# Pure mathematics seminar 2019 Semester 1

The Pure Mathematics Seminar meets Fridays 3:15pm-4:15pm in Peter Hall Room 213.

Organisers: Jesse Gell-Redman, Ting Xue

### Schedule

June 21 Julia Pevtsova (U Washington)

Title: Tensor triangular geometry for modular representation theory

Abstract: Modular representation theory studies representations of a finite group and other algebraic structures such as Lie algebras over a field of positive characteristic. Classifying modular representation up to direct sums – as the classical theory does for complex representations – is often a hopeless task even for such a tiny group as $\mathbb{Z}/3 \times \mathbb{Z}/3$.  I’ll discuss a geometric approach to understanding this wild territory starting with D. Quillen’s classical work on mod p group cohomology and leading to the applications of the ideas of tensor triangular geometry of P. Balmer to modular representation theory.

Joint work with D. Benson, S. Iyengar and H. Krause.

May 24 Martin Taylor (Princeton)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: The Schwarzschild family, discovered in 1915, is the most famous family of solutions of the vacuum Einstein equations of general relativity.  Each member describes a static black hole.  The most basic question one can ask about the family is whether the black hole exterior is nonlinearly stable as a solution of the vacuum Einstein equations.  I will present a theorem on the full finite codimension asymptotic stability of the Schwarzschild family.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of Schwarzschild.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

May 17 Anna Romanov (Sydney)

Title: A Kazhdan–Lusztig algorithm for Whittaker modules

Abstract: Whittaker modules are certain representations of Lie algebras which were first studied by Kostant for their relationship to the Whittaker equation in number theory. From a representation-theoretic viewpoint, Whittaker modules are interesting in their own right, as infinite-dimensional representations with enough imposed structure to make them tractable to study. Milicic and Soergel placed Kostant’s Whittaker modules in a larger category N, and McDowell showed that all irreducible objects in N appear as quotients of certain standard Whittaker modules, which generalize Verma modules. There is also a geometric incarnation of the category N in terms of twisted D-modules on the associated flag variety. Using this geometric description, one can develop an algorithm for computing composition multiplicities of standard Whittaker modules. In this talk, I’ll describe the parallel algebraic and geometric worlds which contribute to this story, then sketch the algorithm which answers the multiplicity question in category N.

May 10 Gwyn Bellamy (University of Glasgow)

Title: The McKay correspondence in higher dimensions

Abstract: The McKay correspondence is a surprising, and beautiful, bijection (defined geometrically by du Val, and algebraically by McKay) between the isomorphism classes of Kleinian surfaces singularities and the simply laced Dynkin diagrams.  I will recall this bijection and explain how it can be used to describe the Kleinian singularities, and their resolution of singularities, as moduli spaces associated to a certain non-commutative algebra. In the final part of the talk, I will describe how this construction can be extended to describe quotient singularities in higher dimensions as moduli spaces. The talk will be totally non-technical. It is based on joint work with Alastair Craw.

May 3 Volker Schlue (Melbourne)

TitleGlobal evolution problems in General Relativity

AbstractGeneral relativity is a theory of gravity that relates the geometry of space and time to the distribution of matter. In fact, even in the absence of matter it can be viewed as a theory for the evolution of the geometry of space in time, and poses a number of challenging problems in the theory of hyperbolic partial differential equations. In this talk, I will elaborate on two global evolution problems for Einstein’s equations: The first is in the regime of a space-time geometry globally close to the flat Minkowski space-time, and can be recast as a scattering problem for nonlinear wave equations (joint with Hans Lindblad). The second relates to models of the expanding universe, and I will describe some of my work on the stability of black hole cosmologies.

Special time and location: 2-3pm at 107 Peter Hall

Title: Topological insulators with non-commutative symmetry groups

Abstract: A recent breakthrough in condensed matter physics was the discovery of so-called topological insulators. These are materials for which a topological non-triviality in their mathematical description forces them to behave “non-local” in a certain sense. We model this by a Riemannian manifold carrying a cocompact action of a discrete sym-metry group G, together with a G-invariant Hamiltonian operator. The question is then whether a certain spectral subspace of $L^2(X)$ has a G-basis of rapidly decaying functions, called “Wannier functions”. We show that this is equivalent to the (non-)triviality of the spectral subspace, when considered as a Hilbert module over thegroup $C^*$-algebra $C^*_r(G)$. This is joint work with Guo Chuan Thiang.

Apr 12 Todd Oliynyk (Monash)

Title: Cosmological Newtonian limits on large spacetime scales

Abstract: Galaxies and clusters of galaxies are prime examples of large scale structures in our universe. Their formation requires non-linear interactions and cannot be analyzed using perturbation theory alone. Currently, cosmological Newtonian N-body simulations are the only well developed tool for studying structure formation. However, the Universe is fundamentally relativistic, and so the use of Newtonian simulations must be carefully justified. This leads naturally to the question: On what scales can Newtonian cosmological simulations be trusted to approximate realistic relativistic cosmologies? In this talk, I will describe recent work done in collaboration with Chao Liu in which we provide a rigorous answer to this question by establishing the existence of 1-parameter families of $\epsilon$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $\Lambda >0$ and a linear equation of state $p=\epsilon^2 K \rho$$0, for the parameter values $0<\epsilon\leq \epsilon_0$. These solutions exist globally on the manifold $M=[0,\infty) \times \mathbb{R}^3$, are future complete, and converge as $\epsilon \searrow 0$ to solutions of the cosmological Poisson-Euler equations. As I shall describe in the talk, these solutions represent inhomogeneous, nonlinear perturbations of a homogeneous fluid filled universe where the inhomogeneities are driven by localized matter fluctuations that evolve to good approximation according to Newtonian gravity. Time permitting, I will briefly discuss a number of new results in hyperbolic partial differential equations that were used to establish the global existence results described in the talk.

Apr 5 Colin Guillarmou (Orsay)

Title: On Liouville quantum field theory on Riemann surfaces.

Abstract: We will discuss some joint work with Rhodes and Vargas on defining a 2 dimensional conformal field theory through path integral (via probabilistic methods) on Riemann surfaces of genus g>1, and we describe the behaviour of the partition function on the moduli space of Riemann surfaces using analysis and hyperbolic geometry (Teichmüller theory). This allows to show the convergence of Polyakov partition function which appeared in 2d gravity.

Apr 4 Marco Maxxucchelli (ENS Lyon)

Special time and location: 2-3pm at 107 Peter Hall

Title: Min-Max Characterzations of Zoll Riemannian manifolds

Abstract: A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all periodic with the same minimal period. This class of manifolds has been thoroughly studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors. It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and only if its unit-speed periodic geodesics all have the same minimal period. In this talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds on the work of Lusternik and Schnirelmann. I will then show an analogous result for certain higher dimensional closed manifolds, including spheres, complex and quaternionic projective spaces: a Riemannian manifold is Zoll if and only if two suitable min-max values in a free loop space coincide. This is based on joint work with Stefan Suhr.

Title: The parabolic geometry of a flying saucer

Abstract: The motion of a flying saucer is restricted by the three-dimensional geometry of the space in which it moves. In this way, various parabolic geometries and Lie algebras emerge from thin air. I shall discuss the geometry of the particular thin air needed so that Engel’s 1893 construction of the exceptional Lie algebra $G_2$ emerges. This is joint work with Pawel Nurowski.

Mar 22 Alexander Dunn (UIUC)

Title: Maass forms and Ramanujan’s third order mock theta function

Abstract: In 1964, George Andrews proved an asymptotic formula (finite sum of terms) involving generalized Kloosterman sums and the $I$-Bessel function for the coefficients of Ramanujan’s famous third order mock theta function. Andrews conjectured that these series converge when extended to infinity, and that it they do not converge absolutely. Bringmann and Ono proved the first of these conjectures in 2006. Here we obtain a power savings bound for the error in Andrews’ formula, and we also prove the second of these conjectures. )

Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on a new estimate which we derive for the Fourier coefficients of such forms which gives a power savings in the spectral parameter as compared to results of Duke and Baruch-Mao.

Title: Stacks, Gerbes, and T-duality

Abstract: I will explain the role of stacks and descent in the theory of principal bundles and bundle gerbes, and talk about various examples and features. I will then describe a new application of non-abelian gerbes to T-duality, in which their stacky nature is of crucial importance.

Mar 1 James Tener (ANU)

Special time and location: 1-2pm at Evan Williams Theatre

Title: Geometric conformal field theory: what, why, and how

Abstract: Since the early days of quantum field theory, mathematicians have worked to find a system axioms which describe the phenomena described by physicists. One of the most successful pieces of this endeavor was Atiyah’s 1988 axiomatization of quantum field theories with topological symmetry via cobordism categories, which initiated a field of research that remains extremely active to this day. Less well known, however, is an earlier definition of Segal for a different flavor of quantum field theories, called conformal field theories, which inspired Atiyah’s definition. The goal of this talk is to introduce and motivate Segal’s definition of a conformal field theory, and to describe how it fits into the broader mathematical landscape. I will also discuss some of the challenges which arose during its development, along with recent and ongoing work to address them.

Feb 22, 2-3pm, Emily Norton (Bonn)

Special location: Evan Williams Theatre

Title: Crystals, Paths, and BGG Resolutions

Abstract: I will explain how the combinatorics of (1) crystals, and (2) paths in alcove geometries, arises in representation theory of the rational Cherednik algebra of the symmetric group. Roughly speaking, (1) describes branching rules, while (2) provides a way to tell when there is a homomorphism between Verma modules (on a certain poset of partitions). Combining these two types of combinatorics, we prove that the character formula of a simple unitary module L of the Cherednik algebra is categorified, so to speak, in the following way: L has a resolution whose terms are direct sums of Verma modules (a “BGG resolution”). These resolutions were conjectured by Berkesch-Griffeth-Sam. This is joint work with Chris Bowman and José Simental.

3:15-4:15pm Soren Galatius (Copenhagen)

Special location: Evan Williams Theatre

Title: New patterns in the cohomology of moduli spaces of curves

Abstract: Complex curves (a.k.a. Riemann surfaces) of positive genus are not rigid objects, but may be deformed and vary in families.  The space $M_g$ is a (3g-3)-dimensional complex variety, whose points are in bijection with isomorphism classes of compact complex curves of genus g.  My talk will survey some old and new patterns in its cohomology, or, equivalently, in the cohomology of the corresponding mapping class group.  Time permitting, I will discuss recent joint work with Kupers and Randal-Williams (arXiv:1805.07187), and with Chan and Payne (arXiv:1805.10186).

Feb 8 Peter Scott (U Michigan)

Title: A relative algebraic torus theorem

Abstract: In joint work with Vincent Guirardel and Gadde Swarup, we have proved a relative version of the algebraic torus theorem of Dunwoody and Swenson. I will explain what the Dunwoody-Swenson result is, after first discussing the necessary background. Only very basic topological knowledge will be needed. Then I will explain the idea of our relative result and why it is a natural extension. There will be essentially no proofs!

Feb 1, 2-3pm, Paul Smith (University of Washington, Seattle)

Title: Elliptic Algebras

3:15-4:15pm, Christoph Boehm (Muenster)

Title: Homogeneous Einstein Metrics on Euclidean Spaces are Einstein Solvmanifolds