Nearly all of the early asymptotic results from FDA are derived under the assumption that the observations are independent and identically distributed. A common source of functional data, however, is when long, continuous records are broken into segments of smaller, perhaps hourly or daily, curves. In this case, successive curves may exhibit dependencies, and Functional Time Series Analysis seeks to provide a flexible, nonparametric framework for studying such data. In this talk, we will first survey some of the central ideas of FDA, including principle component analysis, which is a commonly used technique in dimension reduction. We will then discuss the framework and theory of Functional Time Series Analysis. The talk will conclude with a more detailed application to the one-way functional analysis of variance problem and a study of electricity demand data from Adelaide, Australia.

Projective dimension and Stillman's question 
Paolo Mantero (University of California at Riverside)
December 11, 2014

Abstract: Minimal projective resolution are major tools to study ideals in a polynomial ring. In this talk, we will discuss results and open problems concerning resolutions of ideals, and in particular we will focus on a question raised by M. Stillman concerning their lengths. We will discuss the motivation and some intriguing features of this question, and illustrate recent developments and upcoming results.

Ensemble Trees and CLTs: Using Subsamples to Peek Inside the Black Box
Lucas Mentsch (Cornell University)
December 4, 2014

Abstract: Modern learning algorithms are typically seen as prediction-only tools, meaning that the interpretability and intuition provided by a more traditional modeling approach are sacrificed in order to achieve superior predictions. In this talk, we argue that this black-box perspective need not always be the case. We demonstrate that predictions from ensemble learners like bagged trees and random forests, when built with subsamples in lieu of full bootstrap samples, can be viewed as incomplete, infinite-order U-statistics and as such, are asymptotically normal. Furthermore, we show that the limiting variance depends only on the size of the ensemble relative to the size of the training set and by enforcing a structure on the subsamples used in the ensemble, we can form a consistent estimate of variance at no additional computational cost. This allows for statistical inference to be carried out in practice and in particular, we can produce confidence intervals to accompany predictions and define formal hypothesis tests for both additivity and feature significance. Then a large test set is required, we extend our testing procedures and utilize random projections to accommodate the potential p>>n setting. These tools are illustrated on data provided by Cornell University's Lab of Ornithology.

A full-rank chiral polytope in dimension 4
Javier Bracho (UNAM)
November 13, 2014 

Abstract: An unexpected example of a finite regular polytope of rank 4 which admits a chiral embedding in $R^4$ will be presented and discussed

What types of singular polynomials exist?
Matt Hedden (Michigan State University) 
November 6, 2014

Abstract:  A polynomial in 2 complex variables naturally gives rise to a surface, by considering the points in 2-dimensional complex space where it vanishes. If at one of these points the partial derivatives of the polynomial also vanish, the point (and polynomial) is said to be singular. In this talk I will discuss questions pertaining to the algebro-geometric classification of singular polynomials. It turns out that singular points of complex polynomials are connected to knot theory in a beautiful way, and this connection allows topological methods to attack these classification questions.
This is joint work with Maciej Borodzik and Chuck Livingston.
The colloquium will be followed by a more detailed seminar for the experts.

The details 

Abstract: I'll raise the level of discussion from the previous talk, and tell you about the techniques we use to prove our results which obstruct certain configurations of singularities from arising on an algebraic curve in CP^2.

Combinatorial stability and representation stability
Tom Church (Stanford University)
October 31, 2014 
Friday at 3:05 p.m.
SCEN 407
*note the change of day, time and location

Abstract: How many roots does a random squarefree polynomial f(T) in F_q[T] have? On average, it's a bit less than one root per polynomial. The precise answer depends on the degree of f(T), but as deg f(T) goes to infinity, the expectation stabilizes and converges to 1 - 1/q + 1/q^2 - 1/q^3 + ... = q / (q+1). In joint work with J. Ellenberg and B. Farb, we proved that the stabilization of this combinatorial formula is equivalent to a representation-theoretic stability in the cohomology of braid groups. I will describe how combinatorial stability for statistics of squarefree polynomials, of maximal tori in GL_n(F_q), and other natural geometric counting problems can be converted to questions of representation stability in topology, and vice versa.

Uniform Growth Rate
Jing Tao (University of Oklahoma)
October 23, 2014

Abstract: In an evolutionary system in which the rules of mutation are local in nature, the number of possible descendants of a given object after m mutations grows exponentially in m, but the growth rate depends only on the set of rules, not on the original object. In this talk, I will describe several evolutionary systems and how this principle can be applied to derive some interesting results, such as, the space of homotopy classes of triangulations of the surface S with n vertices related by triangle flips has a growth rate that is independent of S and n; and, given the correct choice of generators, mapping class groups, Out(F_n) and SL(n,Z), all have uniform growth rates.

Growing Our Own Expertise: The Role of Colleagues in Improving Mathematics Instruction
Ilana Horn (Vanderbilt University)
October 16, 2014

Abstract: Research on mathematics teaching and learning points to the importance of helping students learn more than the content but also the processes of mathematical thinking. Most of us did not experience this type of instruction in our own schooling, yet there are good reasons to want to learn to teach in ways that support learning the why of mathematics in addition to the what and how. In this talk, I present a framework for interactive forms of teaching that support the development of students’ mathematical thinking. Then, I talk about some common challenges that arise as we shift instruction in that direction. Finally, I present ways colleagues can support each other in growing this form of practice.

Anisotropic surface energies 
Bennett Palmer (Idaho State University) 
October 2, 2014

Abstract: Anisotropic surface energies are used to model the interface between immiscible materials when at least one of the materials is in an ordered phase (e.g. crystal,liquid crystal). This talk will discuss the variational theory related to this type of energy. In particular, we will discuss results characterizing stable equilibrium surfaces and the corresponding flow.

The talk is aimed at a general audience and will concentrate on motivation.

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics 
Jiri Lebl (Oklahoma State University)
September 25, 2014

Abstract: This is joint work with Dusty Grundmeier and Liz Vivas. I will discuss a generalization of Green's hyperplane restriction theorem (a well-known theorem in commutative algebra) and its application to rigidity of CR mappings between hyperquadrics (a well-known problem in CR geometry). That is, we prove that the rank of a real-analytic function thought of as a Hermitian form is bounded by an expression involving the maximal rank on a complex affine hyperplane. This theorem in turn will be used to show that if $f$ is a CR mapping taking hyperquadric $Q(a,b)$ to hyperquadric $Q(A,B)$, both of which are not spheres, then either $f$ maps into a hyperplane, or $A$ is bounded as a function of $B$. The proof is by extending the ideas of generic monomial ideals to analytic functions.  

Left-orderability and three-manifolds 
Tye Lidman (University of Texas) 
September 18, 2014

Abstract: A group is called left-orderable if it can be given a left-invariant total order. We will study this property and discuss the question of when the fundamental group of a three-manifold is left-orderable. We will also discuss conjectural relationships between the topology and geometry of the manifold and this algebraic condition.

A Conjecture of Chudnovksy 
Louiza Fouli (New Mexico State University) 
September 11, 2014

 

Spring 2014 Colloquia

Oriented matroids and straight-edge embeddings of graphs 
Elena Pavelescu (Oklahoma State University)
May 1, 2014

Abstract: Matroid theory is an abstract theory of dependence introduced by Whitney in 1935. It is a natural generalization of linear (in)dependence. Oriented matroids can be thought of as combinatorial abstractions of point configurations over the reals. To every linear (straight-edge) embedding of a graph one can associate an oriented matroid, and the oriented matroid captures enough information to determine which pairs of disjoint cycles in the embedded graph are linked. In this talk, we will introduce the basics of oriented matroids. Then we show that any linear embedding of K9, the complete graph on nine vertices, contains a non-split link with three components.

Schmidt boundaries of Feerman type space-times 
Sorin Drogamir (Universita' della Basilicata, Potenza, Italy) 
April 8, 2014
*note change of day and location

Abstract: For every smoothly bounded strictly pseudoconvex domain D there is a natural Lorentzian metric F on the product C(M) = M x S of the boundary M of D and the circle S, discovered by C. Fefferman (1976) in his study the boundary behavior of the Bergman kernel of D. There is a natural connection-distribution H (discovered by C.R. Graham, 1987) determined by a fixed contact form on M. Let T and N be respectively the Reeb vector field and the tangent to the S-action on C(M). If T is the H-horizontal lift of T then X = T - S is a time-orientation on C(M), hence (C(M), F, X) is a space-time. One purpose of this talk is to show how Schmidt's boundaries (b-boundaries, conformal and projective boundaries, cf. B.G. Schmidt, 1971) of (C(M), F, X) lead to natural boundary constructions for M. Godel's universe is a solution to Einstein's field equations for an incoherent matter distribution. The static Einstein universe is a solution to the same equations, implying that Mach's principle (according to which the matter distribution of space-time should uniquely determine its geometry) cannot be incorporated into general relativity theory on the ground of the gravitational field equations alone. It is conjectured that an additional boundary condition should be added to distinguish between the two solutions and we speculate that Schmidt boundaries may do the job. The use of Cauchy-Riemann geometry is meant to circumnavigate the notorious lack of computability of Schmidt boundaries by lowering the dimension, an idea suggested by the b-boundary calculations for the Schwartzschild metric (cf. R.A. Johnson, 1977) and for the closed Friedman model (cf. B. Bosshard, 1976).

Closed-range composition operators on Hardy and Bergman spaces 
Pratibha Ghatage (Cleveland State University) 
April 3, 2014

Abstract: Some classical spaces of analytic functions, such as the Hardy or Bergman space on the open unit disk D, are defined in terms of Taylor coefficients of analytic functions. If φ is an analytic function on D whose range is contained in D, then either the Hardy or the Bergman space is closed under composition with φ. It is well-known that such composition operators are injective. We discuss geometric conditions on φ which makes them boundedly invertible on their range.

Regular functions over the quaternions 
Caterina Stoppato (University of Florence, Italy)
March 20, 2014

Abstract: During the last century, several function theories have been introduced over the algebra of quaternions and other hypercomplex algebras. The aim of such constructions was to recover in higher dimensions the refined tools that are available in the complex case through the theory of holomorphic functions. This is by no means a trivial generalization because of all the peculiarities of the noncommutative setting. Among the different approaches to hypercomplex analysis, the one Gentili and Struppa set out in 2006 for the quaternions has rapidly developed into a full-fledged theory and it is the object of current research along with its generalization to a large class of alternative real algebras. The Colloquium will overview the main features of this theory and its applications to open problems from other areas of mathematics.

The number of equations defining an algebraic set 
Anurag Singh (University of Utah) 
March 13, 2014

Abstract: Given an algebraic set, i.e., the solution set of a family of polynomial equations, what is the minimal number of polynomials needed to define the set? The question is surprisingly difficult, with a rich history. We will give a partial survey, and discuss results and questions coming from local cohomology theory.

Models for Large Complex Spatial Datasets 
Avishek Chakraborty (Texas A&M University)
February 3, 2014
Monday at 3:30 p.m.
SCEN 322 
* note change of day and location

Abstract: Modeling and inference for geographically-indexed datasets are getting increasingly frequent in current literature. This type of datasets arises in diverse applications from biology, environmental sciences and engineering. The task of statistical inference for these problems consists of finding the strength of association in measurements across adjacent regions in the maps and utilizing it to enhance the predictive performance of the model. We start our discussion with a classification of the spatially-referenced datasets based on the geographical resolution, nature of available information, existing modeling techniques and challenges. We specifically cover two important features that are of prime interest in current research. First, the variable of interest can exhibit complex pattern of association or clustering across different parts of the region. Second, spatial datasets are often massive - either they come from a large region or they are sampled at a high resolution. This poses a significant challenge for modeling because of the computational demand. We shall discuss approaches that can potentially increase the flexibility of these models and improve computational efficiency. The talk will be complemented with real examples from ecology, atmospheric sciences and petroleum engineering.

Functional Data Analysis Techniques in a Childhood Obesity Study 
Jong Soo Lee (University of Delaware at Newark)
January 31, 2014 
Friday at 3:30 p.m.
SCEN 322 
* note change of day and location

Abstract: In a childhood obesity study, the room respiration calorimeters are used to measure minute-by-minute sleeping energy expenditure (SEE) in 109 children, ages 5-18. The goal is to elucidate the population structure of SEE and to discriminate patterns between obese and non-obese children. In this study, functional data analysis (FDA) techniques are applied to SEE data, which reveal more insights about the SEE then the previous studies. First, a smoothing method is implemented to extract the true SEE pattern for each subject and to store the data for the subsequent use in inference. Then, a functional principle component analysis (FPCA) is used to capture the different patterns of variability in SEE between obese and non-obese children, contributing to our understanding of the structure of sleep among children. Other application of FDA inference are also discussed.

High-Dimensional Inference in Genetical Genomics 
Wei Lin (University of Pennsylvania) 
January 30, 2014 
Thursday at 3:30 p.m.
SCEN 322

Abstract: We consider two high-dimensional statistical inference problems motivated by genetical genomics applications, where high-throughput data on genetic variants and gene expression levels, as well as clinical traits, are available for joint analysis. In the first problem, we aim at identifying and estimating important causal effects of gene expressions on the clinical trait, using genetic variants as instrumental variables. To deal with the high dimensionality and unknown optimal instruments, we propose a two-stage regularization methodology, which extends the classical two-stage least squares method by exploiting sparsity in both stages.
In the second problem, we are concerned with simultaneous dimension reduction and variable selection in the multivariate regression of gene expressions on genetic variants. We introduce a sparse orthogonal factor regression approach to reveal a low-dimensional latent factor structure represented by a sparse singular value decomposition of the regression coefficient matrix. The methodology is formulated as an orthogonality constrained regularization problem, coupled with an efficient algorithm via the alternating direction method of multipliers. In both contexts, we investigate theoretical properties of the regularized estimators in the high-dimensional setting where the dimensionality of genetic variants and gene expressions may be comparable to or much larger than the sample size. The practical performance and usefulness of the proposed methods are illustrated by simulation studies and the analysis of mouse and yeast expression quantitative trait loci (eQTL) data sets.

Dynamically Weighted Particle Filter for Sea Surface Temperature 
Duchwan Ryu (Georgia Regents University)
January 28, 2014 
Tuesday at 3:30 p.m.
SCEN 322

Abstract: The sea surface temperature (SST) is an important factor of the earth climate system. A deep understanding of SST is essential for climate monitoring and prediction. In general, SST follows a nonlinear pattern in both time and locations, and can be modeled by a dynamic system which changes with time and locations. We propose a radial basis function network-based dynamic model which is able to catch the nonlinearity of the data and use the dynamically weighted particle filter to estimate the parameters of the dynamic model. We analyze the SST observed in the Caribbean Islands area after a hurricane using the proposed dynamic model. Comparing to the traditional grid-based approach which requires an extreme amount of computing resource due to its high computational demand, our approach requires much less CPU time and makes negitive time forecast of SST doable even on a personal computer.

Multicategory Angle-based Large-margin Classification 
Chong Zhang (University of North Carolina at Chapel Hill) 
January 27, 2014
Monday at 3:30 p.m.
SCEN 322 
* note change of day and location

Abstract: Large-margin classifiers are popular classification methods in both machine learning and statistics. These techniques have been successfully applied in many scientific disciplines such as bioinformatics. Despite the success of binary large-margin classifiers, extensions to multicategory problems are quite challenging. Among existing simultaneous multicategory large-margin classifiers, a common approach is to learn k different classification functions for a k-class problem with a sum-to-zero constraint. Such a formulation can be inefficient. In this talk, I will present a new Multicategory Angle-based large-margin Classification (MAC) framework. The proposed MAC structure considers a simplex based prediction rule without the sum-to-zero constraint, and consequently enjoys more efficient computation. Many binary large-margin classifiers can be naturally generalized for multicategory problems through the MAC framework. Both theoretical and numerical studies will be discussed to demonstrate the usefulness of the proposed MAC classifiers.

Logistic Gifi: A Logistic Distance Association Model for Exploratory Analysis of Categorical Data 
Gary Evans (UCLA-Department of Statistics) 
January 24, 2014
Friday at 3:30 p.m.
SCEN 322 
* note change of day and location

Abstract: In this work, we explore a distance association method we call Logistic Gifi, for categorical data which advances geometric data analysis techniques in much the same way that homogeneity analysis did with regard to correspondence analysis, multidimensional scaling and general clustering methods. It uses the methods of multidimensional unfolding with a likelihood-based loss measure to create low-dimensional geometric representations of data in which distances correspond in a direct way to the probabilistic structure of the data. As with homogeneity analysis, a central feature of our method is the use of binary indicator matrices and, in some applications, fuzzy-coded (i.e., non-binary, row stochastic) indicator matrices to represent categorical data. This gives us a very versatile method with considerable flexibility in the types of data which can be analyzed. We create and study algorithms to use the method to compute low-dimensional geometric representations of various types of data. We analyze the convergence properties of this complex algorithm and show how minimal polynomial extrapolation can be used to accelerate it. We then study relationships between this logistic distance method and logit-based regressions. We present several applications of the method to visualizations of regression results as well as data types such as roll calls, social networks and Markov chains. Finally, a version of the method with bias parameters is introduced and developed and used to emphasize features of data visualizations. We show how bias constraints can be used to represent certain types of model testing. Last, we study the stability of the method.

A Topologist's View of DNA 
Ken Baker (University of Miami) 
January 9, 2014
Thursday at 4:00 p.m.
SCEN 408
Reception time is 3:30 p.m.
* note change in time

Abstract: DNA is famously pictured as a double helix. What's lesser known is that the central axis of this double helix may form a closed loop, even one that's knotted. Appealing to such knottedness, a topologist may glean insight to how proteins transform strands of DNA. We'll overview Ernst & Sumners' pioneering work in DNA Topology and connect it to current open questions in Low Dimensional Topology.

 

Spring 2013 Colloquia

Complexity of surface homeomorphisms
Chris Leininger (University of Illinois, Champagne/Urbana)
May 2, 2013

Abstract: Thurston classified homeomorphisms of surfaces, up to isotopy, into three types. This is modeled on what happens for the torus in which every isotopy class is represented by a 2 × 2 integral matrix, and is either hyperbolic, parabolic or elliptic. For the torus, the richest structure occurs for a hyperbolic matrix which gives rise to an Anosov diffeomorphism of the torus. The analogue on an arbitrary surface is a pseudo-Anosov homeomorphism.

After recalling the situation for the torus, I will explain what a pseudo-Anosov homeomorphism is through a detailed example. I will also describe the basic measure of complexity for a pseudo-Anosov homeomorphism called its dilatation, and provide some interpretations of it. Then I will explain the motivating problem of finding the least complex pseudo-Anosov homeomorphisms. I will end the talk by describing my work with Farb and Margalit on this problem which has found connections with topology, algebra and geometry.

Arithmetic Combinatorics
Neil Lyall (University of Georgia)
April 11, 2013

Abstract: We will give a friendly introduction to the field of arithmetic (or additive) combinatorics, a rapidly developing area of mathematics with close connections to number theory, combinatorics, harmonic analysis and ergodic theory that includes many beautiful results such as Szemeredi's theorem on arithmetic progressions. We do not intend to discuss the proof of this famous result in any detail beyond reviewing its history, but plan instead to take this opportunity to highlight a selection of intimately related results, a new probabilistic technique introduced to the field by Croot and Sisask, and perhaps even some open problems. The talk should be accessible to all.

Are you sure that's an ellipse? Poncelet ellipses, Blaschke products and other stories
Pam Gorkin (Bucknell University) 
February 28, 2013

Abstract: In this talk, we'll describe a well-known theorem due to Jean Poncelet and a surprising way in which Poncelet ellipses are related to important analytic functions known as finite Blaschke products. Along the way, we'll discuss some related problems from several different fields, including group theory and geometry. 

Left-orderability of 3-manifold groups
Cameron Gordon (University of Texas)
February 21, 2013

Abstract: We will discuss connections between three notions in 3-dimensional topology that are, roughly speaking, algebraic, topological, and analytic, respectively. These are: the left-orderability of the fundamental group of a 3-manifold M, the existence of certain codimension 1 foliations on M, and the Heegaard Floer homology of M.
This is joint work with Steve Boyer and Liam Watson.

Non-unique knot surgery descriptions
Ken Baker (University of Miami)
February 7, 2013

Abstract: Every (closed, compact, connected, oriented) 3-manifold admits infinitely many descriptions as surgery on a link in the 3-sphere. However this is not necessarily the case if we restrict ourselves to links of one component, i.e. knots. Aside from more straightforward obstructions such as homology, it is not readily apparent when a manifold even admits a surgery description on a knot. In this talk we'll survey the history of constructions of manifolds with multiple knot surgery descriptions and reexamine its relevance in modern Low Dimensional Topology.

Criteria for integral dependence
Javid Validashti (University of Illinois)
January 31, 2013

Abstract: Rees type criteria for integral dependence of ideals and modules play a significant role in equisingularity theory, where one would like to use numerical invariants to distinguish between members of a given a family of singularities. These criteria are based on the Hilbert-Samuel or Buchsbaum-Rim multiplicity, which rely on certain finiteness conditions. To explore this problem, we introduce a few notions of multiplicity without any finiteness assumption and we show that these invariants can be used in detecting integral dependence and characterizing equisingularity conditions numerically. This talk is in part based on joint works with Bernd Ulrich (Purdue University) and Steven L. Kleiman (MIT).  

Generalized multiplicities and Hilbert functions
Yu Xie (Georgia State University)
January 24, 2013

Abstract: The study of multiplicities and Hilbert functions is essential in both commutative algebra and algebraic geometry. It has important applications in many areas such as intersection theory and singularity theory. The classical multiplicities and Hilbert functions are obtained by computing the colengths of powers of ideals, hence are only defined for ideals that are primary to the maximal ideal in a Noetherian local ring.  In this talk, we are going to focus on the following: How the generalized multiplicities and Hilbert functions (for arbitrary ideals) defined; why we study them; how they are applied to study Generalized multiplicities and Hilbert functions the Cohen-Macaulayness of the associated graded rings.

Measures of Singularities: Characteristic 0 versus Characteristic p
Lance Miller (University of Utah)
January 17, 2013

Abstract: The equations y^2 - x, y^2 - x^3, y^2 - x^2(x+1), and y^3 - (x^2 + y^2)^2 + 3x^2y all define plane curves. Of these, the last three are singular at (0,0). Is one 'more singular' than another? In this talk we discuss some classic techniques of assigning a numerical measure to singularities and how those measures behave under reduction to positive characteristic.  

 

Fall 2013

Asteroids, parallel parking, and the Heisenberg group: Sub-Riemannian geometry and hypoellipticity
Nate Eldredge (University of Northern Colorado)
October 17, 2013

Abstract: This talk will be a gentle introduction to sub-Riemannian geometry, which can be thought of as a way to describe systems that move around subject to constraints. We'll look at some familiar examples, such as the classic video game Asteroids and parallel parking a car, and perhaps less familiar ones, such as the Heisenberg group and its relationship to planimeters (mechanical devices for measuring area).
We will discuss the Chow-Rashevsky theorem, which gives a criterion for "being able to get there from here". Finally, I'll describe some connections to the theory of hypoelliptic operators and Brownian motion, and some results relating to the heat kernels that play the role of the Gaussian in this setting. Much of the talk will need only basic ideas from multivariable calculus; those who have some familiarity with differential geometry and/or PDEs will get a little bit more. 

Notions of rank for automorphisms of right-angled Artin groups
Ric Wade (University of Utah)
October 15, 2013
*note the change in day

Abstract: When looking at the groups SL_n(Z) and Out(F_n) there is a clear distinction between the case n=2, where both groups are virtually free, and their behavior when n is greater than 2. This is similar to the difference between studying rank one lattices in Lie groups, and lattices of higher rank. Automorphism groups of right-angled Artin groups can behave in a similar way. We look at some examples where the

outer automorphism group of a RAAG is virtually a RAAG (e.g virtually free or virtually free abelian), and give some partial results aiming to describe when this happens in general.

Filling invariants: homological vs. homotopical
Pallavi Dani (Louisiana State University)
October 3, 2013

Abstract: Classical isoperimetric functions give optimal bounds on the minimal-area fillings of loops by disks. There are a number of variations: rather than sticking to loops and disks, one might consider fillings of spheres by balls, or cycles by chains. A natural question then is: for a given space, are these functions the same? Or are there spaces in which a particular type of filling (say filling cycles by chains) is more efficient than other types? What if the space admits a proper cocompact isometric action by a finitely presented group? I will talk about recent work with A. Abrams, N. Brady and R. Young which addresses this question.

Liber accusantionis
Michael McQuillan (University of Rome Tor Vergata)
September 12, 2013

Abstract: It is often supposed that the mathematical work of Alexander Grothendieck only applies to algebraic geometry. Apart from the fact that Grothendieck probably wrote more about functional analysis than algebraic geometry (albeit he directed more work in algebraic geometry); such a view ignores the meta-geometric nature of his mathematics, and its applicability to any type of geometry whatsoever. By way of background in meta-geometry, it's useful to be familiar with the film The Matrix, (especially the first one, the others are less significant) so as to be able to compare mathematics to The Matrix, i.e., a device to enslave mathematicians. The goal of the talk will be to extend this comparison with a view to not only better understanding the thought and methodology of Grothendieck, but also to expose (at considerable personal risk) the plot of the Mafia ring (cf. the sentinels in the film) to systematically obscure and distort Grothendieck's work.

 

Fall 2012 Colloquia

Title and abstract TBA
Jennifer Halfpap (University of Montana)
November 29, 2013

Generalizations of the Cauchy Integral Formula to Higher Dimensions
R. Michael Range (State University of New York at Albany)
November 15, 2013

Abstract: The Cauchy kernel and the corresponding integral formula are central in classical complex analysis. In this talk I will discuss several extensions to higher dimensions, as well as some relevant complex analytic/geometric features and obstructions that are not visible in dimension one. At the end I will discuss a recent new construction that may lead to significant applications in cases not covered by the previously known integral kernels.

Assessing mathematical meanings that high school teachers have and attempt to convey—as distinct from assessing performances they exhibit or what they know
Patrick Thompson (Arizona State University)
November 8, 2013

Abstract: The construct teachers' pedagogical content knowledge for teaching mathematics (PCK) has morphed over recent years into the construct mathematical knowledge for teaching (MKT). In neither case did researchers of these constructs explicate what they mean by knowledge. An outcome of using "knowledge" without explication is that assessments of teachers MKT focus on teachers' performances (solving problems, diagnosing student errors) without addressing whence their performances (why teachers do what they do) or how assessment results can be used diagnostically in professional development. I will report our attempt to develop a method for discerning high school mathematics teachers' mathematical meanings and ways of thinking with regard to specific concepts.

On Supnorm Estimates for ∂ on infinite type convex domains in C2
Yuan Zhang (Indiana University-Purdue University at Fort Wayne)
October 25, 2013

Abstract: In this talk, we study ∂ equation on some smooth convex domains of infinite type in C2 . In detail, we show that supnorm estimates hold for those infinite exponential type domains provided the exponent is less than 1. This is a joint work with John Erik Fornaess and Lina Lee.

Title and abstract TBA
Sean Bowman (Oklahoma State University)
October 18, 2013

Title and abstract TBA
Matt Paitz
October 11, 2013

Spring 2012

Weighted Composition Operators Between "Mobius-Invariant Analytic Function Spaces"
Flavia Colonna (George Mason University)
April 26 (date changed from April 19), 2013

Abstract: An interesting question in operator theory is: Given two Banach spaces of analytic functions X and Y on the open unit disk D in the complex plane and a linear operator T : X → Y , what is a minimal collection of functions in the range of T whose boundedness (respectively, convergence to 0) in norm in Y guarantees the boundedness (respective, the compactness) of T ? For the case of the composition operator Cϕ : f → f ◦ϕ (where ϕ is a fixed analytic self-map of D), Tjani proved that for several analytic function spaces, a class of functions of this type is {CϕS : S conformal automorphism of D}. In this talk, we study this problem in the case of the weighted composition operator Wψ,ϕ : f →ψ(f ◦ϕ) (where ψ, ϕ are fixed analytic functions on D, and ϕ(D) ⊆ D) between several Mobius-invariant spaces of analytic functions defined on the open unit disk D in the complex plane. 

Best Estimates for Derivatives of Multivariate Polynomials
Larry Harris (University of Kentucky) 
March 29, 2013

Abstract: In this talk, we study ∂ equation on some smooth convex domains of infinite type in C2 . In detail, we show that subnorm estimates hold for those infinite exponential type domains provided the exponent is less than 1. This is a joint work with John Erik Fornaess and Lina Lee.

Title and abstract TBA
Sean Bowman (Oklahoma State University)
October 18, 2013

Title and abstract TBA
Matt Paitz
October 11, 2013

Spring 2012

Weighted Composition Operators Between "Mobius-Invariant Analytic Function Spaces"
Flavia Colonna (George Mason University)
April 26 (date changed from April 19), 2013

Abstract: An interesting question in operator theory is: Given two Banach spaces of analytic functions X and Y on the open unit disk D in the complex plane and a linear operator T : X → Y , what is a minimal collection of functions in the range of T whose boundedness (respectively, convergence to 0) in norm in Y guarantees the boundedness (respective, the compactness) of T ? For the case of the composition operator Cϕ : f → f ◦ϕ (where ϕ is a fixed analytic self-map of D), Tjani proved that for several analytic function spaces, a class of functions of this type is {CϕS : S conformal automorphism of D}. In this talk, we study this problem in the case of the weighted composition operator Wψ,ϕ : f →ψ(f ◦ϕ) (where ψ, ϕ are fixed analytic functions on D, and ϕ(D) ⊆ D) between several Mobius-invariant spaces of analytic functions defined on the open unit disk D in the complex plane. 

Best Estimates for Derivatives of Multivariate Polynomials
Larry Harris (University of Kentucky) 
March 29, 2013

Abstract: In this talk, we study ∂ equation on some smooth convex domains of infinite type in C2 . In detail, we show that subnorm estimates hold for those infinite exponential type domains provided the exponent is less than 1. This is a joint work with John Erik Fornaess and Lina Lee.

Title and abstract TBA
Sean Bowman (Oklahoma State University)
October 18, 2013

Title and abstract TBA
Matt Paitz
October 11, 2013

Spring 2012

Weighted Composition Operators Between "Mobius-Invariant Analytic Function Spaces"
Flavia Colonna (George Mason University)
April 26 (date changed from April 19), 2013

Abstract: An interesting question in operator theory is: Given two Banach spaces of analytic functions X and Y on the open unit disk D in the complex plane and a linear operator T : X → Y , what is a minimal collection of functions in the range of T whose boundedness (respectively, convergence to 0) in norm in Y guarantees the boundedness (respective, the compactness) of T ? For the case of the composition operator Cϕ : f → f ◦ϕ (where ϕ is a fixed analytic self-map of D), Tjani proved that for several analytic function spaces, a class of functions of this type is {CϕS : S conformal automorphism of D}. In this talk, we study this problem in the case of the weighted composition operator Wψ,ϕ : f →ψ(f ◦ϕ) (where ψ, ϕ are fixed analytic functions on D, and ϕ(D) ⊆ D) between several Mobius-invariant spaces of analytic functions defined on the open unit disk D in the complex plane. 

Best Estimates for Derivatives of Multivariate Polynomials
Larry Harris (University of Kentucky) 
March 29, 2013

Abstract: In 1892, the famous mathematician A. A. Markov proved that the Chebyshev polynomial of degree has the largest rst derivative of any polynomial of degree n that has absolute value at most 1 on the unit inter-val Three years later, his younger brother Vladimir, then a student at St. Petersburg University, published a small book in which he proved that the same result holds for arbitrary derivatives. This result remains a major and deep theorem in the theory of polynomials.
In 1954, Michal of Caltech stated a straightforward extension of the V. A. Markov theorem for real normed linear spaces but gave a hopelessly erroneous proof. In the intervening years, a number of partial results were proved until 2005 when V. I. Skalyga published a daunting proof of the general result in Izvestiya.
In this talk, I will outline a proof of the general V.A. Markov theorem. We begin by reducing to the case of two real variables and then apply Lagrange interpolation in two variables, a Christo el-Darboux formula and a simple argument of Rogosinski. In particular, this gives a new proof of the classical theorem.

Link homology and surfaces
Adam Lowrence (University of Iowa)
March 15, 2013

Abstract: A link homology is an invariant that typically generalizes a classical link polynomial. Two important examples of link homologies are Khovanov homology, which generalizes the Jones polynomial, and knot Floer homology, which generalizes the Alexander polynomial. In this talk, we will discuss the relationship between these link homologies and two surfaces associated to a link, the Seifert and Turaev surfaces. Specifically, we will show how certain gradings in these link homologies give information about the genus of the associated surfaces.

The d-bar equations on "bad" domains
Debraj Chakrabarti (Indian Institute at Bombay)
February 23, 2013

Abstract: We discuss the problem of establishing regularity of the d-bar problem in Sobolev spaces on certain domains, on which the classical methods (using the d-bar Neumann problem, or the existence of Stein neighborhoods) do not apply. In particular, we consider product domains, and the Hartogs triangle as such model "bad" domains. This is joint work with Mei-Chi Shaw.

How to degenerate the Jacobian
Jesse Kass (University of Michigan)
February 2, 2013

Abstract: Many questions about a compact Riemann surface can be investigated through an associated topological group, the Jacobian. In turn, the Jacobian can be studied using the technique of degeneration. In my talk, I will explain what a Jacobian is, what degeneration is, and how two different approaches to degeneration are related.

Legendrian knots, contact homology, and generating families 
Dan Rutherford (University of Arkansas)
February 1, 2013
Location change: SCEN 501

Abstract: Legendrian knots play an important role in low-dimensional topology. They are fundamental objects in contact topology and also shed new light on the study of ordinary (smooth) knots. This talk will focus on Legendrian knots in standard contact 3-space where front projections may be used to give a combinatorial formulation of Legendrian isotopy.
After providing a brief introduction to Legendrian knot theory, I will focus on Legendrian contact homology, which assigns a differential graded algebra (DGA) to a Legendrian knot, and also discuss invariants arising from generating families. (The generating family invariants are due to Jordan-Traynor and Pushkar, and the DGA was introduced in this context by Chekanov and Eliashberg.) Several connections between these two classes of knot invariants exist. In particular, a joint result of the speaker and D. Fuchs relates linearized contact homology groups and generating family homology.

Folding, accessibility, and dimension theory for free groups
Lars Louder (University of Michigan)
January 31, 2013
Location change: SCEN 101

Abstract: I will describe my work on Nielsen equivalence in surface groups, an accessibility theorem for hierarchies of splittings of (relatively) hyperbolic groups, and describe the role the notions of folding and accessibility play in the proof that algebraic varieties defined over free groups have finite Krull dimension.

Automorphisms of surfaces and free groups
Matthew Clay (Allegheny College)
January 26, 2013
Location change: SCEN 501

Abstract: The dynamical behavior of a homeomorphism of a surface was classified by Nielsen in the 1920s-1940s. His work was overlooked until the 1970s when Thurston, unaware of Nielsen's papers, rediscovered this classification. Since this time, the Nielsen-Thurston classification of surface homeomorphisms has become an indispensable tool in the theory of surfaces and three dimensional manifolds.
Bestvina and Handel, among others, applied Thurston's approach to the study of automorphisms of free groups. As a result, these two theories have developed in parallel, oftentimes with insights from surfaces leading to insights in free groups.
I will describe the dynamical picture in both settings and some recent developments.

The Giroux Correspondence
Jeremy Van Horn Morris (Stanford University)
January 24, 2013
Location change: SCEN 501

Abstract: In 2002, Giroux unveiled a strange relationship between contact structures on (2n+1) dimensional manifolds and open book decompositons: (2n-1) dimensional submanfolds with nice fibrations of their complement. In dimension 3, this has an easier description as an open book decomposition is just a fibered knot or link. It is a particularly powerful relationship which has opened up the tools of contact and symplectic geometry and low-dimensional topology to each other. We'll discuss the history of both sides, how they relate now, and what we can hope for in the future.

Spectral properties of the Fibonacci Hamiltonian
David Demanik (Rice University)
January 20, 2013
Location change: SCEN 501

Bayesian graphical models for multivariate functional data
Hongxiao Zhu (Duke University)
January 10, 2013
Location change: SCEN 408

Abstract: In a broad variety of application areas there is interest in inferring the dependence structure in multivariate functional data. For data in vector form, conditional independence relationships between variables can be inferred through allowing zeros in the precision matrix through a Gaussian graphical model. Bayesian methods can be used to allow unknown locations of zeros, with a hyper inverse-Wishart prior chosen for the covariance. We generalize these methods to define a new class of Gaussian process graphical models for multivariate functional data. We focus on models with decomposable graph structures with a single precision matrix encoding the conditional independence between the functions. We also discuss the more general class of non-decomposable graphs. Properties of the proposed process are considered, and two efficient algorithms are developed for posterior computation relying on Markov chain Monte Carlo. The methods are evaluated through simulation studies and applied to EEG data.

Why We Do What We Do
Mark Daniels (University of Texas)
January 9, 2013
Location change: SCEN 408

Abstract: A discussion of inquiry-based mathematics teaching methods used in support of Project Based Instruction is presented based upon classroom instructional practices employed by some faculty in mathematics at the University of Texas at Austin. The philosophy behind using such methods, examples of materials used, and research support for these practices in undergraduate mathematics courses for mathematics majors and pre-service mathematics majors will be presented discussed.

 

Fall 2011

Geometry of Teichmüller space
Kasra Rafi (Oklahoma University)
November 18, 2011
SCEN 408
*note the day and location

Abstract: The Teichmüller space of a surface S is the space of all hyperbolic structures on S up to isotopy. Teichmüller theory is an old but still very active area of mathematics with connections to a wide range of fields, such as algebraic geometry, number theory and dynamical systems. We, however, concentrate on the geometric properties of Teichmüller space and the behavior of geodesics in the Teichmüller metric. We give a close examination of the case when S is a torus and see how things generalize to Teichmüller space of a surface with higher genus.

Estimation and testing for spatially distributed curves
Piotr Kokoszka (Colorado State University)
November 17, 2011

Abstract: In many fields, most notably in environmental science and geosciences, data have the form of temporal curves available at unevenly distributed spatial locations. It is often of interest to combine such data to evaluate global or regional temporal trends. The research I will present is motivated by records of space physics data available at globally distributed observatories and reaching back into the 1960s. These data have been used to evaluate the hypothesis of "global cooling'' in the ionosphere. We cast such data structures into the framework of functional data analysis, and show how the spatial dependence can be used to estimate their main parameters: the mean function and the functional principal components. These estimation techniques are then used to construct a test of independence of two families of such curves. The test is applied to assess the impact of long term changes in the internal magnetic field of the Earth on ionospheric trends. We conclude with a brief discussion of asymptotic results. 

Cauchy integrals and Möbius geometry in complex analysis
Mike Bolt (Calvin College)
November 10, 2011

Abstract: A basic construction in complex analysis is the Cauchy integral, which provides a representation for holomorphic functions using their values along a curve. In higher dimensions, there are various generalizations of the Cauchy integral. These typically take advantage of the geometry of the underlying domain.
Focusing on dimension two, we'll consider one of the constructions, the Leray integral, along with its basic analytic properties. Using Möbius invariance, we'll suggest that it's the most natural extension of the Cauchy integral to higher dimensions. This leads to the problem of understanding the Möbius geometry of a real hypersurface in C^n.

Conformally invariant Systems of Differential Equations
Leticia Barchini (Oklahoma State University)
November 4, 2011

Abstract: The wave operator in Minkoswski space R^{3;1} is a classic example of a conformally invariant operator. The Lie algebra so (4; 2) acts on R^{3;1} via a multiplier representation . When acting on sections of an appropriate line bundle over R^{3;1} the elements X 2 so(4; 2) satisfy [c(X),Ω] = c(X)Ω, where c(X) is a continuous function on R^{3;1}.
Several systems of partial differential equations are associated to each complex simple Lie algebra of rank greater than one. Each system is conformally invariant under the given Lie algebra. The systems so constructed yield explicit reducibility results for a family of scalar generalized Verma modules attached to two-step nilpotent parabolic subalgebras of the given Lie algebra. At the same time the systems give homomorphisms between generalized Verma modules that often are non-standard.

The Cubic Dirac Operator
Roger Zierau (Oklahoma State University)
November 3, 2011
*note this talk will take place in SCEN 402

Abstract: Many interesting homogeneous spaces are of the form G/H with G and H both reductive Lie groups. Examples include the symmetric spaces (both rie-mannian and pseudo-riemannian), flag manifolds and all homogeneous spaces for a compact Lie group. These spaces are known to possess many Dirac operators. Of special interest is the case of a riemannian symmetric space G/K where an invariant connection gives rise to an invariant Dirac operator on sections of finite rank vector bundles. In the 1970s, Parthasarathy and Schmid-Atiyah proved that the space of L2 harmonic spinors gives a realization of the discrete series representations of G. In the late 1990s, Kostant introduced the 'cubic' Dirac operator DG/H on (almost) all homogeneous spaces G/H, with G and H both reductive. I will discuss some of the many wonderful properties of this operator. An integral formula, in the spirit of the Poisson transform, which produces solutions to DG/H F = 0 will be presented.

Testing in functional data analysis and other problems
Dan Spitzner (University of Virginia)
Thursday, October 20, 2011

Abstract: The theme of this talk is hypothesis testing, and particular attention is paid to testing in functional data analysis, which is especially interesting because the context is high-dimensional, and the data model involves an assumption of smoothness. Basic concepts for modeling and theoretical evaluation in functional data analysis are discussed through several examples. The approach involves orthogonal basis decomposition combined with "tapering,’’ or down-weighting high frequency patterns, in order to model smoothness. It will be seen that working with functional data highlights various challenges for testing, generally, including those of incorporating prior information (such as a smoothness assumption) and accounting for multiplicities. Subsequently, I will discuss a new approach to Bayesian testing that substitutes for the well-known Bayes factor a novel form of evidence assessment called a “neutral-data comparison.” It will be shown that the neutral-data comparison provides a coherent, flexible multiple-testing tool, with which functional-data and other common testing problems are readily solved and interpreted. It moreover provides a means to supplement Bayesian estimation with a well-defined testing component, even when relying on vague prior information.

Invariants of Legendrian Knots
Dan Rutherford (University of Arkansas)
Thursday, September 22, 2011

Abstract: Legendrian knots arise naturally in low-dimensional topology. When studied in their own right, they provide an interesting variation on the classical theory of smooth knots in 3-space. In this talk I will give an introduction to Legendrian knot theory which will focus on relationships between some invariants of a Legendrian knot, L, and the underlying smooth knot type of L.

Small Generating Sets of the Torelli Group
Andy Putman (Rice University)
Thursday, September 8, 2011
*note the speaker will continue his discussion in the topology seminar immediately following the colloquium

Abstract: The Torelli group is the subgroup of the mapping class group of a surface that acts trivially on the first homology group of the surface. It has many remarkable properties. The focus of this talk is a theorem of Dennis Johnson which asserts that if the genus of the surface is large, then the Torelli group is finitely generated. We'll explain why this result is surprising and then discuss a new theorem which shows that the Torelli group has a generating set that is "small."

 

Spring 2011

On Rarita-Schwinger type operators on the sphere
Carmen Judith Vanegas (Universidad Simon Bolivar, Caracas, Venezuela)
April 21, 2011

Abstract: Rarita-Schwinger operators are generalizations of the Dirac operator. In the last ten years they have been studied in Euclidean spaces by many authors. We can also construct Rarita-Schwinger type operators together with their fundamental solutions on the sphere. Using similar arguments as those used in the case of Euclidean spaces we establish the conformal invariance for the projection operators and the spherical Rarita-Schwinger type equations under the Cayley transformation. In turn, Stokes' Theorem, Cauchy's Theorem, Borel-Pompeiu Formula and Cauchy Integral Formula are proved for the sphere.
This research is being jointly carried out with Junxia Li and John Ryan.

From undergraduate Calculus to Cartan Geometries 
Jan Slovak (Masaryk University, Brno, Czech Republic)
April 19, 2011

Abstract: First, I will reinterpret the usual partial derivatives in view of Klein's approach to affine geometry, and provide the basic theorem of Calculus (on the existence of primitive functions) in terms of logarithmic derivatives on Lie groups. Then it is easy to introduce the Cartan's curved version of the Klein geometries and come up to indications of the BGG calculus for parabolic geometries. On the way, the whole story will be illustrated by some examples from conformal, projective and CR geometries.

Out (Fn) and Outer Space 
Patrick Reynolds (University of Illinois at Urbana-Champaign)
April 14, 2011

Abstract: The group Out (Fn) of outer of the rank-n free group Fn has been given considerable attention for the past century. In the past two decades, a new viewpoint has emerged for studying Out (Fn); this approach, combining topological, geometric, and dynamical tools (analogous to certain tools used for studying mapping class groups of surfaces), has enjoyed considerable success. The beginning of this modern approach is easily identified as a paper of Culler-Vogtmann, in which a "good topological model space" for Out (Fn) was defined; 
nowadays, this space is known as (rank-n) Outer space, denoted CVn.
This talk is an introduction to Outer space; after describing CVn, we will mention the key topological properties established in the seminal paper of Culler-Vogtmann and their (co)homological consequences for Out (Fn). Afterwards, we will briefly mention the "Thurston-type" compactification of Outer space, indicating how the dynamics of the action of Out (Fn) on this compactification sheds light onto subgroup structure in Out (Fn).

Conformal foliations
Michael Eastwood (Australian National University in Canberra)
April 5, 2011

Abstract: In Euclidean three-space, there are some special foliations called conformal. They enjoy remarkable properties and can often be constructed from holomorphic functions of two complex variables. I shall explain this construction and its geometric interpretation via twistor theory. The ideas are well-known but this exposition is joint work with Paul Baird.

Coverings of non-orientable manifolds: Orientable or not orientable? 
Jair Remigio Ju’arez (University of Arkansas)
March 3, 2011

Abstract: Let M be a non-orientable k-manifold and f:N-->M be a covering space of M. In this talk we present an algorithm to determine when M is orientable or not. This algorithm is based in some special subgroups of S_k (the symmetric group on k elements) called imprimitive subgroups, so in the way we will discuss the definition of imprimitive subgroup and the relation between this class of subgroups and covering spaces. Also we will have some examples of the algorithm working on a non-orientable surface and a quick review about manifolds and covering spaces.

An Application of Representation Theory to Statistics
Michael Schein (Bar-Ilan University, Isreal)
January 31, 2011

Abstract: Suppose that we are given a function f that is known to lie in a predetermined n-dimensionalspace of functions. Measuring the value of f at any n distinct points will, generically, provide enough information to determine the function. Suppose that, as in the real world, there are errors of measurement: repeated evaluation of f at a given point gives rise to a distribution of values. A choice of n points is called an optimal design if it minimizes the number of measurements necessary to determine f to a specified degree of confidence.
This talk is the result of a number theorist's dabbling in such foreign fields. We will show how some basic facts about the representation theory of finite groups may be used to obtain results about optimal designs. In particular, we find a short proof of a result of Lee (using work of de Boor) about minimality of B-splines.
No background, either in statistics or in representation theory, is assumed.

Colloquia is usually on Thursdays from 3:30 p.m. to 4:20 p.m. in SCEN 407. Alternate times and dates will be noted. 

A reception takes place before each colloquium at 3 p.m. in SCEN 350.

The events are open to the public.

Fall 2010

The d-bar Neumann Laplacian and the Daubechies wavelet
Siqi Fu (Rutgers University)
December 2, 2010

Abstract:  This is an expository talk on spectral theory of the d-bar Neumann Laplacian.  We will discuss the interplay between spectral behavior of the d-bar Neumann Laplacian and geometry of the underlying space.  In particular, we will explain how compactly supported wavelet constructed by Daubechies plays a role in the theory.

Pushing fillings into subgroups
Noel Brady (University of Oklahoma)
November 18, 2010

Abstract: In the field of geometric group theory one considers finitely generated groups as geometric objects. One obtains interesting group invariants by considering geometric questions. For example, the notion of an isoperimetric inequality (best area fillings of loops of a given length) gives the Dehn function of a group.
In this talk we will discuss examples which exhibit the relation between Dehn functions of an ambient group G and a kernel group K in a short exact sequence
               1 -> K -> G -> Z -> 1
We will examine a result of Gersten and Short in the case that G is a hyperbolic group, and a result of Dison and of Abrams-Brady-Dani-Duchin-Young in the case that G is a right-angled Artin group. If time permits we, will discuss other examples. 

Confessions of a young Computational Scientist: Career Talk by Pat Quillen, MathWorks
Pat Quillen (MathWorks)
November 04, 2010

Abstract: What can I do with my computer besides surf the web?  What role does Mathematics actually play in this?  How can I learn more and get some experience doing these things?  Our speaker will touch on these topics as more as he describes some applications of numerical analysis and shares his personal experience---the good, the bad, and the ugly---in writing mathematical software for The Masses.

Quantitative isoperimetric inequalities in good domains
Kai Rajala (University of Jyvaskyla)
October 28, 2010

Abstract: By the sharp isoperimetric inequality, balls are the unique minimizers of the surface measure among sets in euclidean space with fixed volume. Quantitative isoperimetric inequalities are estimates showing that sets which almost minimize the surface measure are almost balls with respect to some geometric quantity. We present some basic estimates of this kind, and show that while the isoperimetric inequality is in general more stable in the plane than in higher dimensions, there is a large class of domains (John domains) for which good metric estimates can be proved in all dimensions.
The classical Bonnesen inequality states that planar Jordan domains with small isoperimetric deficit are almost disks in a strong metric sense. More recent works of Hall, Fusco-Maggi-Pratelli, and others show that although Bonnesen's inequality does not in general hold in higher dimensions, there are other natural quantitative inequalities that hold in all dimensions.

Stable commutator length in the free group
Danny Calegari (Caltech)
October 21, 2010

Abstract: SCL answers the question: "what is the simplest surface in a given space with prescribed boundary?" where "simplest" is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, SCL is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, SCL is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). I will discuss how SCL in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi-dimensional continued fractions and Klein polyhedra (as time permits).

Geometric Discrepancy Theory
Jill Pipher (Brown University)
October 07, 2010

Abstract: The first part of this talk will be an introduction to both the classical ideas and some modern developments in discrepancy theory. I will describe a variety of open problems as well as some progress on specific problems in two dimensions (joint work with D. Bilyk, X. Ma, C. Spencer). Discrepancy theory has connections with number theory, harmonic analysis, probability, approximation theory and numerical methods (quasi Monte Carlo simulation).

Spring 2010

Lens space surgery: orders of fundamental groups versus Seifert genera

Toshio Saito (USCB)
April 29, 2010

Abstract: A Dehn surgery is a topological operation to modify a 3-dimensional manifold. In this talk we will discuss the Goda-Teragaito conjecture on Dehn surgeries yielding lens spaces. Specifically I will verify the conjecture by computer and consider improvability of the conjecture.

Quasiconformality, homeomorphisms preserving quasi-minimizers between metric measure spaces and uniform density property
Niko Marola (University of Helsinki, Finland)
April 20, 2010

Abstract: We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.
Joint work with R. Korte (Helsinki) and Nageswari Shanmugalingam (Cincinnati).

Flipping Bridge Surfaces
Maggy Tomova (University of Iowa)
March 18, 2010
 
Abstract: Recently the "stabilization conjecture," an important question in 3-manifolds, was resolved. In a joint paper with Jesse Johnson we gave a generalization of this result by allowing knots in the manifold. I will present our results in the special case of a knot in the three sphere and give an idea of how this special case can be generalized to knots in any 3-manifold.

On-line inference in autoregressions, mixtures of autoregressions and state-space autoregressions with structured priors
Raquel Prado (University of California, Santa Cruz)
March 11, 2010

Abstract: This work is motivated by the analysis of multiple brain signals recorded during an experiment that aimed to characterize mental fatigue in subjects performing a cognitive task continuously for an extended period of time. The recorded brain signals can be modeled via mixtures of autoregressive process with structured priors.  More specifically, we follow a Bayesian approach that imposes structured prior distributions on the reciprocal roots of the characteristic polynomials that define the AR processes.  Such prior structure allows modellers to include scientifically meaningful information related to various states of mental alertness.  We focus on the implementation of sequential Monte Carlo methods for on-line parameter learning within the following model classes: structured AR models, mixtures of structured AR models and structured AR plus noise models.

The closed range property and boundary regularity for the Cauchy-Riemann equations
Mei-Chi Shaw (University of Notre Dame)
March 9, 2010

Abstract: In this talk we will study the closed range property and boundary regularity of the Cauchy-Riemann equations on domains in complex Euclidean spaces or complex manifolds. When the domain is pseudoconvex in the complex Euclidean space, one has the celebrated   H\"ormander's $L^2$ existence theorems and Kohn's boundary regularity results.  We will discuss the case for an annulus between two pseudoconvex domains in $C^n$ as well as the recent results on product domains (joint work with Debraj Chakrabarti). Some results and open problems on domains in complex projective spaces will also be discussed.

The Stretch Conjecture
Al Baernstein (Washington University in St. Louis)
March 4, 2010

Wavelets and Semigroups
Swanhild Bernstein, Institut für Angewandte Analysis (TU Bergakademie Freiberg, Germany)
March 2, 2010

Embedded Surfaces in 3-Manifolds
Jesse Johnson (Oklahoma State University)
February 11, 2010

Abstract: I will give an overview of the current state of research studying isotopy classes of surfaces in 3-dimensional manifolds, with a focus on open problems and possible future directions.

Hierarchical spatial models for predicting forest variables over large heterogeneous domains
Dr. Sudipto Banerjee, Associate Professor, Division of Biostatistics (School of Public Health of the University of Minnesota)
January 28, 2010

Abstract: We are interested in predicting one or more continuous forest variables (e.g., biomass, volume, age) at a fine resolution (e.g. pixel-level) across a specified domain. Given a definition of forest/non-forest, this prediction is typically a two-step process. The first step predicts which locations are forested. The second step predicts the value of the arable for only those forested locations. Rarely is the forest/non-forest predicted without error. However, the uncertainty in this prediction is typically not propagated through to the subsequent prediction of the forest variable of interest. Failure to acknowledge this error can result in biased and perhaps falsely precise estimates. In response to this problem, we offer a modeling framework that will allow propagation of this uncertainty. Here we envision two latent processes generating the data. The first is a continuous spatial process while the second is a binary spatial process. We assume that the processes are independent of each other. The continuous spatial process controls the spatial association structure of the forest variable of interest, while a binary process indicates presence of a "measurable'' quantity at a given location. Finally, we explore the use of a predictive process for both the continuous and binary processes to reduce the dimensionality of the data and ease the computational burden. The proposed models are motivated using georeferenced National Forest Inventory (NFI) data and coinciding remotely sensed predictor variables.
This is joint work with Andrew O. Finley (Department of Forestry and Geography, Michigan State University)

Rank and rank gradient of groups that split
Jason DeBlois (University of Illinois, Chicago)
January 26, 2010

Abstract: "The rank of a finitely generated group G -- the minimal cardinality of a generating set -- provides a rough measure of the complexity of G.  In particular, results like Grushko's theorem relate the ranks of groups that "split" to the objects involved in their decompositions.  I will discuss applications of such theorems to questions about rank gradient, which measures the growth rate of rank in families of finite-index subgroups, and relate them to the "rank vs. Heegaard genus" question for 3-manifolds.  Then I will describe how geometric methods can be used to improve estimates in some cases, and show why this matters for rank gradient."

On Evolution Equations
Cristina Caputo (University of Texas at Austin)
January 25, 2010

Abstract: This talk will be made accessible to a general audience of mathematicians (graduate students included). Existence, regularity, and other issues will be described for solutions of certain evolution equations. Similarities and differences of the behavior of such solutions will be discussed.

Backwards uniqueness for the Ricci flow and the non-expansion of the isometry group.
Brett Kotschwar (Massachusetts Institute of Technology, Department of Mathematics)
January 19, 2010

Abstract:  I will discuss some recent work on the problem of backwards uniqueness or unique-continuation for the Ricci flow, and show that two solutions of uniformly bounded curvature that agree at some non-initial time must agree identically at all previous times.  A particular consequence is that the flow does not sponsor the generation of new isometries within the lifetime of a solution, nor permit a solution to become Einstein or self-similar in finite time.

 

Fall 2009

Geometry through the School Years
Richard Askey (University of Wisconsin)
November 19, 2009

There has been a lot written about teaching arithmetic to children and the National Mathematics Advisory Panel had a focus on algebra and preparing students to take it. Geometry has been ignored, yet it is the part of mathematics where our students do most poorly once one gets beyond the stage of names for different figures. This talk with start with elementary school geometry.
One essential difference between a triangle and a quadrilateral is that a triangle is rigid while a quadrilateral is not. This can easily be illustrated with fingers. Other topics will include such things as how the same figures can be used to get the area of a triangle, the angle sum of a triangle and later even the addition formula for sines and cosines, and why there is a factor of 1/2 in the formula for the area of a triangle but 1/3 in the formula for the volume of a pyramid.

Is Width Additive?
Ryan Blair (University of California, Santa Barbara)
November 17, 2009

Width is an invariant of knots that depends on the number of minima and maxima of the knot as well as their relative positions. The behavior of width with respect to connected sum has been studied for three decades. Originally, it was conjectured that width is additive under the operation of connect sum. I will outline classic results pertaining to the question and discuss ongoing joint work with Maggy Tomova that suggests width is not additive.

Homogenization of Elliptic Boundary Value Problems
Zhongwei Shen (University of Kentucky)
October 22, 2009

Recent progress on boundary value problems in Lipschitz domains for a family of second order elliptic equations with rapidly oscillating coefficients, arising the theory of homogenization.
This is a joint work with Carlos Kenig.

Nonlinear Geometric Optics
Jeffrey Rauch (University of Michigan)
October 19, 2009

Geometric optics is a family of methods to construct approximate solutions to partial differential equations. The solutions have a small parameter often representing a wavelength. The approximations are accurate in the short wavelength limit. In this limit, direct numerical simulation is not possible. The history is long and rich. In this talk we recall some of the history and basic ideas of the subject and recent advances, notably for nonlinear hyperbolic equations. For the compressible Euler equations there are solutions with three incoming wave trains and with outgoing waves with velocities dense in the unit sphere.
My work in this area is with J.L. Joly and G. Metivier.

Complex Analysis beyond One Dimension
Mehmet Celik (University of Arkansas at Fort Smith)
October 8, 2009

The study of functions defined on multidimensional complex space is called the theory of several complex variables. Classical complex analysis is the study of functions defined on one dimensional complex space. The main object of study in both theories is complex analytic functions. Some parts of the theory of analytic functions, such as the maximum principle, normal limit of nowhere-zero analytic functions, and Cauchy's estimates for derivatives, are essentially the same in all dimensions. The most interesting parts of the multidimensional complex theory are the features that differ from the one dimensional theory. At the beginning of the presentation, we will mention some differences between these two theories from different points of view. Then, we will focus on one particular difference and explore the world of complex analysis beyond one dimension.
The talk is accessible to any graduate students in mathematics.

On The Critical Group of Finite Projective Planes
Stuart Shirell (University of Arkansas)
September 17, 2009

The critical group of a graph on n vertices is defined to be $Z^n/(Image(M))$, where M is the Laplacian matrix associated to the graph. This is an isomorphism invariant and is largely viewed as relatively robust.
Relatively little is known about the critical group in general, however. For example the critical groups of complete graphs, bipartite complete graphs, and the n-cube are (mostly) known, but little else is known about these groups or how they reflect the symmetry properties of the underlying graph. We determine completely the critical groups of non-degenerate, finite projective planes and show that all projective planes of a given order have isomorphic critical groups. 

Growth Networks
Raja Kali and Javier Reyes (Department of Economics, University of Arkansas)
September 10, 2009

Is there a way to kindle acceleration in the economic growth rate of a country? What role do trade and comparative advantage play in this process? These are among the most enduring and important questions in economics. This research project aims to use a novel, network-based approach to international trade to make progress in decoding the mystery of growth acceleration.
One of the general results of the literature in complex networks is that high performance networks in many settings (biological, technological, social, economic) have the "small world" property. In other words, in several contexts, the small world seems to be an "optimal" topology. A small world is a network whose topology combines high clustering among nodes with high connectivity (short path length) across nodes. Due to high clustering, such networks are likely to have strong spillovers between nodes, and short path length provides the potential for long range leaps across the network. Both features are advantageous in the context of economic development and growth. Could it be that the key to growth acceleration is whether the pattern of product specialization of a country develops a "small world" topology before the take-off? This could come about because product space and the pattern of product specialization of a country, which are both evolving over time, overlap so as to create the conditions for a small world. If true, then this implies that the country's location in product space and its pattern of product specialization matter for its likelihood of experiencing a growth acceleration. Our research aims to marshal evidence to examine this insight.

Weights in generalizations of Serre's conjecture
Michael Schein (Bar-Ilan University, Isreal)
August 27, 2009

Given a prime p and a modular form, one can associate to it a two-dimensional mod p representation of the absolute Galois group of Q. Given such a mod p representation, J.-P. Serre conjectured in the 1970's when it should arise from a modular form in this way, and, if so, what the weights and levels of the associated modular forms are. We will explain these notions and discuss generalizations of the conjecture to totally real fields and to Galois representations of arbitrary dimension, as well as theorems toward these new conjectures. 

Spring 2009

Contact geometry, genus and low-dimensional topology

John Etnyre (Georgia Institute of Technology)
April 30, 2009

Abstract: Recently Giroux has completed a story begun by Thurston and Winkelnkemper over 30 year ago. This story describes a relation between geometric objects (contact structures on 3-manifolds) and topological objects (open book decompositions of 3-manifolds). This relation has resulted in many breakthroughs in our understanding of contact geometry and low dimensional topology. In this talk I will give an overview of contact geometry and its history and then discuss Giroux's relation between contact structures and open book decompositions and explore some of its implications to contact geometry, symplectic geometry and to low dimensional topology.

Incompressible one-sided Heegaard splittings for hyperbolic once-punctured torus bundles
April 23, 2009

Abstract: After reviewing some basic notions of 3-manifolds and important surfaces in them, I will briefly discuss two ways to study such surfaces: minimal surface theory for Riemannian 3-manifolds and normal surface theory for triangulated 3-manifolds. The rest of the talk will focus on the standard ideal triangulation of hyperbolic once-punctured torus bundles. Using the normal surface theory, I will give a classification of incompressible one-sided Heegaard splittings for such manifolds in terms of principal/Hecke congruence subgroup of level 2 in SL(2,Z). This is a joint work with Bus Jaco.

The Theorem of Frobenius: An Analysis Perspective
Brian Street (University of Toronto)
March 26, 2009

Abstract: Informally, the theorem of Frobenius (from differential topology) states that if one has vector fields X_1, ..., X_q such that [X_i, X_j] is in the span of X_1, ..., X_q, then for each point of the ambient manifold, there is a submanifold passing through that point whose tangent space is precisely the span of X_1, ..., X_q.  This theorem is "soft" in the sense that it gives no control on the coordinate charts defining the submanifold.  In this talk, we discuss a certain choice of charts where one does have good control.  We then discuss applications of this to questions in harmonic analysis.

Tire tracks geometry, hatchet planimeter and Menzin's conjecture
Sergei Tabachnikov (Pennsylvania State University)
March 16, 2009

Abstract: The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel, fixed on the bicycle frame); the same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and hence its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping of a circle to a circle is a Mobius transformation, a remarkable fact that has various geometrical and dynamical consequences. Mobius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. I shall outline a proof of a 100 years old conjecture: if the front wheel track is a convex curve with area at least pi then the respective monodromy is hyperbolic. Time permitting, I shall also discuss bicycle motions such that the track of the rear wheel coincides with that of the front wheel.  I shall explain why such ''unicycle" tracks become more and more oscillating in forward direction and cannot be infinitely extended backward.

Refocusing College Algebra
Prof. Don Small (West Point)
March 5, 2009

Abstract: A national movement is growing to refocus college algebra away from the 1960s symbolic, algorithmic, and skill based program towards a course focused on the present and future needs of students. Major factors fueling this change will be discussed. The philosophy and goals of a refocused program will be presented along with several examples to illustrate the scope and tone of the proposed program. Implementation and results from other schools will be discussed.

Marginalized Random Effects Models for Multivariate Longitudinal Binary Data
Keunbaik Lee (School of Public Health at LSU)
February 26, 2009

Pigeons, Parties and Progressions: a Gentle Introduction to Arithmetic Combinatorics
Neil Lyall (University of Georgia)
February 19, 2009

Closed range composition operations on Bergman and Bloch spaces
Pratibha Ghatage (Cleveland State University)
January 29, 2009

Optimal Contrast in Visual Cryptography
Kenneth Wantz (Southern Nazarene University)
January 22, 2009

Regularity of minimal graphs in the Heisenberg groups
Giovanna Citti (Universita' di Bologna, Italy)
January 20, 2009

Abstract: We will discuss recent progress on the regularity of minimal intrinsic graphs in the Heisenberg groups. We will focus on minimal graphs that arise as limits of Riemannian minimal surfaces. As a motivation we will describe a mathematical model for the first layer of the visual cortex which makes use of such graphs.

Fall 2008

Shock reflection, free boundary problems, and degenerate elliptic equations.
Mikhail Feldman (University of Wisconsin)
November 20, 2008

Abstract: In this talk we will start with discussion of Riemann problem for multidimensional systems of conservation laws, and shock reflection phenomena. Then we describe recent results on existence and regularity of global solutions to shock reflection for potential flow, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the sonic line). This is a joint work with Gui-Qiang Chen.

Stein Neighborhood Bases on Non-Smooth Domains
Phillip Harrington (University of South Dakota)
November 18, 2008

Abstract: Approximation by smooth domains is a basic technique when doing analysis on non-smooth domains.  In several complex variables, non-smooth pseudoconvex domains are characterized by the fact that they can be exhausted from within by smooth pseudoconvex domains.  This can be used to obtain existence results for solutions to the inhomogeneous Cauchy-Riemann equations.  Less well understood is the special case when pseudoconvex domains have a neighborhood base of pseudoconvex domains (known as a Stein neighborhood base).  When they exist, such neighborhood bases provide additional information about the boundary regularity of solutions to the Cauchy-Riemann equations.  Unfortunately, there are known counter-examples, even for smooth domains.  This talk will review the basic theory of the Cauchy-Riemann equations on pseudoconvex domains, and then discuss sufficient conditions for the existence of Stein neighborhood bases on non-smooth domains.

Analytic Methods for Image Processing
Joseph Lakey (New Mexico State University)
November 6, 2008

Abstract: This expository talk will review methods for addressing two basic problems in image processing, namely denoising and segmentation.  The denoising method amounts to filtering coefficients taken with respect to some unitary transformation of the image.  The question then is which transformation and how to filter. Segmentation refers to separation of geometric information such as edges from textures and noise.  This is usually accomplished by minimizing an energy functional that accounts for geometric and oscillating components.  Several variational models will be reviewed along with their mathematical properties and minimization will be illustrated through several numerical examples. Finally, more recent work connecting transformational methods and energy models will be outlined.

Symmetry Analysis of PDEs: Classical, Nonclassical and Beyond
Danny Arrigo (University of Central Arkansas)
October 23, 2008

Abstract: Symmetry analysis has played an important role in the construction of exact solutions to nonlinear partial differential equations.  Based on the original work of Lie (1881) it provides a unified explanation for the seemingly diverse and ad-hoc integration methods still used to solve ordinary differential equations.  In this talk, both classical and nonclassical methods will be discussed with a focus on nonlinear reaction-diffusions equations.  It will be shown that for several classes of one-dimensional nonlinear diffusion and reaction, the equations admit nontrivial symmetries.  Recent results indicate that these symmetries can be obtained through compatibility with lower order equations. The heat equation serves to highlight this for quasilinear evolution equations while the Boussinesq equation, a 2+1 dimensional nonlinear diffusion equation and a nonlinear wave equation system serve to show that this is true in general.  Results will be presented illustrating the connection between compatibility (sometimes referred to as Charpit’s method) of first order PDEs and nonclassical contact symmetries.  Generalized compatibility for a class of 2+1 nonlinear diffusion equation and the nonlinear Cubic Schrödinger equation will also be discussed.

L^p-summability of Riesz means for the sublaplacian on complex spheres
Marco Peloso (Universita' di Milano, Italy)
August 29, 2008

Abstract: The classical problem of the Lp-boundedness of the Bochener-Riesz multiplier is equivalent to the Lp-convergence of sz means for the Laplacian in Rn.  In this talk we briefly review some of the classical theory for elliptic operators on compact manifolds and for the sublaplacian on the Heisenberg group.  Then we study the question of the Lp-summability for the Riesz means S#R for the sublaplacian on complex spheres.  In particular we show that the critical exponent for the convergence depends on the topological dimension and not on the homogeneous dimension on the underlying space.

Fall 2007

Neuromorphometry and Image-Guided Surgery
John W. Haller (National Institute of Biomedical Imaging and Bioengineering, National Institutes of Health)
November 29, 2007

Abstract: Innovations in biomedical imaging critically depend on the mathematical and statistical sciences. This presentation will focus on the application of mathematics and statistics for post-processing of medical images.  Application of patient-specific models for diagnosing disease and guiding interventions (specifically, image-guided surgery) will be described. This talk will be especially accessible to graduate students and non-experts in the field.
Mathematics and statistics became vital for x-ray computed tomography (CT) and positron emission tomography (PET) imaging in the 1970s.  In the 1980s, MRI provided yet another method for exquisitely characterizing soft tissue.  A critical problem for medical imaging in the 21st century is moving from what is primarily a qualitative science, where health care workers make a subjective determination of the presence or absence of disease, to a quantitative science, where probabilities are assigned, change is quantified, and the quantitative sciences are used to provide greater diagnostic specificity.
To advance to a more quantitative science of medical imaging, computer-based mathematical and statistical methods are being explored in a variety of clinical applications, including real-time, image-guided interventions. An image-guided intervention is defined as a patient encounter where images are used during a minimally-invasive procedure for guidance, navigation and orientation to reach a specified target in the body. Common requirements for all image-guided interventions are a source of images, a real-time interactive display linked to the intervention with a means of target definition in the context of real 3D space of the patient’s anatomy (as distinguished from the virtual image space).

Identifying Protein Biomarkers From Mass Spectrometry Data with Ordinal Outcomes
Deukwoo Kwon (National Cancer Institute)
October 25, 2007

Abstract: In recent years, there has been an increased interest in using protein mass spectroscopy to identify molecular markers that discriminate diseased from health individuals. Existing methods are tailored towards classifying observations into nominal categories. Sometimes, however, the outcome of interest may be measured on an ordered scale. When we ignore this natural ordering it results in some loss of information. We propose a Bayesian model for the analysis of mass spectrometry data with ordered outcome. The method provides a unified approach for identifying relevant markers and predicting class membership. This is accomplished by building a stochastic search variable selection method within an ordinal outcome model. We apply the methodology to mass spectrometry data on ovarian cancer cases and healthy individuals. We also utilize wavelet-based techniques to remove noise from the mass spectra prior to analysis. We identify protein markers associated with being healthy, having low grade ovarian cancer, or being a high grade case. For comparison, we repeated the analysis using conventional classification procedures and found improved predictive accuracy with our method.

Beyond the Cauchy-Riemann equations
Craig Nolder (Florida State University)
October 18, 2007

On the nonexistence of smooth Levi flat real hypersurfaces in the complex projective space
Andrei Iordan (Universite' Paris 6, France)
October 11, 2007

Abstract: In 1993 D. Cerveau conjectured the non-existence of smooth Levi-flat real hypersurfaces in in the complex n-dimensional projective space, with n greater than or equal to 2, ie. of real hypersurfaces admitting a local foliation by complex-analytic hypersurfaces. If n is greater than or equal to 3, this problem was solved for real analytic hypersurfaces by A. Lins Neto and for smooth hypersurfaces by Y.-T. Siu.
In this lecture we will discuss this problem and we will give an improvement of the regularity in Siu's theorem.

Hawaiian Earrings, Their Homotopy and Homology Groups
Satya Deo (Harish-Chandra Research Institute)
September 20, 2007

Abstract: Hawaiian earrings and their r-dimensional generalizations, r >1, have recently attracted a lot of attention of algebraic topologists for several reasons. These are compact metric spaces which are locally nice everywhere except at one point, and consequently have quite complicated fundamental groups. Their homology groups and homotopy groups are yet not fully computed though several interesting partial results have been obtained. Barrat and Milnor were first to use these spaces to give an example of a space whose homology groups have no respect for the dimension of the space. They also left open several questions which are currently being tackled. The talk will highlight some of the recent results on these problems.

Statistical Mechanics of Population Coding in the Brain
Eitan Gross (University of Arkansas)
September 13, 2007

Abstract: The response of a single neuron to a sensory stimulus is extremely noisy and only weakly coupled to changes in the stimulus. Consequently, the information carried by a single neuron is very low. The brain overcomes this limitation by distributing the information across a large number of neurons which together carry more accurate information about the stimulus. Population coding is thus a central paradigm for information processing in the brain. This paradigm has evoked numerous studies on the thermodynamic efficiency of population codes. Of particular interest is the way in which the amount of information about the stimulus depends on the size of the neuron population that participates in the response, as well as on the transfer function of the individual neuron. In this project, we developed a computer model of a neuronal population system made of stochastic, statistically independent elements; and studied the mutual information between several types of stimuli and our model system. We analyzed the properties of the mutual information (MI) in the limit of a large system size N, using statistical mechanics. For discrete-valued stimuli, MI saturates exponentially with N. For continuous-valued stimuli, MI increases logarithmically with N and is related to the logarithm of the Fisher information of the system. Furthermore, we found that the exponent of MI saturation scales as the Renyi distance between response probabilities induced by different stimuli. I will also demonstrate how our model can be used to solve for singularities in the binding problem. Our model provides a tractable tool to study brain response mechanisms to neuronal representations of sensory, motor, and cognitive events.

Spring 2007

Viscosity Solutions of the p-Laplace Equation
Juan Manfredi (University of Pittsburgh)
May 3, 2007

Abstract: Consider the p-Laplacian equation 0 = div(|rv|p-2rv) and its parabolic counterpart @v @t = div(|rv|p-2rv) These equations can be studied as divergence form equations using the notion of weak or distributional solution, and also as non-divergence form equations using the notion of viscosity solution. During the first part of the talk we will discuss the relationships between these two notions of generalized solution, which fortunately agree for bounded functions. During the second part of the talk we will present some recent regularity results for viscosity supersolutions and their spatial gradients. We give a new proof of the existence of rv in Sobolev’s sense and of the validity of the equation ZZ-v @' @t + h|rv|p-2rv, r'idx dt 0 (1) for all test functions ' 0. Here is the underlying domain in Rn+1 and v is a bounded viscosity supersolution in . The first step of our proof is to establish (1) for the so-called infimal convolution va. The function v has the advantage of being differentiable with respect to all its variables x1, x2, · · · , xn, and t, while the original v is merely lower semicontinuous to begin with. The second step is to pass to the limit as ! 0. It is clear that v ! v but it is delicate to establish a sufficiently good convergence of the rv’s. This is joint work with Peter Lindqvist at Trondheim.

The geometric structure of the visual cortex
Giovanna Citti (University of Bologna)
May 2, 2007

Statistical Mechanics of Population Coding in the Brain
Eitan Gross (University of Arkansas, Dept. of Physics)
May 1, 2007

Abstract: The response of a single neuron to a sensory stimulus is extremely noisy and only weakly coupled to changes in the stimulus. Consequently, the information carried by a single neuron is very low. The brain overcomes this limitation by distributing the information across a large number of neurons which together carry more accurate information about the stimulus. Population coding is thus a central paradigm for information processing in the brain. This paradigm has evoked numerous studies on the statistical efficiency of population codes. Of particular interest is the way in which the amount of information about the stimulus depends on the size of the neuron population that participates in the response, as well as on the transfer function of the individual neuron. In this project, we developed a computer model of a neuronal population system made of stochastic, statistically independent elements; and studied the mutual information between several types of stimuli and our model system. We analyzed the properties of the mutual information (MI) in the limit of a large system size N, using statistical mechanics. For discrete-valued stimuli, MI saturates exponentially with N. For continuous-valued stimuli, MI increases logarithmically with N and is related to the logarithm of the Fisher information of the system. Furthermore, we found that the exponent of MI saturation scales as the Renyi distance between response probabilities induced by different stimuli. I will also demonstrate how our model can be used to solve for singularities in the binding problem. Our model provides a tractable tool to study brain response mechanisms to neuronal representations of sensory, motor, and cognitive events.

A Short Introduction to Cauchy-Riemann Theory
Roman Dwilewicz (University of Missouri-Rolla)
April 26, 2007

Abstract: After a short introduction to the Cauchy-Riemann (CR) theory, some problems of the theory will be presented: holomorphic extensions and approximations of CR functions and, if time allows, d-bar problem in complex fiber bundles and applications to vector bundles over complex tori and relations to theta functions. This all will be done in an elementary way and no prior knowledge of complex analysis will be expected.

Random walks and groups
Joseph Maher (Oklahoma State)
April 2, 2007

Abstract: We will start from the simplest examples of random walks, which are the nearest neighbor random walks on graphs. After each unit of time, you jump to one of the adjacent vertices in the graph with equal probability. The standard examples of random walks on the integer lattice in R, or R^n, are of this form. If we interpret the integer lattice in R^n as the Cayley graph of the abelian group Z^n, we can relate properties of the random walk to algebraic properties of groups. We will mention how random walks have given new information about braid groups and the mapping class groups of surfaces.

Carnot-Carathéodory metrics and viscosity solutions
Frederica Dragoni (Istituto Nazionale d'Alta Matematica, Italy; visiting University of Pittsburgh)
April 19, 2007

Abstract: In the first part I’ll give some basic notions about Carnot-Carathéodory metrics and viscosity solutions. I’m going to show some examples in order to understand why those theories are introduced. Moreover I’ll quote some known results of existence and uniqueness for evolutive Hamilton-Jacobi equations, introducing the Hopf-Lax formula (in the classic setting) and showing an important link to the calculus of variation. In the second part I’ll give an existence result theorem in the context of semicontinuous initial data and Hörmander-Hamiltonians. The key to prove this result is to solve the associated eikonal equation. In the third part I’ll prove a convergence theorem which generalizes a known result for the usual inf-convolutions to the metric setting. In this part I’ll use a Large Deviation Principle in the hypoelliptic case, got by an easy new proof, using some techniques of measure theory.

Taking a wavy Euclidian royal road to relativity and quantum theory: Occam’s razors and Evenson’s lasers
William Harter (University of Arkansas, Dept. of Physics)
April 12, 2007

Abstract: A concise, lucid, precise (as well as colorful) derivation of special relativity and quantum theory is possible by Euclidean ruler&compass logic. The trick is to look carefully at the geometry of simple wave interference and apply Occam’s razor to Einstein’s 1905 postulates regarding the speed of light. This shows how light makes its own coordinate frame to position itself in a kind of mini-GPS and then tells us some things about matter.
The Earth Global Positioning System (GPS) is one result of a renaissance in experimental optical precision begun by co-workers of Ken Evenson (1932-2002) who made ultra-precise measurements of speed of light: c= 299,792,458m•s-1 in 1972 at the Time and Frequency Section of the Boulder lab of the National Institute of Standards. Evenson’s work was honored in the 2005 Nobel Prize in Physics.
This talk is an attempt to show logical clarity and precision worthy to accompany experimental precision of Evenson metrology, and like his work, exploits wave resonance and spectral properties of light and matter to an uncommon degree.

A Filtering Approach to Abnormal Cluster Identification
Zhengxiao Wu (University of Wisconsin, Madison)
March 29, 2007

Abstract: A series of events $X₁,X₂,\ldots$ occur at times $\tau₁,\tau₂,\ldots$. Each event is either "normal'' or "abnormal.” We model the observations as a marked point process with a randomly initiated and growing cluster which represents the "abnormal'' events. Our goal is to compute the conditional probability that an observed event is abnormal in real time.
Employing filtering techniques, we derive versions of the Zakai and Kushner-Stratonovich equations in our setting. This framework is applied in earthquake occurrence modeling. Such filtering model performs well in declustering.

Capturing local spatial behavior and speciation through hybrid Dirichlet Process semi-parametric modeling
Michele Guindani (University of Texas M.D. Anderson Cancer Center)
February 22, 2007

Abstract: Spatial heterogeneity plays an important role in a diverse set of applications. For example, in ecology, heterogeneous environments promote camouflaged prey species and disruptive selection; in economics, local characteristics determine regional policies; in health sciences, histopathological heterogeneity characterizes certain cancer tissues within and among tumor types.
Recent Bayesian modeling of univariate spatial data has considered mixed effect models, where a residual stationery (homogeneous) Gaussian effect is assumed. Arguably, one might prefer the flexibility of a nonstationery, non-Gaussian specification. In a nonparametric setting, this can be accommodated by mixture of Dirichlet process (DP) models. The DP is an example of a species sampling prior, which are typically used to describe diversity of different ecological groups of species under different environmental conditions.
However, a limitation of the mixture of DP models is that the latent factor driving species sampling is globally defined and may fail to account for spatial heterogeneity. In this work, we introduce a novel class of prior distributions, the hybrid Dirichlet Processes (hDP), which generalize the DP and overcome this limitation. In a spatial setting, the hDP are defined as mixtures of Gaussian random fields with spatially varying weights. A crucial feature of this specification is the possibility to model local speciation and hybrid clustering.
We illustrate the procedure by means of a simulated example and an application to the analysis of hippocampal atrophy in brains of patients affected by Alzheimer's disease.
This is joint work with Alan Gelfand (Duke University, USA) and Sonia Petrone (Bocconi University, Italy).

A spatialy-adjusted Bayesian additive regression tree model to merge two datasets
Song Zhang (University of Texas M.D. Anderson Cancer Center)
February 12, 2007

A presentation for the automorphisms of the 3-sphere that preserver a genus two Heegard splitting
Erol Akbas (University of Arkansas)
February 08, 2007

A partnership between a middle school mathematics teacher and a university researcher centered on the content
Anthony Fernandes (University of Arizona)
January 29, 2007

Abstract: This talk will outline a case study between a middle school mathematics teacher and a university researcher as we had discussions centered on the mathematics content and teaching. The study seeks to understand the nature of this partnership, how it evolved over time, the constraints on the partners, the benefits to the teacher’s work, and the evolution of the cognitive demand of tasks. In the talk I will share my reasons for doing the study, selected background literature, preliminary results and possible future directions of this research.

Making Sense of the Infinite: A Study Investigating the Learning and Teaching of Infinite Series
Brian Lindaman (University of Kansas)
January 23, 2007

Abstract: This study used results from a preliminary survey to create and implement a teaching experiment specifically targeting students' understanding of infinite series. The experiment compared computational performance and conceptual understanding of two groups of calculus students. One group received traditional instruction on series; the other group received instruction which included classroom reform strategies such as writing during class, working in pairs, and an emphasis on visualization. Results and conclusions will be shared along with descriptions of classroom assessments and activities related to this project.

Mathematics textbooks and state curriculum standards: an analysis of the alignment between the K-8 written and intended curricula
Shannon Dingman (University of Missouri)
January 17, 2007

 

Fall 2006

A Presentation For The Automorphisms Of The $3$-Sphere That Preserve A Genus Two Heegaard Splitting
Erol Akbas (Georgia State University)
November 30, 2006

Abstract: Scharlemann constructed a connected simplicial 2-complex $\Gamma$ with an action by the group ${\mathcal H_{2}}$ of isotopy classes of orientation preserving homeomorphisms of $S3$ that preserve the isotopy class of an unknotted genus $2$ handlebody $V$. We prove that the 2-complex $\Gamma$ is contractible. Therefore we get a finite presentation of ${\mathcal H_{2}}$.

Tunnel number 1 knots and the disk complex
Darryl McCollough (University of Oklahoma)
November 20, 2006

Abstract: A knot K in the 3-sphere is said to be a tunnel number 1 knot when there exists an arc meeting K in its endpoints so that a neighborhood of the union of K and the arc, necessarily a genus-2 handlebody, can be moved to the standard (unknotted) position. The tunnel number 1 knots are an interesting and well-studied class, which includes all 2-bridge knots and torus knots. In this talk, we present the basic ideas of a new theory which describes all the tunnels of tunnel number 1 knots using a tree derived from the collection of all disks in the standard genus-2 handlebody.
The theory makes fundamental use of recent results of M. Scharlemann and E. Akbas.

Spectral analysis of Laplacians on the Heisenberg group
Fulvio Ricci (Scuola Normale, Pisa, Italy)
November 14, 2006

Abstract: The Heisenberg group $H_n$ is a non-commutative Lie group, and Fourier analysis on it in principle requires tools from representation theory that make it hard to handle in many instances. However, if attention is restricted to functions or operators that are invariant under unitary transformations of $H_n$, the Fourier analysis becomes essentially commutative, and various aspects of classical analysis on $R^n$ can be recovered. This applies to functional calculus on left-invariant differential operators that commute with unitary transformations, namely the central derivative, the sub-Laplacian and polynomials in these two operators. Recently these methods have been pushed forward so to cover other operators, including Hodge Laplacians acting on differential forms.

On Hermitian geometry of complex surfaces
Massimiliano Pontecorvo (University of Rome 3, Italy)
November 6, 2006

Abstract: We present some recent results on anti-self-dual Hermitian metrics and relate them to more general questions on the geometry of compact complex surfaces.

On Hardy type inequalities
Alexander Balinsky (University of Cardiff, United Kingdom)
November 2, 2006

Abstract: The aim of this talk is to present some recent results on Hardy and Sobolev type inequalities for magnetic Dirichlet forms with multiple singularities in dimension d=2. Our approach is based on the conformal invariance of magnetic Dirichlet forms with Aharonov-Bohm potentials. The strategy is first to establish Hardy-type inequalities for doubly connected domains using uniformization, and second to use Morse theory to decompose R2 into doubly connected domains with explicit uniformizations.

Uniformization results in conformal geometry
Andrea Malchiodi (SISSA, Trieste, Italy)
October 26, 2006

Abstract: The classical uniformization theorem asserts that every compact surface admits a conformal metric with constant Gauss curvature. This result has a counterpart in dimension greater or equal to three, when one is interested in finding conformal metrics with constant scalar curvature, which is known as the Yamabe problem. I will review these results and discuss some other conformally invariant objects of higher order, the $Q$-curvature and the Paneitz operator, which play a role in the study of four-dimensional manifolds.

Spring 2006

Finding global minimizers of segmentation and denoising functionals
Selim Esedoglu (University of Michigan)
May 2, 2006

Abstract: Segmentation is a fundamental procedure in computer vision. It forms an important preliminary step whenever useful information is to be extracted from images automatically. Given an image depicting a scene with several objects in it, its goal is to determine which regions of the image contain distinct objects.
Variational segmentation models, such as the Mumford-Shah functional and its variants, pose segmentation as finding the minimizer of an energy. The resulting optimization problems are often non-convex, and may have local minima that are not global minima, complicating their solution. We will show that certain simplified versions of the Mumford-Shah model can be given equivalent convex formulations, allowing us to find global minimizers of these non-convex problems via convex minimization techniques. In particular, we will show that a recent convex duality based algorithm due to A. Chambolle, which was originally developed for Rudin, Osher, and Fatemi’s total variation denoising model, can be adapted to the segmentation problem.

 

Quasiconformal geometry of fractals
Mario Bonk (University of Michigan)
April 27, 2006

Abstract: Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston’s characterization of rational functions with finite post-critical set.
In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpinski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will report on some recent developments in this area.

Local Isometric embedding of surfaces in 3-space
Qing Han (Notre Dame)
April 11, 2006

Abstract: Schaefli asserted in 1883 that 2 dimensional Riemannian manifolds admit local isometric embeddings in the 3 dimensional Euclidean space. It was verified by Janet in 1925 that it is true if the 2 dimensional Riemannian manifolds are analytic. The smooth case remains open. It turns out that Gauss curvature plays an important role. It is well known that Schaefli’s assertion holds if the Gauss curvature does not vanish.
In this talk, I will present some new results concerning Gauss curvature which is allowed to vanish.

Circle Packing and the Experimental Imperative
Ken Stephenson (University of Tennessee)
March 30, 2006

Abstract: The talk is nominally about circle packings – that is, about configurations of circles having prescribed patterns of tangency. As it turns out, however, these packings provide a discrete entre into a world which begins to look very familiar, the world of conformal maps, analytic functions, and Riemann surfaces. So the talk is really about this conformal geometry, right? Well....no. Actually, the talk is about experiments, about being captured by experiments; it’s about road trips and touring. This owl and its fingerprint (see pdf file linked below) will be just one of our sightings.
Illustration: [PDF]

Boundedness of Fourier integral operators on Hardy spaces
Marco Peloso (Politecnico di Torino, University of Missouri-Columbia, and Washington University, St Louis)
March 9, 2006

Abstract: We consider the class of Fourier integral operators with non-degenerate phase and classical symbol a in S^m. In their seminal result, Seeger-Sogge-Stein proved that such an operator T maps L^p boundedly into itself if and only if |1/p− 1/2| −m/(n − 1). After illustrating a few examples, we prove that T maps the local Hardy space h^p boundedly into itself when 1/p − 1/2 < −m/(n − 1).
This is joint work with S. Secco.

A meta-analysis of effects of standards-based curricula on student's achievement in mathematics classes
Xiaobao Li (Texas A&M University)
March 1, 2006

Abstract: This study employs meta-analysis method to

1. synthesize results of individual studies to investigate the magnitude of effects of standards-based curricula on mathematics achievement in students
2. analyze the variation of such magnitude due to grade, curricula, length of implementation of standards-based curricula, publication types, and teachers' professional development.

What wavelets are and do
Guido Weiss (Washington University at St. Louis)
February 23, 2006

Abstract: I plan to give a talk that describes the field of Wavelets as a pure mathematical subject and its properties that make it a good source for applications in many fields in science and engineering. I strongly believe that this subject is a very beautiful mathematical field. I hope to convince a rather wide audience of this. Students, Engineers, Scientists and Mathematicians having little background in Analysis should be able to understand the message I will try to give.

Motion of Surfaces by Curvature
Gieri Simonett (Vanderbilt University)
February 16, 2006

Abstract: Several geometric evolution laws that describe the motion of curves and surfaces driven by curvature will be introduced. In the examples considered, surfaces will evolve in such a way as to minimize certain geometric quantities (surface area, total curvature). I will discuss questions related to the existence and uniqueness of solutions and their geometric properties. Numerical simulations will be provided, showing that some surfaces will smoothly evolve into spheres (thereby reaching a final destination), while others may develop singularities. Many questions raised by the numerical simulations are wide open and still await a rigorous mathematical treatment.

Complex Symmetric Operators
Stephen Garcia (University of California, Santa Barbara)
February 10, 2006

Abstract: We discuss Hilbert space operators which have symmetric matrix representations with respect to an orthonormal basis. This surprisingly large class includes many well-known classes of operators of interest in both pure and applied mathematics. We highlight several recent theorems and a few applications.

Yang-Baxter Equations and Their Super Solutions
Gizem Karaali (University of California, Santa Barbara)
February 9, 2006

Abstract: I will start with a brief overview of the Yang-Baxter equations and their relationship to quantum groups. I will then explain the super analog of these concepts. In particular, I will discuss certain results which generalize the nongraded case and then concentrate on some examples which are peculiar to the super case. This talk should be accessible to anyone who is somewhat intrigued by this abstract but not necessarily sure of what a quantum group is.

Fourier Series: Past, Present and Future
M. Lacey (Georgia Tech)
January 26, 2006

Abstract: Carleson’s celebrated 1965 theorem on the pointwise convergence of Fourier series has over the last ten years become part of an emerging theory in Harmonic Analysis. I’ll survey the history of Carleson’s theorem, explain why the theorem is hard, why one might be interested in it, and elements of the new theory that has grown up around this Theorem.

 

Fall 2005

Some Zero-Sum Problems
Francesco Pappalardi (Universita Rome III)
November 17, 2005

A prototype of zerosum theorems, the well-known theorem of Erdos, Ginzburg and Ziv says that for any positive integer n, any sequence a₁, a₂, . . . , a_2n−1 of 2n − 1 integers has a subsequence of n elements whose sum is 0 modulo n.
Appropriate generalizations of the question, especially that for (Z/pZ)^d, generated a lot of research and still have challenging open questions. After having reviewed some classical results, I will propose a new generalization of the Erdos-Ginzburg-Ziv Theorem and prove it in some basic cases.
This is a joint project with S.D. Adhikari, Y.G. Chen, J.B. Friedlander and S.V. Konyagin.

A closer look at certain strongly Cohen-Macaulay ideals and residual intersections
Christine Cumming (Tulane University)
November 11, 2005
3:30pm

Strongly Cohen-Macaulay ideals often have nice properties. I will present a few cases when strong Cohen-Macaulayness is clear and another case when it is not clear. Such an ideal satisfies a condition on the depth of the powers of that ideal. I use this to show when residual intersections (a generalization of linkage) are Cohen-Macaulay.

On square-free monomial ideals
Tai Huy Ha (Tulane University)
November 10, 2005
3:30pm, SCEN 322

Let I be a polynomial ideal generated by square-free monomials. The ideal I can be associated to a graph (if the generators of I are of degree 2), or more generally, to a simplicial complex (otherwise). In this talk, I will give a survey on current studies in establishing a dictionary between algebraic properties of I and combinatorial structures of the associated graph or simplicial complex.

Visualizing one dimensional Teichmuller space
Yasushi Yamashita (Nara Women’s University)
November 8, 2005

We present a method for drawing Bers embeddings of the Teichmuller space of once punctured tori in which Jorgensen’s theory on the quasifuchsian space of once punctured tori is applied. We made a computer program of this algorithm to produce the pictures. In this talk we show the pictures of the Bers embeddings.
We observe the self-similarity of the boundary of a Bers slice in our pictures.
This is a joint work with Yohei Komori, Toshiyuki Sugawa and Masaaki Wada.

 

 

Spring 2005

Heat Operator on Some Noncompact Spaces
Thalia Jeffres (Wichita State University)
April 28, 2005

Abstract: I will describe some joint work with Paul Loya in which we studied differentiability properties of the heat operator on manifolds with conical singularities and manifolds with cylindrical ends. We applied these results to obtain existence of solutions to some nonlinear parabolic differential equations on these manifolds.
This talk will be for general audience, including graduate students, and I will discuss the heat equation and some applications in Rⁿ.

Uniqueness in the inverse conductivity problem and regularity of the conductivity
Russell Brown (University of Kentucky)
April 21, 2005

Two and Three D modeling, analysis of Navier-Stokes Equations
Kumud Altmeyer (University of Arkansas Pine Bluff)
April 20, 2005

When are 3 dimensions not 3-dimensional? Understanding the topology of 3-manifolds
Ryan Derby-Talbot (University of Texas)
Tuesday April 12, 2005

Abstract: In studying the topological nature of n-manifolds, the number n = 3 is somehow unique. While n-manifolds are completely classified for n < 3 and cannot be classified for n > 3, the answer is still unknown for n = 3.  This seems strange given that we can visualize 3 dimensions. Since 3-manifolds actually reside in 4, 5 and 6 dimensional space, it seems there is just enough room for these manifolds to be  complicated while not losing all of our visual intuition.   Using lower dimensional analogues as our guide, we will explore some of the intricacies that arise in attempting to understand 3-manifolds. In particular, we will see how our ability to visualize 3 dimensions plays an important role in studying 3-manifolds, via such objects as knots and Heegaard splittings.

Title and abstract TBA
Tao Li (Oklahoma State University
Friday April 8, 2005
2:30 pm

Approximation and Geometric Function Theory in Complex and Hypercomplex Variables
Sorin Gal (University of Oradea, Romania)
April 7, 2005
Thursday at 2:30 pm

Equilibrium distribution of a charge in presence of an external field
Igor Pritsker (Oklahoma State University)
March 31, 2005

Abstract: Assuming that the unit circumference is made of a conducting wire, it is easy to guess that the equilibrium distribution of a charge located on this wire is uniform (in angular sense). However, the problem
of equilibrium distribution becomes much more difficult if there is an external field acting on our charge. We shall discuss how to solve this problem for rather general external fields, and shall also consider some
applications. Despite its seemingly special nature, potential theory with external fields has been actively used in approximation theory, orthogonal polynomials, random matrices, combinatorics and number theory.

Space-Time Analysis of Extreme Values
Gabriel Huerta (University of New Mexico)
March 18, 2005
Friday at 2:30 pm

Bayesian Multivariate Spatial Models for Roadway Traffic Crashing Mapping
Joon Jin Song (University of Massachusetts)
March 17, 2005
Thursday at 2:30 pm

Abstract: We consider several Bayesian multivariate spatial models for estimating the crash rates from different kinds of crashes. Conditional autoregressive (CAR) model is considered for the spatial effect model and is generalized for the multivariate case. A general theorem for each case is provided to ensure posterior propriety under noninformative prior. The different models are compared according to some Bayesian criterion. Markov chain Monte Carlo (MCMC) is used for computation. We illustrate these methods with Texas Crash Data.

On the Geometrical Structure of the Visual Cortex
Professor Giovanna Citti (Universita' di Bologna, Italy)
March 16, 2005
Wednesday at 2:30 pm

Clifford Analysis on Dirac Bundles and Applications
Mircea Martin (Baker University, Kansas)
March 3, 2005

Abstract: The goal of this talk is to show that the singular integral operators associated with the fundamental solutions of Dirac operators on Dirac bundles are controlled by certain maximal operators. The inequality that quantifies the link involves an absolute constant that can be explicitly computed. As consequences of that inequality we will derive several quantitative Hartogs–Rosenthal-type theorems for Dirac operators concerning monogenic approximation on compact sets with respect to different natural norms on the space of sections of a Dirac bundle.

Extremal Problems in Hardy and Bergman Spaces
Catherine Beneteau (Seton Hall University)
March 1, 2005

Largest Circuits and Cocircuits in Matroids
Nolan McMurray (University of North Carolina at Wilmington)
February 24, 2005

A Glimpse into Cofiniteness and Local Cohomology
Janet Vassilev (University of Arkansas)
February 22, 2005

Abstract: Local cohomology has been used successfully as a problem solving tool in commutative algebra and algebraic geometry since it's development in the 1960s. Surprisingly, finding local cohomology modules which do not satisfy various finite-ness properties has been quite difficult. We will present some recent results of the speaker on I-cofinite modules M (i.e. modules for which Ext^p_R(R/I,M) are finitely generated for all p). Special attention will be given to a class of local cohomology modules, H^q_I(M), which are not I-cofinite.

A generalization of Gram-Schmidt orthogonalization generating all Parseval frames
Gitta Kutyniok (University of Giessen, Germany)
February 17, 2005

Multivariate Lattice Models for Areal Data with Application to Multiple Disease Mapping
Xiaoping Jin (University of Minnesota)
January 27, 2005

Abstract: The last decade has seen an explosion of interest in disease mapping, with increasing availability of Geographic Information System (GIS) technology and spatial databases. For example, the databases from the National Center for Health Statistics (NCHS) or from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute, publicly available to anyone with a web browser, provide an enormous supply of georeferenced data. Conditionally autoregressive (CAR) models (Besag et al., 1991) have been widely used for single disease mapping with such data. But when we simultaneously map multiple diseases, a multivariate areal model may be needed to permit modeling of dependence between diseases while maintaining spatial dependence across regions.  Existing methods for multivariate areal data (see, e.g. Kim et al., 2001; Carlin and Banerjee, 2003; Gelfand and Vonatsou, 2003) typically suffer from unnecessary restrictions on the covariance structure. In this talk, we propose a class of Bayesian hierarchical models for multivariate areal data that avoids these restrictions, permitting flexible modeling of correlations both between diseases and across areal units. Our framework encompasses a rich class of multivariate conditionally autoregressive (MCAR) models that are computationally feasible via modern Markov chain Monte Carlo (MCMC) methods. We illustrate the strengths of our approach over existing models using simulation studies, and also offer a real-data application to disease mapping which involves annual lung, larynx, and esophagus cancer death rates in Minnesota counties between 1990 and 2000.
This work is joint with Bradley P. Carlin and Sudipto Banerjee (Division of Biostatistics, University of Minnesota).

Empirical Bayesian Analysis for High-Dimensional Computer Output
Dorin Drignei (Iowa State University)
January 24, 2005

Robust transmission/disequilibrium test for incomplete family genotypes
Gulhan Alpargu (University of Massachusetts)
January 20, 2005

Abstract: Computer experiments are increasingly used in scientific investigations as substitutes for physical experiments in cases where the later are difficult or impossible to perform. A computer experiment consists of several runs of a computer model for the purpose of better understanding the input-output relationship. The practical difficulty in some situations is that a single computer model run may use a prohibitive amount of computational resources. A recent approach proposes to use statistical models as less expensive surrogates for such computer models; these provide both point predictors and uncertainty characterization of the outputs. This talk describes a two-stage statistical method for computer experiments which produce multivariate output on a spatio-temporal grid with large time.

 

Fall 2004

Intersection Multiplicity
Izuru Mori (SUNY, Brockport)
December 3, 2004

Schrodinger equations with time dependent potentials
Virginia Naibo (University of Kansas)
December 2, 2004

The theorem of Busemann-Feller-Alexandrov for convex functions in Carnot groups
Nicola Garofalo (Purdue University)
November 18, 2004

G-compactness of elliptic systems
Leonid Kovalev (Washington University)
September 30, 2004

 

 

Spring 2004

Sparse Fourier representations via sampling
Martin Strauss (AT&T Labs and the University of Michigan)
April 20, 2004

The Radon transform in texture analysis
Swanhild Bernstein (University of Mining and Technology, Freiberg, Germany)
April 8, 2004

Dense surface groups in Lie groups
Juan Souto (Mathematisches Institut Rheinische Friedrich-Wilhelms-Universitdt Bonn, Germany)
April 6, 2004

Abstract: We show that every Lie group $G$ whose identity component $G_0$ is not solvable and with $G/G_0$ finitely generated contains a dense subgroup isomorphic to the fundamental group of a surface. In the particular case that $G$ is connected and semi-simple, the surface may be chosen to have genus 2.
This is a joint work with E. Breuillard, T. Gelander and P. Storm.

Slepian functions as the solution of energy concentration problem: A brief history and new development
Xiaping Shen (University of Ohio)
April 1, 2004

Abstract: Because of the band-limiting nature of physical devices, the function space which consists of all band-limited functions of finite energy is the most interesting one for electrical engineers. On the other hand, functions with finite time duration, like Daubechies wavelets, possess desirable properties such as high computing efficiency. However, a non-trivial function cannot have time limiting and frequency limiting properties simultaneously. The next best thing one can ask is the following: among all possible band-limited functions with a given bandwidth $\sigma$, which function maximizes the fraction of energy over the prescribed time interval $[-\tau ,\tau ]$? Studied by Slepian and his collaborators at Bell labs in the 1960s, the continuous prolate spheroidal wave functions (PSWFs or Slepian functions), are special functions that lead to the optimal solution of this energy concentration problem. They have many potential applications in areas such as communication and signal processing. In this talk, we will briefly review the historical development 
of PSWFs in both the areas of mathematics and engineering, and report our most recent research results in connection with Shannon sampling theory and wavelet analysis.
This is joint work with Gilbert Walter.

Integrable field theories, meromorphic loops and the Riemann-Hilbert problem
Edwin Beggs (University College of Wales, Swansea, UK)
March 30, 2004

Efficient Parameterization and Estimation of Spatio-Temporal Dynamic Models                                                                                                     
(Bill) Ke Xu (Department of Statistics, University of Missouri – Columbia)
March 29, 2004

Efficient Parameterization and Estimation of Spatio-Temporal Dynamic Models                                                                                                     
(Bill) Ke Xu (Department of Statistics, University of Missouri – Columbia)
March 29, 2004

Conjugate functions and semi-conformal mappings
Michael Eastwood (University of Adelaide, Australia)
March 23, 2004

Abstract: Suppose f is a smooth function of two variables. Is there a smooth function g such that |grad f| = |grad g| and <grad f,grad g> = 0? The answer is yes if and only if f is harmonic. What about the same question for a function of three or more variables? Joint work with Paul Baird derives a differential inequality that must be satisfied by f and a differential equation in the case of a function of three variables. When f admits a conjugate, the pair (f,g) provides a semiconformal mapping into R^2. In particular, harmonic morphisms provide examples of conjugate pairs but there are more besides. The problem of finding a conjugate is conformally invariant so it not surprising that our constraints are also conformally invariant.

How far can a convex function stretch the unit disk from being a disk?
Roger Barnard (Texas Tech University)
March 4, 2004

Invertible substitutions on the line and the projection method
Edmund Harris (Imperial College, London UK)
February 26, 2004

Discrimination Measures for Locally Stationary Time Series Using the Excess Mass Functional
Gabriel Chandler (Department of Statistics, University of California, Davis)
February 20, 2004

Benchmark Estimation for Markov Chain Monte Carlo Samples
Subharup Guha (Department of Statistics, The Ohio State University)
February 16, 2004

Fall 2003

Why is the isotropic correlation model so popular in spatial statistics?
Chunsheng Ma (Wichita State University)
November 20, 2003

Complex-valued planar harmonic functions and regions of constant valence
Genevra Neuman (Kansas State University)
November 13, 2003

Extending bounded holomorphic functions
John McCarthy (Washington University)
November 6, 2003

Configurations of Lines
Javier Bracho (UNAM, Mexico)
October 23, 2003

 

 

Spring 2003

Application of Bayesian Methods to Spatial Econometrics
James P. LeSage (University of Toledo, Ohio)
May 1, 2003

Reproducing Kernels and Invariant Subspaces
Stefan Richter (University of Tennessee)
April 24, 2003

An inversive approach to the Cauchy integral
Michael Bolt (University of Michigan at Ann Arbor)
March 27, 2003

Extremal Problems in Hardy and Bergman Spaces
Catherine Beneteau (Seton Hall University at New Jersey)
March 13, 2003

The inverse mapping theorem on stratified groups
Valentino Magnan (Scuola Normale Superiore, Pisa, Italy)
March 4, 2003

Step functions, harmonic measure, and planar domains
Lesley Ward (Harvey Mudd College)
February 24, 2003

Density of wavelet frames
Gitta Kutyniok (University of Paderborn, Germany)
January 23, 2003

Fall 2002

Distortion of dimension by quasisymmetric maps
Jeremy Tyson (University of Illinois, Urbana Champaign)
December 5, 2002

On projective duality and osculating spaces
Sergey Lvovskiy (Independent University of Moscow)
November 21, 2002

Geometry of nilpotent Lie groups
Michael Cowling (University of New South Wales)
November 19, 2002

Some Applications of Wavelets in Statistics
Marina Vannucci (Texas A&M)
October 31, 2002

Function theory on the Unit Sphere in C2
John Wermer (Brown University)
October 17, 2002

Electromagnetic wavelets and conformal space-time transformations
Gerald Kaiser (Virginia Center for Signals and Waves)
October 10, 2002

The strange but true history of the Poincare Conjecture
Mark Brittenham (University of Nebraska, Lincoln)
September 26, 2002

 

 

Spring 2002

Analysis of spherical symmetries in Clifford analysis
Yakov Krasnov (Bar-Ilan University, Israel)
May 28, 2002

The space of monogenic BMO-functions on the unit sphere
Swanhild Bernstein (Bauhaus University, Weiman, Germany)
May 27, 2002

Analytic capacity and Calderon-Zygmund Theory
Joan Verdera (Universitat Autonoma de Barcelona/UCLA)
April 23, 2002

Inverse Eigenvalue problems for quadratic matrix and operator pencils
Biswa Nath Datta (Northern Illinois University)
April 18, 2002

Zeros of hypergeometric functions
Peter Duren (University of Michigan at Ann Arbor)
April 16, 2002

On the conformal Martin boundaries
Nageswari Shanmugalingam (University of Texas at Austin)
April 4, 2002

Abstract: Recent research in potential theory has focused on the Martin boundary constructed using the fundamental solutions of the Laplacian operator. Such constructions yield a class of minimal harmonic functions that play the role of the Poisson kernel of general domains in Riemannian manifolds; they provide a potential theoretic boundary for the domain in question. The talk will focus on an analogous construction of Martin boundary for the $n$-Laplacian operator on hyperbolic $n$-dimensional Riemannian manifolds using the singular solutions of the operator $\Delta _n u :=-{\div}(|\grad u|^{n-2}\grad u)$ and study the correspondence between the conformal Martin boundary and the geometric boundary of bounded uniform domains in an $n$-dimensional hyperbolic manifold.
This is joint work with Ilkka Holopainen and Jeremy Tyson.

The geometry of modules over a complete intersection
David Jorgensen (University of Texas at Arlington)
April 1, 2002

Abstract: This talk will provide a glimpse of the study of free resolutions of finitely generated modules defined over a commutative Noetherian ring.  Perhaps the earliest result on this topic is Hilbert's syzygy theorem, for finitely generated modules over a polynomial ring. To complement this theorem of Hilbert, we will focus on free resolutions of finitely generated modules defined over a certain type of quotient of a polynomial ring, called a graded complete intersection. We will describe how to associate to a given finitely generated module over a graded complete intersection a special affine algebraic set, called the support set of the module, which encodes subtle homological information about the module. Finally, we will discuss previous and new results on the geometry of these support sets.

Some remarks concerning integrals of curvature for curves and surfaces
Stephen Semmes (Rice University)
March 28, 2002

Multiplicities in Local Algebras 
C.-Y. Jean Chan (Purdue University)
March 25, 2002

Abstract: Algebraic geometers often describe a non-smooth point by its multiplicity and view it as an invariant of the variety.  The multiplicities are defined in an algebraic method under such considerations.  In this talk, I would like to give an geometric interpretation of Hilbert-Samuel and Serre's multiplicities of subvarieties in the affine n-space. These different multiplicities can be viewed intuitively as the intersection numbers obtained by self-intersecting or intersecting with another subvariety.
I will also present my recent results which make connections between these and other multiplicities defined by Buchsbaum-Rim and Mori-Smith respectively.

Recent Advances on Function Spaces, Harmonic Analysis and Boundary Value Problems
Osvaldo Mendez (University of Texas at El Paso)
March 12, 2002

Minimal area problems in conformal mapping
Alex Solynin (Texas Tech University)
March 7, 2002

Abstract: We discuss some minimization problem for the Dirichlet integral of analytic and one-to-one functions. A general problem of this  type consists in finding the minimal Dirichlet norm among all univalent functions satisfying a given set of restrictions on coefficients. Two problems of this type with minimal non-trivial restictions on the functions from the standard class  S were solved by D. Aharonov, H. Shapiro and A. Solynin. Continuing this project we solved recently similar problems for the class of convex functions.

Factorization of almost periodic matrix functions and its applications
Ilya Spitkovsky (William and Mary College)
March 1, 2002

Abstract: We will give an overview of the current state of the factorization problem in the case when the matrix function involved is almost periodic (in Bohr sense). Existence of factorization will be discussed along with algorithms of its actual construction. Applications include (but are not limited to) convolution type equations on finite intervals.

On the partial derivatives of the fundamental solution of the Euclidean  Cauchy-Riemann operator in  R^{n+1} and their associated Eisenstein series in Clifford Analysis
Soeren Krausshar (Ghent State University, Belgium)
February 21, 2002

Aspects of Liaison theory
Uwe Nagel (University of Paderborn, Germany)
February 18, 2002

Abstract: Liaison theory started with the idea to study a curve by  relating it to a simpler curve. The modern point of view is to consider it as a classification theory for curves, surfaces etc. In the talk some of the ideas, recent results and problems will be discussed. In particular, the interplay between geometric and algebraic methods will be explained. Finally, an application to simplicial polytopes will be described.

A history of primary decomposition
Tom Marley (University of Nebraska at Lincoln)
February 14, 2002

Complex cobordisms and the embeddability of CR-manifolds
Bruno De Oliveira (University of Pennsylvania)
Tuesday February 12, 2002

Viscosity Solutions on Grushin type Planes
Tom Bieske (University of Michigan at Ann Arbor)
February 7, 2002

 

Fall 2001

How hard is it to decide whether a collection of polynomials has a common zero?
Carlos Berenstein (University of Maryland)
December 7, 2001

Bayesian unit root tests in Stochastic volatility models
Sujit Ghosh (North Carolina State University)
November 29, 2001

On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols
Alexey Karapetyants (University of Arkansas)
November 15, 2001

 

Null Lagrangians
Tadeusz Iwaniec (Syracuse University)
November 8, 2001

Potential theory of the furthest-point distance function
Igor Pritsker (Oklahoma State University)
October 26, 2001

Subelliptic harmonic maps from Carnot groups
Changyou Wang (University of Kentucky)
October 11, 2001

Torsion in the group of homeomorphisms of the long line
Satya Deo (R.D. University Jabalpur, India)
October 4, 2001

 

Fixed points of holomorphic mappings
Steven Krantz (Washington University)
September 20, 2001

Nonparametric Minimal Surfaces in the Heisenberg Group
Scott Pauls (Dartmouth College)
September 13, 2001
Thursday at 2:30 p.m.

 

 

 

Spring 2001

On the Set of Solutions to Some Nonlinear Equations
Dimiter Vassilev (University of Arkansas)
January 18, 2001
Thursday at 3:30 p.m.

 

Bayesian Methods for Change-Point Detection in Long-Range Dependent Processes
Bonnie Ray (New Jersey Institute of Technology)
February 8, 2001
Thursday at 3:30 p.m.

An Elementary Approach to Symmetric Spaces of Rank One
Adam Koranyi (Lehman College, New York)
March 8, 2001
Thursday at 3:30 p.m.

 

Mathematical Uncertainty Principles: Old and New
Joe Lakey (New Mexico State University)
March 27, 2001
Tuesday at 3:30 p.m.

On the Motion of the Interface Between Two Fluids
Sijue Wu (University of Maryland)
March 28, 2001
Wednesday at 3:30 p.m.

The Harmonic Analysis for \box_b Operators
Lihe Wang (University of Iowa)
April 5, 2001
Thursday at 3:30 p.m.

One-dimensional symmetry of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions
Nicola Garofalo (Johns Hopkins/Purdue University)
April 12, 2001
Thursday at 3:30 p.m.

Lee Form on Quaternionic Kaehler Manifolds
Loius Pernas (University of Picardie, France)
April 19, 2001
Thursday at 3:30 p.m.

Recent Progress on the Bethe-Sommerfeld Conjecture
Zhongwei Shen (University of Kentucky)
April 25, 2001
Wednesday at 3:30 p.m.

Meromorphic functions and Picard's Theorem
Elias Saleeby (University of Arkansas)
April 26, 2001
Thursday at 3:30 p.m.

Bergman's Coordinates at Corners
David Barrett (University of Michigan
May 3, 2001
Thursday at 3:30 p.m.

 

Fall 2000

The Inverse Conductivity Problem with Less Regular Conductivities
Russell Brown (University of Kentucky)
November 10, 2000
Thursday at 3:30 p.m.

Pretzel, a Proof by Computer Animation
Professor Eric Sedgwick (DePaul University)
November 3, 2000
Thursday at 3:30 p.m.

The Homeomorphism Problem and Triangulation
William Jaco (Oklahoma State University)
October 26, 2000
Thursday at 3:30 p.m.

A Class of Differential Equations with Singular Perturbations
Qing Han (Notre Dame University)
October 12, 2000
Thursday at 3:30 p.m.

Test Ideals in One Dimensional Domains
Professor Janet Vassilev (University of Arkansas)
October 5, 2000
Thursday at 3:30 p.m.

Spring 2000

A Notion of Rectifiability Modeled on Carnot Groups
Scott Paul (Rice University)
May 4, 2000
Thursday at 3:30 p.m.

NSF: Program, Funding in Mathematical Sciences and the Future
Joe Jenkins (Program Director for Mathematical Analysis, National Science Foundation, Washington DC)
April 27, 2000
Thursday at 3:30 p.m.

Spectral Properties of Elliptic Layer Potentials on Curvilinear Polygons
Irena Mitrea (University of Minnesota)
April 14, 2000
Friday at 3:30 p.m.

Circles, Triangles and Billiards
Eugene Gutkin (University of Southern California)
March 9, 2000
Thursday at 3:30 p.m.

Interactions between Several Complex Variables and Clifford Analysis
Swanhild Berstein (Bauhaus University, Weimar, Germany)
March 7, 2000
Tuesday at 3:30 p.m.

Hardy Spaces in Non-Smooth Domains: Recent Progress
Marius Mitrea (University of Missouri at Columbia)
March 1, 2000
Wednesday at 3:30 p.m.

Variation of the Spectrum
Thomas Ransford (University of Laval, Quebec, Canada)
February 17, 2000
Thursday at 3:30 p.m.

Topology of Cyclic Configuration Spaces and Periodic Trajectories of Multi-Dimensional Billiards
Michael Farber (Tel Aviv University, Israel)
February 10, 2000
Thursday at 3:30 p.m.

Reversible Cellular Automata
Jarkko Kari (University of Iowa)
February 3, 2000
Thursday at 3:30 p.m.

 Fall 1999

Koenig's map of analytic self-maps of the disc
Pietro Poggi-Carradini (Kansas State University)
November 18, 1999

Lipschitz extensions on the Heisenberg group
Thomas Bieske (Visiting Assistant Professor, University of Arkansas)
November 11, 1999

Equations with critical growth on Carnot groups suggested by problems in CR geometry
Nicola Garofalo (Purdue University)
October 28, 1999

Clifford analysis techniques in multidimensional operator theory
Mircea Martin (Baker University)
October 14, 1999
Thursday at 3:30 p.m.

Stability of Sobolev spaces with zero boundary values
Lars Hedberg (Linkoping University, Sweden)
October 4, 1999
Monday at 3:30 p.m.

Unknotting knots: new codes and the Dynnikov algebra
Alexei Sossinsky (Independent University of Moscow)
September 23, 1999
Thursday at 3:30 p.m.

Bounded Point Evaluations and Polynomial Approximation
Jim Thomsen (Indiana University)
September 16, 1999
Thursday at 2:30 p.m.

Spring 1999

Robin Capacity
Peter Duren (University of Michigan at Ann Arbor)
April 22, 1999

Minimal Area Problems with Side Conditions
Dov Aharonov (Technion, Haifa)
April 13,1999

The role of genetic catastrophes in the origin of species: reconstruction of evolutionary history of mammals by analysis of the genetic texts
Andrei Gudkov (Professor of Biological Sciences, University of Illinois at Chicago)
April 8, 1999

The story of a pentagonal tiling and a "pentagonal" number
Ken Stephenson (University of Tennessee)
March 25, 1999

Estimating a Life Distribution Based on Ages and Ages of Departure
Mark Rothmann (Department of Statistics, University of Iowa)
March 12, 1999

Topology and Three-Dimensional Magnetic Fields: From Gauss and Maxwell to Modern Computational Electromagnetics
P. Robert Kotiuga (Boston University)
March 11, 1999

Nonparametric Bqyesian Modeling of Long Memory Time Processes
Giovanni Petris (Department of Statistics, Carnegie Mellon University)
March 8, 1999

Exact Bootstrap Moments of an L-estimator
Michael Ernst (Division of Statistics, University of Florida)
March 5, 1999

Duality in Complex Analysis
Lev Aizenberg (Bar-Ilan University, Israel)
March 4, 1999

Calculus Reform: Trickle Up and Trickle Down
Janet Woodland (University of Arkansas)
March 2, 1999

Squares of Vector Lattices
Gerard Buskes (University of Mississippi)
February 25, 1999

Boundary behavior of harmonic functions and conformal geometry of sub-Laplacians
Nicola Garofalo (Purdue University)
February 22, 1999

A Singular Perturbation Approach to a Two Phase Parabolic Free Boundary Problem Arising In Flame Prapagation
Donatella Danielli (Purdue University)
February 15, 1999

Systems of Analytic Functions that are Simultaneously Orthogonal over Two Domains
Harold S. Shapiro (Royal Institute of Technology, Stockholm)
January 28, 1999

Fall 1998

Dual Varieties, Local Differential Geometry, and a Classical Problem in Linear Algebra
Joseph Landsberg (Paul Sabatier University, Toulouse)
November 19, 1998

Stein-Weiss Operators, Spectra and Ellipticity
Tom Branson (University of Iowa)
November 12, 1998

Fluid Flows with Moving Boundary, Integrals of Motion and Algebraic Geometry
Pavel Etingof (Harvard University)
November 5, 1998

Nonlinear Singular Integral equations Involving the Hilbert Transform in Clifford Analysis
Swanhild Bernstein (University of Arkansas)
October 29, 1998

Spring 1998

Tight closure, or why characteristic p>0 can be better
Ian Aberbach (University of Missouri at Columbia)
April 23, 1998
Thursday at 2:30 p.m.

Congruences of lines in 3-space
Serge Lvovsky (Moscow Independent University, Russia)
April 21, 1998

Nevanlinna-Pick kernels
John McCarthy (Washington University)
April 9, 1998

The minmax sphere eversion
John Sullivan (University of Illinois, Champagne-Urbana)
March 27, 1998

Approximation of Cauchy type integrals by rational functions with prescribed poles
Genrik Tumarkin (University of LA)
March 25, 1998

Legendrian tangles
Lisa Traynor (Bryn Mawr College)
March 12, 1998

On regular hypercomplex elementary functions and boundary value problems
Wolfgang Sproessig (Technical University of Freiberg, Germany)
March 10, 1998
Tuesday at 4:30 p.m.

The $bar\partial$ and $bar\partial_b$ problems on nonsmooth domains
Mei-Chi Shaw (University of Notre Dame)
March 5, 1998

Construction of high order general linear methods for ordering differential equations
Zdzislaw Jackiewicz (Arizona State University)
February 12, 1998

Partial Differential equations in Carnot Caratheodory spaces
Luca Capogna (Courant Institute)
January 15, 1998

 

Fall 1997

Spectral theory and harmonic analysis
Alan McIntosh (Macquarie University, Australia)
November 21, 1997

Weil classes on abelian varieties
Yuri Zahrin (Pennsylvania State University)
Thursday, November 6, 1997
2:30 pm

Some theorems on strictly ordered but aperiodic structures in euclidean space
Ludwig Danzer (University of Dortmund, Germany)
Friday, October 24, 1997

Modeling and control issues concerning magnetostrictive materials
Ralph Smith (Iowa State University)
Friday, October 17, 1997

Hypergeometric functions and geometry of Grassmanians
Vladimir Retakh (University of Arkansas)
Thursday, October 2, 1997

Spring 1997

Interpolation by bounded analytic functions
John Wermer (Brown University)
April 17, 1997
SE 109

Derivations into Banach modules
Garth Dales (University of Leeds, U. K.)
April 16, 1997

Finite type invariants of 3-manifolds
Thang Le (SUNY at Buffalo)
April 3, 1997

Zoo of primitive Vassilev knot invariants
Serge Chmutov (Russian Academy of Sciences and Fields Institute, Canada)
February 20, 1997

Decidable Theories
Matthew Valeriote (McMaster University, Canada)
February 13, 1997

Semilinear semigroups and the KdV equation
Jerry Goldstein (University of Memphis)
January 24, 1997
Friday at 3:30 p.m.

The Favard class for a nonlinear parabolic problem
Gisele Goldstein (University of Memphis)
January 24, 1997
Friday at 12:30 p.m.

Fall 1996

On damping in elastic systems: myths, models and mathematics
John Burns (VPISU)
November 19, 1996
Tuesday at 3:30 p.m.

Hyperbolic n-manifolds via Clifford algebras
Peter Waterman (University of Northern Illinois at DeKalb)
November 14, 1996
Thursday at 4:30 p.m.

Billiard dynamics: a survey
Eugene Gutkin (University of Southern California)
November 12, 1996

Compact composition operators of some Moebius invariant Banach spaces
Maria Tjani (University of Arkansas, Fayetteville)
November 7, 1996

Singular integrals on star shaped Lipschitz surfaces and generalizations
Tao Qian (University of New England, Armidale, Australia)
October 24, 1996

Aperiodic tilings with n-fold symmetry
Ludwig Danzer (Dortmund, Germany)
October 3, 1996

Invariant subspaces of spaces of analytic functions
Dinesh Singh (University of Delhi)
October 2, 1996

On self-dual locally compact abelian groups
Karl Hofmann (Darmstadt, Germany and Tulane)
September 20, 1996

Quantitative Approximation Theory
Stephen Fisher (Northwestern University)
September 5, 1996

 

 

Spring 1996

TBA
Peter Ebenfelt (University of California at San Diego)
April 18, 1996

Hardy Spaces on Lipschitz Domains, Clifford Algebras and Compensated Compactness
Marius Mitrea (University of Minnesota, Minneapolis)
March 14, 1996

Statistical Analysis of Mixtures and the Empirical Probability Measure
Philippe Barbe (CNRS, Toulouse, France)
March 13, 1996

On the Projective Dimension of the Cauchy Fueter System and Applications to the Theory of Regular Functions in Several Quaternionic Variables
Daniele Struppa (George Mason University at Virginia)
February 22, 1996

Symbolic Powers and Cohen-Macauley Rees Algebras
Susan Morey (University of Texas at Austin)
February 15, 1996

Wavelets - The Angle between Past and Future
Sasha Volberg (Michigan State University)
Thursday, January 20, 1996