Abstracts
Week of September 19, 2021
Group Actions and Dynamics  Local mixing of diagonal flows on Anosov homogeneous spaces 
4:00pm 
LOM 206

Strong mixing results, such as the ones proved by Babillot and Winter for convex cocompact rank one spaces, have many applications to problems in homogeneous dynamics. In the infinite measure setting, one hopes for local mixing. We will discuss a local mixing result generalizing their results in the context of Anosov homogeneous spaces. 
Geometry, Symmetry and Physics  Moduli spaces of local Galois representations and deformed affine Springer fibers 
4:30pm 
Zoom

Abstract: The moduli spaces of representations of Galois groups of padic fields with padic Hodge theoretic conditions play a pivotal role in the study of arithmetic properties of automorphic forms. Despite this, the geometry of these spaces is poorly understood, perhaps for good reason: conjecturally their complexity is bounded below by the modular representation theory of finite groups of Lie type. In this talk, I will survey some progress on the study of these moduli spaces in some special but important cases, where it turns out that they are closely related to degenerations of flag varieties into certain (deformed) affine Springer fibers. This is based on joint work with (various subsets of) D. Le, B. Levin and S. Morra. Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode. 
Geometry & Topology  SLLN for asymptotic translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces 
4:15pm 
LOM 214

In this talk, we discuss random walks on isometry groups of Gromov hyperbolic spaces and Teichmüller spaces. We prove that nonelementary random walks exhibit at least linear growth of asymptotic translation lengths without any moment condition. As a corollary, it follows that almost every samplepath on a mapping class group eventually becomes pseudoAnosov. Moreover, we show that if the underlying measure has a finite first moment, then the growth is linear, as Strong Law of Large Number. This is joint work with Hyungryul Baik and Inhyeok Choi. 
Applied Mathematics  Generalization Error Lower Bounds for Nonlinear Learning Models 
2:30pm 
https://yale.zoom.us/j/2188028533

Abstract: Deep learning algorithms operate in regimes that defy classical learning theory. Neural networks architectures often contain more parameters than training samples. Despite their huge complexity, the generalization error achieved on real data is small. In this talk, we aim to study generalization properties of algorithms in high dimensions. Interestingly, we show that algorithms in high dimension require a small bias for good generalization. We show that this is indeed the case for deep neural networks in the overparameterized regime. In addition, we provide lower bounds on the generalization error in various settings for any algorithm. We calculate such bounds using random matrix theory (RMT). We will review the connection between deep neural networks and RMT and existing results. These bounds are particularly useful when the analytic evaluation of standard performance bounds is not possible due to the complexity and nonlinearity of the model. The bounds can serve as a benchmark for testing performance and optimizing the design of actual learning algorithms. 
Undergraduate Seminar  Putnam Seminar 
4:00pm 
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214

The Putnam seminar meets every Wednesday from 4 to 5:30 in LOM 214. As always, everyone is warmly welcomed to come to hang out, learn more cool math, and meet folks. The seminar is casual, and folks can come and go as they like. See Pat Devlin’s webpage (and/or contact him) for more information. Folks can sign up for the mailing list here: https://forms.gle/nYPx72KVJxJcgLha8 
Algebra and Geometry lecture series  Quantizations in charateristic p. Lecture 3 
4:00pm 
https://yale.zoom.us/j/99019019033 (password was emailed by Ivan)

This is the third lecture in the series. Details can be found here: Algebra and Geometry lecture series (yale.edu) 
Geometric Analysis and Application  Noncollapsed degeneration and desingularization of Einstein 4manifolds  2:00pm  
Abstract: We study the moduli space of unitvolume Einstein 4manifolds near its finitedistance boundary, that is, the noncollapsed singularity formation. We prove that any smooth Einstein 4manifold close to a singular one in a mere GromovHausdorff (GH) sense is the result of a gluingperturbation procedure that we develop and which handles the presence of multiple trees of singularities at arbitrary scales. This sheds some light on the structure of the moduli space and lets us show that spherical and hyperbolic orbifolds which are Einstein in a synthetic sense cannot be GHapproximated by smooth Einstein metrics. 
YUMS  Differential operators, algebraically 
5:00pm 
LOM 214

Linear differential operators are ubiquitous in Analysis: much of the subject of PDE is to understand their solutions in various function spaces. Little more surprisingly, differential operators are also important in Algebra, in particular, in Algebraic geometry and Representation theory. In this talk, we will discuss basic algebraic features of linear differential operators with polynomial coefficients.
