# 36th Spring Lecture Series

"Conformal Differential Geometry and its Interaction with Representation Theory." A series of five lectures by Michael Eastwood (Australian National University).

April 7-9, 2011

### Participants

Ian Robinson (Utah State University)
Robin Graham (University of Washington)
Gloria Mari-Beffa
(University of Wisconsin)
Irina Kogan
(North Carolina State University)
J.M. Landsberg
(Texas A&M University)
Colleen Robles
(Texas A&M University)
Jan Slovak
(Masaryk University, Czech Republic)
Vladimir Soucek (Charles University, Czech Republic)
Maciej Dunajskl (University of Cambridge, United Kingdom)
Igor Zelenko (Texas A&M University)

### 2011 Organizers

Andrew Raich (University of Arkansas)
John Ryan (University of Arkansas)

The University of Arkansas Spring Lecture series are conferences organized every spring by the Department of Mathematical Sciences of the University of Arkansas. Each conference is focused on a specific topic chosen among the current leading research areas in Mathematics; a principal lecturer delivers a short, five-lecture course and selects a number of specialists who are invited to give talks on subjects closely related to the topic of the conference. Short talks by young Ph.D.s and finishing graduate students are solicited to complement the conference. Each Lecture Series has grown into an ideal opportunity for specialists and young researchers to meet and exchange ideas about topics at the forefront of modern mathematics.

The Spring Lectures are usually sponsored by the NSF jointly with the University of Arkansas. The proceedings of several conferences have appeared in the series "University of Arkansas Lecture Notes in the Mathematical Sciences," published by John Wiley & Sons.

Ian Anderson (Utah State University)
Title: Symmetric Criticality and Invariant Boundary Terms in the First Variational Formula
Abstract: [PDF] I will begin this talk with some simple examples & historical remarks regarding Palais' Principle of Symmetric Criticality. This principle asserts that, under certain conditions, “symmetric critical points are critical symmetric points”. I shall give a precise formulation of this principle within the setting of the calculus of variations on jet spaces and state the known obstructions to the principle of symmetric criticality in this context. This leads the question of the existence of invariant boundary terms in the first variational formula for invariant Lagrangians. The status of this question will be reviewed.

Maciej Dunajsk (University of Cambridge, United Kingdom)
Title: Kahler metrics in conformal geometry
Abstract: [PDFLet (M, g) be Riemannian four-manifold. Does there exist a non-zero function f:M->R such that 1) f^2 g is flat?   and  2) f^2 g satisfies Einstein equations?    Most people know the answer to #1. Nobody (really) knows the full answer to #2.
In this talk I will provide the answer to a third f^2 g: is Kahler for some Kahler form?

Michael Eastwood (Australian National University, Canberra, Australia)
Title: Introduction to conformal differential geometry (Lecture 1)
Abstract: [PDF] This talk will introduce conformal differential geometry in dimensions three and higher. I'll start by discussing how conformal geometry arises (sometimes unexpectedly). Then I'll describe the "flat model" for this geometry and the surrounding mathematical landscape.

Michael Eastwood, Australian National University, Canberra, Australia
Title: Conformally invariant differential operators (Lecture 2)
Abstract: [PDF] What does "conformally invariant" actually mean? There are several closely related formulations but on the flat model (the round sphere if you will) they all agree. I'll explain how the representation theory of Lie groups can be used to classify the invariant operators on the flat model, most of which fit together into the BGG (Bernstein-Geland-Gelfand) complex.

Michael Eastwood (Australian National University, Canberra, Australia)
Title: Higher symmetries of the Laplacian (Lecture 3)
Abstract: [PDF] Which linear differential operators preserve harmonic functions? Even on Euclidean space, this is a deceptively simple question. The answer maybe expressed in terms of conformal geometry, Lie theory, and the AdS/CFT correspondence.

Michael Eastwood (Australian National University, Canberra, Australia)
Title: Twistor theory and the harmonic hull (Lecture 4)
Abstract: [PDF] Harmonic functions are real-analytic and so automatically extend as functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated conformal geometry. I shall base the constructions on a formula of Bateman from 1904.
This is joint work with Feng Xu.

Michael Eastwood (Australian National University, Canberra, Australia)
Title: The X-ray transform on projective space (Lecture 5)
Abstract: [PDF] There are several related transforms on both real and complex projective space. I'll discuss how they fit together and how twistor theory, the Penrose transform, representation theory, and the BGG complex can be used to establish their properties.
This is based on joint work with various authors including Toby Bailey, Robin Graham, and Hubert Goldschmidt.

Robin Graham (University of Washington)
Title: Ambient metrics and exceptional holonomy
Abstract: [PDF] Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of interest in recent years.  This talk will outline a construction of metrics in dimension 7 whose holonomy is the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.
This is work with Travis Willse and generalizes results of Leistner and Nurowski.

Toshihisa Kubo (Oklahoma State University)
Title: Systems of third-order invariant differential operators of Heisenberg parabolic type
Abstract: [PDF] The wave operator $\Box$ in Minkowski space $R^{3,1}$ is a classical example of a conformally invariant differential operator.  The Lie algebra $so(4,2)$ acts on $R^{3,1}$ via a multiplier representation $\sigma$. When acting on sections of an appropriate bundle over $R^{3,1}$, the elements of $so(4,2)$ are symmetries of the wave operator; that is, for $X \in so(4,2)$, we have $[\sigma(X), \Box]=C(X)\Box$ with $C(X)$ a smooth function on $R^{3,1}$.  The notion of conformal invariance of operators was generalized by Konstant in the 1970's. Recently, Barchini, Kable, and Zierau generalize this notion of conformal invariance for systems of differential operators.
In this talk we discuss examples of systems of $k$-th order differential operators. We present in detail a construction of a system of third order differential operators invariant under $sl(3,C)$.

J.M. Landsberg (Texas A&M University)
Title: A surprising connection between a question in quantum mechanics and complexity theory
Abstract: [PDF] I will describe joint work with my students Yang Qi and Ke Ye on tensor network states. Tensor network states are an attempt to describe "feasible" states of quantum mechanical systems. I will discuss a question posed to me regarding tensor network states that amazingly is related both to the complexity of matrix multiplication and the GCT program of Mulmuley and Sohoni to prove a variant of P \neq NP.

Junxia Li (University of Arkansas)
Title: Some Rarita-Schwinger Type Operators
Abstract: [PDF] In this paper we study a generalization of the classical Rarita-Schwinger type operators and construct their fundamental solutions. We give some basic integral formulas related to these operators. We also establish that the projection operators appearing in the Rarita-Schwinger operators and the Rarita-Schwinger equations are conformally invariant. We further obtain the intertwining operators for other operators related to the Rarita-Schwinger operators under actions of the conformal group. This is joint work with Charles Dunkl, John Ryan and Peter Van Lancker.

Gloria Mari-Beffa (University of Wisconsin)
Title: Geometric realizations of KdV equations in semisimple parabolic manifolds
Abstract: [PDF] In this talk we will describe how solutions of systems of KdV equations (flows that model waves in shallow water, for example) can be realized as flows of curves in semisimple parabolic manifolds with a |1|-gradation, including the Mobius conformal sphere, the manifold of pure spinors, etc.  We will describe the spinor case in detail.

Marian Munteanu (Michigan State University)
Title: Canonical directions on surfaces in M2(c) x R
Abstract: [PDF] The geometry of surfaces in 3-dimensional spaces, especially of the form M x R, has enriched in last years. The most interesting situations occur when M is a surface of constant Gaussian curvature and hence a lot of classification results are obtained. It is proved that for a constant angle surface in E3, S2 x R or in H2 x R, the projection of R(t) onto the tangent plane of the immersed surface, denoted by T, is a principal direction with the corresponding principal curvature identically zero. The main topic of the present talk is to investigate surfaces in H2 x R for which T is a principal direction connecting these results with the previous ones concerning constant angle surfaces.

Ed Pegg (Wolfram Research)
Title: Instant Fame [Public Lecture]
Abstract: Using 2011-era Computers on Old Problems: Over the last few years, puzzlers, mathematicians, computer scientists and hobbyists have been revisiting old puzzles and problems, exploring solutions that would have been impossible to discover just a few years ago. With new tools, anyone can join in the fun at home! The talk will focus on a variety of computational problems and methods.

Colleen Robles (Texas A&M University)
Title: Homological rigidity of Schubert varieties in compact Hermitian symmetric spaces
Abstract: [PDF] The integral homology of a compact Hermitian symmetric spaces (CHSS) is generated by the homology classes of its Schubert varieties.  Most Schubert varieties are singular.  In 1961 Borel & Haefliger asked: when can the homology class [X] of a singular Schubert variety be represented by a smooth subvariety Y of the CHSS? Remarkably, the subvarieties Y with [Y] = [X] are integrals of a differential system.  I will discuss recent work with Dennis The in which we give a complete list of those Schubert varieties X for which there exists a first-order obstruction to the existence of a smooth Y.

Title: Orthogonally separable coordinates on the 3-sphere
Abstract: [PDF] Integrable Killing tensors are used for the classification of coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables.  Based on representation theoretic methods we establish purely algebraic equations characterizing the integrability of Killing tensors on constant curvature manifolds and solve these explicitly on the 3-sphere.  We use this to obtain the algebraic variety of integrable Killing tensors and to classify all separable coordinate webs.  Several generalizations of this result will be proposed.

Jan Slovak (Masaryk University, Czech Republic)
Title: Fefferman construction related to free CR distributions
Abstract: [PDF] There are only few exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be $PSU(n+1,n)/P$ with a suitable maximal parabolic subgroup $P$. While the case $n=1$ reproduces the classical case of 3-dimensional real hypersurfaces in $\Bbb C^2$,  for all bigger $n$ the corresponding generic $(2n+n^2)$--dimensional submanifolds in $\mathbb C^{n+n^2}$ enjoy the fundamental invariant is of torsion type, which allows for explicit computation.
The presentation will focus on the Fefferman construction of a circle bundle with a $|1|$-graded parabolic geometry, completely analogously to the hypersurface type CR geometry.

Abraham Smith (McGill University, Montréal)
Title: Intrinsic Geometry on Second-Order PDEs
Abstract: [PDF] I will introduce an intrinsic geometry on second-order scalar PDEs that provides a clear invariant definition of integrability.  Depending on one's preferred perspective, the intrinsic geometry is a conformal spin geometry, a G-structure, a conical structure, or a parabolic geometry. It allows a local classification of second-order scalar PDEs that is somewhat coarse but of finite type.

Petr Somberg (Charles University, Czech Republic)
Title: Ambient construction and finite reflection groups
Abstract: [PDF] I will describe one variation on the theme of ambient construction, based on finite reflection symmetry group as a subgroup of the conformal group.

Vladimir Soucek (Charles University, Czech Republic)
Title: A new normalization of tractor covariant derivatives and its applications
Abstract: [PDF] It is possible to introduce a new normalization of the tractor covariant derivatives on a tractor bundle V, which is adapted to the bundle V. As an application, we shall show how to construct the prolongation of the first operator  in the curved BGG sequences for a general parabolic geometry.  The result will be demonstrated on various examples for  basic parabolic geometries. The result is based on standard techniques of the BGG machinery.

Jingzhi Tie (University of Georgia)
Title: The sub-Laplacian comparison theorem in a complete pseudo-Hermittian 3-manifold
Abstract: [PDF] This is the joint work with Jianguo Cao (Notre Dame) and Shu-Cheng Chang (Taiwan). We prove the sub-Laplacian comparison theorem and obtain a CR diameter estimate and volume growth estimate in a complete pseudo-Hermittian 3-manifold.

Carmen Judith Vanegas (Universidad Simón Bolívar, Caracas, Venezuela)
Title: Rarita-Schwinger Operators on Spheres
Abstract: [PDF] Using similar methods to define the Rarita-Schwinger operators in $\mathbf{R}^n$, we can define the spherical Rarita-Schwinger operator on the sphere based on the spherical Dirac operator. Then we construct their fundamental solutions and establish that the projection operators appearing in the spherical Rarita-Schwinger operators and the spherical Rarita-Schwinger equations are conformally invariant under the Cayley transformation. Further, we obtain some basic integral formulas related to the spherical Rarita-Schwinger operators.

Travis Willse (University of Washington)
Title: Parallel tractor extension
Abstract: [PDF] Motivated by a problem relating Nurowski's conformal structures to examples of metrics with exceptional holonomy $G_{2(2)}$, I have shown that parallel conformal tractor tensors can be suitably extended to the Fefferman-Graham ambient manifold:  On odd-dimensional manifolds, a parallel tractor tensor can be extended to a tensor on the ambient manifold parallel to infinite order, and this extension is unique to infinite order.  On even-dimensional manifolds, one is guaranteed an extension parallel to order $(n - 2) / 2$, but extension to higher order is generally obstructed; if they exist, extensions parallel to higher order are unique to infinite order under some conditions.
This is joint work with R. Graham, and is a sequel to his talk.

Igor Zelenko (Texas A&M University)
Title: On geometry and symmetries of nonholonomoic distributions and curves of flags
Abstract: I will discuss the following two problems in local differential geometry and the interplay between them: the equivalence of vector distributions (w.r.t. the action of the group of diffeomorphisms) and the equivalence of curves of special flags in a linear space w.r.t. the action of a subgroup $G$ of the General Linear Group with fixed grading on its Lie algebra $g$. For the first problem the procedure of construction of canonical frame (coframe) was described by Tanaka in terms of the algebraic prolongation of the symbol of a distribution. The second problem can be treated in a similar way via the notion of a symbol of a curve of flags (which is a degree -1 element of $g$) and an appropriate notion of its algebraic prolongation in $g$ (as a generalization of Se-ashi works).
The main point of the talk is that the first problem can be reduced in essence to the second one (with the Group $G$ being the Linear Symplectic Group and flags being flags of isotropic/coisotropic subspaces) via so-called symplectification procedure taking its origin in Optimal Control Theory. In this way we introduce the notion of the Jacobi symbol of a distribution, which is a degree -1 element of a graded symplectic algebra and we describe the procedure of construction of canonical frame (coframe) for all distributions with the given Jacobi symbol in terms of natural algebraic operations with the Jacobi symbol in the category of graded Lie algebras. The main advantages of using Jacobi symbols of distributions compared to the classical Tanaka symbols are that the set of Jacobi symbols is discrete and all Jacobi symbols can be easily classified. Besides, the Jacobi symbol of a distribution is much coarser characteristic than its Tanaka symbols: distributions with different Tanaka symbols and even with different small growth vectors may have the same Jacobi symbol. The talk is based on the joint works with Boris Doubrov.

SLS 2011 Video Archive Playlist

Michael Eastwood (Australian National University)
"Introduction to Conformal Differential Geometry (lecture 1)"
SLS 2011 - 01

Ian Anderson (Utah State University)
"Symmetric Criticality and Invariant Boundary Terms in the First Variational Formula"

SLS 2011 - 02

Junxia Li (University of Arkansas)
"Some Rarita-Schwinger Type Operators"
SLS 2011 - 03

Carmen Judith Vanegas (Universidad Simón Bolívar, Caracas, Venezuela)
"Rarita-Schwinger Type Operators on Spheres"
SLS 2011 - 04

Michael Eastwood (Australian National University, Canberra, Australia)
"Conformally Invariant Differential Operators (lecture 2)"
SLS 2011 - 05

Robin Graham (University of Washington)
"Ambient Metrics and Exceptional Holonomy"
SLS 2011 - 06

Travis Willse (University of Washington)
"Parallel Tractor Extension"
SLS 2011 - 07

"Intergrable Killing Tensors on 3-Sphere"
SLS 2011 - 08

Ed Pegg, Jr. (University of Colorado, Boulder)
Math Games
SLS 2011 - 09

Michael Eastwood (Australian National University)
"Higher Symmetries of the Laplacian (lecture 3)"
SLS 2011 - 10

Maciej Danajski (University of Cambridge, United Kingdom)
"Kahler metrics in conformal geometry"
SLS 2011 - 11

"A new normalization of tractor covariant derivatives and its applications"
SLS 2011 - 12

Michael Eastwood (Australian National University)
"Twistor theory and the harmonic hull (lecture 4)"
SLS 2011 - 13

Petr Somberg (Charles University, Czech Republic)
"Ambient construction and finite reflection groups"
SLS 2011 - 14

J.M. Landsberg (Texas A&M University)
"A surprising connection between a question in quantum mechanics and complexity theory"
SLS 2011 - 15

Gloria Mari-Beffa (University of Wisconsin)
"Geometric realizations of KdV equations in semisimple parabolic manifolds"
SLS 2011 - 16

Michael Eastwood (Australian National University)
"The X-ray transform on projective space (lecture 5)"
SLS 2011 - 17

Colleen Robles (Texas A&M University)
"Homological rigidity of Schubert varieties in compact Hermitian symmetric spaces"
SLS 2011 - 18

Jan Slovak (Masaryk University, Czech Republic)
"Fefferman construction related to free CR distributions"
SLS 2011 - 19

Igor Zelenko (Texas A&M University)
"On geometry and symmetries of nonholonomoic distributions and curves of flags"
SLS 2011 - 20

Marian Munteanu (Michigan State University)
"Canonical directions on surfaces in M2(c) x R"
SLS 2011 - 21

Abraham Smith (McGill University, Montréal)
"Intrinsic Geometry on Second-Order PDEs"
SLS 2011 - 22

Jingzhi Tie (University of Georgia)
"The sub-Laplacian comparison theorem in a complete pseudo-Hermittian 3-manifold"
SLS 2011 - 23

Toshihisa Kubo (Oklahoma State University)
"Systems of third-order invariant differential operators of Heisenberg parabolic type"
SLS 2011 - 24