Department of Mathematics

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Week of September 25, 2022

September 26, 2022
Applied Mathematics	Riemannian Geometry in Machine Learning	Isay Katsman - Yale University	3:00pm - AKW 200
Group Actions, Geometry and Dynamics	Discrete subgroups of PU(2,1) with large critical exponents	Subhadip Dey - Yale	4:00pm - LOM206
Geometry, Symmetry and Physics	Microlocal sheaves and affine Springer fibers	Pablo Boixeda Alvarez - Yale	4:30pm - LOM 214

September 27, 2022
Geometry & Topology	Some remarks on the geometry of arithmetic locally symmetric spaces	Sebastian Hurtado-Salazar - Yale University	4:15pm - LOM 214

September 28, 2022
Colloquium	Balancing covariates in randomized experiments	Daniel A. Spielman - Yale University	4:00pm - DL220 (10 Hillhouse Avenue)

September 29, 2022
Algebra and Geometry lecture series	Cohomology of Hitchin moduli spaces and the P=W conjecture.	Junliang Shen - Yale University	4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214

September 30, 2022
Friday Morning Seminar	Friday Morning Seminar		10:30am - LOM 205
Graduate Student Seminar	From curves to sheaves: a tale of two moduli spaces	Weite Pi - Yale	11:30am - LOM

Abstracts

Week of September 25, 2022

September 26, 2022
Applied Mathematics	Riemannian Geometry in Machine Learning	3:00pm - AKW 200	Abstract: Although machine learning researchers have introduced a plethora of useful constructions for learning over Euclidean space, numerous types of data in various applications benefit from, if not necessitate, a non-Euclidean treatment. In this talk I cover the need for Riemannian geometric constructs to (1) build more principled generalizations of common Euclidean operations used in geometric machine learning models as well as to (2) enable general manifold density learning in contexts that require it. Said contexts include theoretical physics, robotics, and computational biology. I will cover one of my papers that fits into (1) above, namely the ICML 2020 paper "Differentiating through the Fréchet Mean." I will also cover two of my papers that fit into (2) above, namely the NeurIPS 2020 paper "Neural Manifold ODEs" and the NeurIPS 2021 paper "Equivariant Manifold Flows." Finally, I will briefly discuss directions of relevant ongoing work. Bio: Isay Katsman is a first year PhD student in applied mathematics at Yale advised by Prof. Anna Gilbert. His research interests include Riemannian geometry arising in the context of machine learning, with particular emphasis on applications in mathematical physics. Before Yale, Isay obtained his master's degree in computer science from Cornell University, where he also conducted his undergraduate studies. Throughout his master's and undergraduate careers, Isay was advised by Prof. Christopher de Sa. His work is supported by an NSF Graduate Research Fellowship.
Group Actions, Geometry and Dynamics	Discrete subgroups of PU(2,1) with large critical exponents	4:00pm - LOM206	Let Γ be a discrete subgroup of PU(2,1), the isometry group of the complex-hyperbolic space CH² of (complex) dimension two. Under a suitable normalization of the Riemannian metric of CH², the critical exponent δ(Γ) of Γ, when Γ is a lattice, is 4. In this talk, we discuss an example of a sequence (Γₙ) of discrete subgroups of PU(2,1) such that, for all n∈N, δ(Γₙ) < 4, but δ(Γₙ) → 4, as n → ∞. This example shows that Corlette’s gap theorem on critical exponents of discrete isometry groups of the quaternionic-hyperbolic spaces does not hold for CH². This talk is based on joint work with Beibei Liu.
Geometry, Symmetry and Physics	Microlocal sheaves and affine Springer fibers	4:30pm - LOM 214	The resolutions of Slodowy slices $\widetilde{\mathcal{S}}_e$ are symplectic varieties that contain the Springer fiber $(G/B)_e$ as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves. In this talk I will mostly focus on the case of the homogeneous element $ts$ for $s$ a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

September 27, 2022
Geometry & Topology	Some remarks on the geometry of arithmetic locally symmetric spaces	4:15pm - LOM 214	We discuss an arithmetic version of the classical (Zassenhaus or Margulis) lemma regarding the topology of the thin part of a locally symmetric space. We give several applications including bounds on the thin part of arithmetic locally symmetric spaces and some results regarding arithmetic groups generated by torsion elements. Based on works with Mikolaj Fraczyk and Jean Raimbault, and with David Fisher. All notions will be explained.

September 28, 2022
Colloquium	Balancing covariates in randomized experiments	4:00pm - DL220 (10 Hillhouse Avenue)	In randomized experiments, we randomly assign the treatment that each experimental subject receives. Randomization can help us accurately estimate the difference in treatment effects with high probability. It also helps ensure that the groups of subjects receiving each treatment are similar. If we have already measured characteristics of our subjects that we think could influence their response to treatment, then we can increase the precision of our estimates of treatment effects by balancing those characteristics between the groups. We show how to use the recently developed Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett to efficiently assign treatments to subjects in a way that balances known characteristics without sacrificing the benefits of randomization. These allow us to obtain more accurate estimates of treatment effects to the extent that the measured characteristics are predictive of treatment effects, while also bounding the worst-case behavior when they are not. This is joint work with Chris Harshaw, Fredrik Sävje, and Peng Zhang. Special Note: the colloquium will start at 4pm. (and not at 4:15pm as usual).

September 29, 2022
Algebra and Geometry lecture series	Cohomology of Hitchin moduli spaces and the P=W conjecture.	4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214

September 30, 2022
Friday Morning Seminar	Friday Morning Seminar	10:30am - LOM 205	A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar	From curves to sheaves: a tale of two moduli spaces	11:30am - LOM	Moduli spaces parametrize certain kinds of geometric objects, and they are central objects in algebraic geometry. In this talk, we look at two different moduli spaces: one of smooth curves of fixed genus, and the other of one-dimensional sheaves on the projective plane. The former is a classical subject, dating back to Bernhard Riemann, while the latter relatively new, motivated by enumerative geometry, mathematical physics and so on. Despite the different nature of the objects they parametrize, we will show how their ‘intersection rings’ share surprisingly similar features. This talk is based on joint work with Junliang Shen.