Published on Department of Mathematics (https://math.yale.edu)

Week of September 20, 2020

September 21, 2020
Applied Mathematics	Generation by Decomposition for Handwritten Text and 3D Shapes	Hadar Averbuch-Elor - Cornell	2:30pm - https://us02web.zoom.us/j/89627764901
Group Actions and Dynamics	Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps	Wenyu Pan - University of Chicago	4:00pm - Zoom
Geometry, Symmetry and Physics	Higgs bundle metrics, QFT and higher symmetries	Andrew Neitzke - Yale	4:30pm - https://yale.zoom.us/j/99433355937 (password was emailed by Ivan on 9/11, also available from Andy by email)

September 22, 2020
Geometry & Topology	Classical and quantum traces coming from $SL_n(C)$ and $U_q(sl_n)$	Daniel Douglas - Yale	4:00pm - https://yale.zoom.us/j/96501374645

September 23, 2020
Colloquium	Modular representations of semisimple Lie algebras	Ivan Loseu - Yale University	4:15pm -

September 24, 2020
Analysis	A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations	Jacob Bedrossian - Univ. of Maryland	4:15pm -

September 25, 2020
Clusters and Geometry	Kasteleyn operators from mirror symmetry	Harold Williams - UCDavis	1:00pm -

Abstracts

Week of September 20, 2020

September 21, 2020
Applied Mathematics	Generation by Decomposition for Handwritten Text and 3D Shapes	2:30pm - https://us02web.zoom.us/j/89627764901	Abstract: Deep learning has revolutionized our ability to generate novel images and 3D shapes. Neural networks are typically trained to map a high-dimensional latent code to full realistic samples. In this talk, I will present two recent works focusing on generation of handwritten text and 3D shapes. In these works, we take a different approach and generate image and shape samples using a more granular part-based decomposition, demonstrating that the whole is not necessarily “greater than the sum of its parts”. I will also discuss how our generation by decomposition approach allows for a semantic manipulation of 3D shapes and improved handwritten text recognition performance.
Group Actions and Dynamics	Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps	4:00pm - Zoom	Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. We establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Previously, such results were proved by Naud, Stoyanov and Sarkar-Winter for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for $\Gamma$ with large critical exponent. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$. This is a joint work with Jialun Li.
Geometry, Symmetry and Physics	Higgs bundle metrics, QFT and higher symmetries	4:30pm - https://yale.zoom.us/j/99433355937 (password was emailed by Ivan on 9/11, also available from Andy by email)	I will briefly review a conjectural picture of the hyperkahler metrics on moduli spaces of Higgs bundles which arose from quantum field theory, and then describe some recent progress toward proving these conjectures; this is work of various people including Mazzeo-Swoboda-Weiss-Witt, Fredrickson, Dumas-N, Fredrickson-N. A recent surprise is that for Higgs bundles of rank > 2, one needs to use “higher symmetries” which generalize the covariance of Higgs bundles under holomorphic vector fields.

September 22, 2020
Geometry & Topology	Classical and quantum traces coming from $SL_n(C)$ and $U_q(sl_n)$	4:00pm - https://yale.zoom.us/j/96501374645	Let S be a punctured surface. The $SL_2$-skein algebra of S is a non-commutative algebra, whose elements are represented by knots K in the thickened surface S x [0,1] subject to certain relations. The skein algebra is a quantum deformation of the $SL_2(C)$-character variety of S, depending on a deformation parameter q. Bonahon and Wong constructed an injective algebra map--called the quantum trace map--from the skein algebra of S into a simpler non-commutative algebra, the latter of which can be thought of as a quantum Teichmüller space of S. This mapping associates to a knot K in S x [0,1] a Laurent polynomial in q-commuting variables $X_i$, recovering in the classical case $q=1$ the polynomial expressing the traces of monodromies of hyperbolic structures on S when written in terms of Thurston's shear-bend coordinates for Teichmüller space. In the '00s, Fock and Goncharov, among others, developed a classical and quantum ''higher Teichmüller theory'', which naturally leads to a $SL_n$-version of this invariant.

September 23, 2020
Colloquium	Modular representations of semisimple Lie algebras	4:15pm -	Abstract: Representation theory seeks to understand ways in which a given algebraic object (a group, an associative algebra, a Lie algebra etc.) can be represented via linear operators on a vector space over a field. What the representations are going to look like very much depends on the field in question, and, in particular, on its characteristic. Many important questions are settled in characteristic 0, for example, when we work over the complex numbers. But in the case of positive characteristic fields, which the word “modular” refers to, even basic questions are wide open. In my talk I will concentrate on one of the most important algebraic objects, semisimple Lie algebras, and explain what we know about about their irreducible (=no subrepresentations) modular representations. I will start with the case of sl_2 explaining the results of Rudakov and Shafarevich from 1967 describing the irreducible representations. Then I will talk about recent work on the general case including my paper with Bezrukavnikov from 2020, where we get the most explicit description of irreducible representations available to date. Our primary tool is relating the modular representations of semisimple Lie algebras to the (affine) Hecke category, the fundamental object in modern Representation theory.

September 24, 2020
Analysis	A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations	4:15pm -	Abstract: In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a quantitative version of Hörmander’s hypoelliptic regularity theory in an L1 framework which estimates this (degenerate) Fisher information from below by a fractional, L1-based Sobolev norm using the associated Kolmogorov equation. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and that this class includes the Lorenz 96 model in any dimension greater than or equal to 5, provided the additive stochastic driving is applied to any consecutive pair of oscillators. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96 (stochastically driven or otherwise), despite the overwhelming numerical evidence. If time permits, I will also discuss application of the method also to finite dimensional truncations of the classical shell models of hydrodynamic turbulence, GOY and SABRA.

September 25, 2020
Clusters and Geometry	Kasteleyn operators from mirror symmetry	1:00pm -	Abstract: In this talk we explain an interpretation of the Kasteleyn operator of a doubly-periodic bipartite graph from the perspective of homological mirror symmetry. Specifically, given a consistent bipartite graph G in T^2 with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (G,E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C)^2. The second is to take a certain Lagrangian surface L in T^ T^2 canonically associated to G, equip it with a brane structure prescribed by E, and pass to its homologically mirror coherent sheaf. This lives on a toric compactification of (C*)^2 determined by the Legendrian link which lifts the zig-zag paths of G (and to which the noncompact Lagrangian L is asymptotic). As a corollary, we obtain a complementary geometric perspective on certain features of algebraic integrable systems associated to lattice polygons, studied for example by Goncharov-Kenyon and Fock-Marshakov. This is joint work with David Treumann and Eric Zaslow.