Benson Farb (University of Chicago)
Title: Cohomology, Polynomials, and Representations: an Eternal Golden Braid

Abstract: My goal in these talks is to explain some fascinating interconnections among the four (yes, four) mathematical topics in the title. The story will take us from spaces of configurations of points on a manifold, to Arnold's topological proof that the quintic can't be solved by radicals, to representation theory of the symmetric group, to cohomology theory, to congruence subgroups, to counting points on varieties over finite fields, and beyond. Most of the material in these lectures should be understandable to an undergraduate math major.

Ravi Vakil (Stanford University)
Title: The Mathematics of Doodling (Public Lecture)

Abstract: Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today.

Tara Brendle (University of Glasgow)
Title: Combinatorial models for mapping class groups

Abstract: In 1997 Ivanov proved that the automorphism group of the complex of curves associated to a surface S is isomorphic to the extended mapping class group Mod(S), for most surfaces. His work generated a flurry of activity, with similar results obtained by several different authors for various other simplicial complexes associated to surfaces. Ivanov then posed a "metaconjecture" stating that every "sufficiently rich" complex associated to a surface S has Mod(S) as its group of automorphisms. In this talk we will discuss a resolution of Ivanov's metaconjecture for a wide class of complexes. This is joint work with Dan Margalit.

Allen Hatcher (Cornell University)
Title: Rational Homology of Automorphism Groups of Free Groups

Abstract: We describe geometric constructions for the known nontrivial rational homology classes in the automorphism and outer automorphism groups of free groups, as well as for many candidates for further nontrivial classes. This is part of a general approach to understanding the homology of these automorphism groups by assembling homology classes coming from small rank free groups into classes for larger ranks. This is from joint work with Jim Conant, Martin Kassabov, and Karen Vogtmann.

Eriko Hironaka (Florida State University)
Title: Dual Digraphs and Entropy

Abstract: A 2-complex with semi-flow has a corresponding dual directed graph. In this talk, I will explain how to obtain from this information a polynomial invariant that is independent of homotopy equivalences preserving the semi-flow. Under certain conditions, we show that this is analogous to McMullen's Teichmueller polynomial, and computes the mondromy entropies of maps from the fundamental group of the complex to the infinite cyclic group. This work is joint with Yael Algom-Kfir and Kasra Rafi.

Alexander Kupers (Stanford University)
Title: En-cells and their applications

Abstract: When studying objects with additional algebraic structure, e.g. algebras over an operad, it can be helpful to consider cell decompositions adapted to these algebraic structures. I will talk about joint work with Jeremy Miller on the relationship between En-cells and homological stability. Using this theory, we prove a local-to-global principle for homological stability, as well as give a new perspective on homological stability for various spaces including symmetric products and spaces of holomorphic maps.

Dan Margalit (Georgia Institute of Technology)
Title: Algebraic, geometric, and dynamical aspects of surfaces

Abstract: To each homeomorphism of a surface we can associate a real number - called the entropy - which encodes the amount of mixing being effected. This number can be studied from topological, geometrical, dynamical, analytical, and algebraic viewpoints. We will start by explaining Thurston's beautiful insight for computing the optimal entropy within a homotopy class and explain a new, fast algorithm based on his ideas. Then we will discuss some classical results and recent progress on the problem of understanding phenomena related to small entropy. One theme is that algebraic complexity and geometric complexity both imply dynamical complexity. This work is joint with Ian Agol, Benson Farb, Chris Leininger, and Oyku Yurttas.

Jeremy Miller (Stanford University)
Title: A generalization of non-abelian Poincaré

Abstract: Salvatore and Lurie's non-abelian Poincaré duality theorem equates the topological chiral homology of a group-like En-algebra with a space of sections of a certain bundle. I will describe how to generalize non-abelian Poincaré duality to the case of En-algebras which are not group-like. While non-abelian Poincaré duality for group-like En-algebras can be viewed as the minimal theorem simultaneously generalizing Poincaré duality and May’s recognition principle for iterated loop spaces, this generalization can be viewed as the minimal generalization of the group-completion theorem and McDuff’s scanning result for configuration spaces of unordered particles in an open manifold.

Rohit Nagpal (University of Wisconsin)
Title: FI-modules and the cohomology of modular Sn representations

Abstract: We will talk about the category of finitely generated FI-modules which was introduced by Church, Ellenberg and Farb to study stability patterns in the representation theory of the symmetric groups Sn. An FI-module V over a commutative ring k encodes a sequence (Vn)n≥0 of representations of the symmetric groups (Sn)n≥0 over k. We outline a proof of the periodicity of the cohomology groups Ht(Sn, Vn) when k is a field of characteristic p and V is a finitely generated FI-module over k. As a direct application, we show that if M is compact, connected, oriented manifold of dimension ≥ 2 and confn(M) is the configuration space of unordered n-tuples of distinct points in M then the dimensions of the mod-p cohomology groups Ht(confn(M), k) are eventually periodic in n with period a power of p. We point out some open problems and other recent works in this direction.

Andrew Putman (Rice University)
Title: Integrality in the Steinberg module and the top-dimensional cohomology of SLn(O)

Abstract: I will discuss the relationship between the arithmetic of the ring of integers in an algebraic number field O and the high-dimensional cohomology of SLn(O). This is joint work with Thomas Church and Benson Farb.

Vic Reiner (University of Minnesota)
Title: Sharp representation stability for configurations of points in d-space (joint work with Patricia Hersh)

Abstract: T. Church proved a representation theory stability result for the cohomology of the configuration space of n distinct labelled points in a connected, oriented d-manifold having finite-dimensional cohomology. He showed that the symmetric group representation on the i-th cohomology of such a configuration space eventually stabilizes in a precise sense, and that the stabilization will occur for n beyond 4i when d=2, and for n beyond 2i when d is at least 3.

In the very special case when the manifold is d-dimensional Euclidean space, we show that the stabilization starts earlier, sharply at n=3*i/(d-1) for odd d, and at n = 1 + 3*i/(d-1) for d even. We suspect this is the earliest that stabilization for d-manifolds.

The crux of the analysis is a reasonably explicit description of the relevant symmetric group representations due to S. Sundaram and V. Welker. In addition, we prove some conjectures on these representations due to J. Wiltshire-Gordon.

Ravi Vakil (Stanford University)
Title: Cutting and pasting in algebraic geometry

Abstract:
Given some class of "geometric spaces", we can make a ring as follows.

1. (additive structure) When U is an open subset of such a space X, [X] = [U] + [(X \ U)];

2. (multiplicative structure) [X × Y] = [X] [Y]

In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising “stabilization” structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.

Contributed Talks

David Cohen (Rice University)
Title: The large scale geometry of strongly aperiodic subshifts of finite type

Abstract: Given a group G and a finite set A, a subshift of finite type consists of all functions from G to A which avoid some finite set of forbidden patterns. A subshift of finite type is said to be strongly aperiodic if each element has trivial stabilizer. We discuss the problem of which groups admit strongly aperiodic subshifts of finite type, and show that this property is closely linked to geometric group theory. In particular, groups with at least two ends have no strongly aperiodic subshifts of finite type, and having such a subshift is a QI-invariant for torsion free finitely presented groups.

Sean Howe (University of Chicago)
Title: Point counting, stability, and the complement of the discriminant locus

Abstract: A result of Poonen implies that the number of rational points on a random smooth degree d hypersurface in Pn over a finite field approaches a binomial random variable as d goes to infinity. We explain how to translate this result into conjectures on the stable cohomology of the complement of the discriminant locus with twisted coefficients. In some cases the conjectures are known -- for n=1 they are due to results of Church, Ellenberg, and Farb on representation stability of configuration spaces, and for general n and constant coefficients they are due to results of Tommasi on the stable cohomology of the complement of the discriminant locus -- but in other cases they are wide open! Time permitting we will also describe connections with the stable cohomology of moduli of curves.

Rita Jiménez Rolland (Universidad Nacional Autónoma de México)
Title: The cohomology of Mg,n as an FI-module

Abstract: In this talk we consider the action of symmetric group Sn on the i-th rational cohomology group of Mg,n, the moduli space of genus g Riemann surfaces with n marked points. We discuss how this sequence of Sn-representations has the underlying structure of a finitely generated FI-module and ``stabilizes'' in a certain sense once n is large enough. As a consequence we obtain that, for n large enough with respect to i, the i-th Betti number of Mg,n is a polynomial in n of degree at most 2i.

Balazs Strenner (University of Wisconsin)
Title: Algebraic degrees of pseudo-Anosov mapping classes

Abstract: Thurston showed that the algebraic degrees of stretch factors of pseudo-Anosov maps on the orientable surface of genus g are bounded by 6g-6. He claimed that his Dehn twist construction produces examples of maximal 6g-6 degree stretch factors, but he did not give a proof. We will discuss a construction based on Penner’s construction of pseudo-Anosov maps that produces many of maximal degree examples. The same construction can also be used to generate stretch factors of all even degrees less than 6g-6.

Jenny Wilson (Stanford University)
Title: Representation stability and classical Weyl groups

Abstract: Over the past several years Church, Ellenberg, Farb, Nagpal, and others have developed machinery for studying sequences of representations of the symmetric groups, using a concept they call an FI--module. I will describe how their work generalizes to sequences of representations of the Weyl groups in type B/C and D. I will outline some applications, including stability results for the cohomology of certain families of groups and spaces related to the (pure) braid groups.

Jesse Wolfson (University of Chicago)
Title: Spaces of Rational Functions, Representation Stability, and Statistics over Finite Fields

Abstract: In the late '70s, Graeme Segal showed that the homology of the space of rational functions of fixed degree on a Riemann surface stabilizes as the degree tends to infinity. In ongoing joint work with Benson Farb, we revisit these spaces through the lens of representation stability, and use this make computations of their rational cohomology. These computations carry implications for point counts and related statistics for rational functions of curves over finite fields. In this talk, I'll aim to give background and context for our investigation, and describe some of the phenomena we are encountering.

SLS 2015 Video Archive Playlist

Benson Farb
"Cohomology, Polynomials, and Representations: an Eternal Golden Braid (Lecture 3)"
SLS 2015 - 01

Benson Farb
"Cohomology, Polynomials, and Representations: an Eternal Golden Braid (Lecture 4)"
SLS 2015 - 02

Benson Farb
"Cohomology, Polynomials, and Representations: an Eternal Golden Braid (Lecture 5)"
SLS 2015 - 03

Tara Brendle
"Combinatorial models for mapping class groups"
SLS 2015 - 04

Allen Hatcher
"Rational Homology of Automorphism Groups of Free Groups"
SLS 2015 - 05

Eriko Hironaka
"Dual Digraphs and Entropy"
SLS 2015 - 06

Alexander Kupers
"En-cells and their applications"
SLS 2015 - 07

Dan Margalit
"Algebraic, geometric, and dynamical aspects of surfaces"
SLS 2015 - 08

Jeremy Miller
"A generalization of non-abelian Poincaré"
SLS 2015 - 09

Andrew Putman
"Integrality in the Steinberg module and the top-dimensional cohomology of SLn(O)"
SLS 2015 - 10

David Cohen
"The large scale geometry of strongly aperiodic subshifts of finite type"
SLS 2015 - 11

Balazs Strenner
"Algebraic degrees of pseudo-Anosov mapping classes"
SLS 2015 - 12