## Week of October 3, 2021

 Applied Mathematics Uncertainties and inference in physical dynamics 10:30am - https://yale.zoom.us/j/2188028533 Abstract: We discuss two statistical problems that arise in the course of physical modeling. First, we consider an experiment where a low-dimensional dynamical system is observed as a high-dimensional signal, e.g., a video of a chaotic pendulums system. Assuming that we know the dynamical model, but do not know the observation function (the experimental design) - can we estimate the true parameters from the experiment? The key information lies in the temporal inter-dependencies between the signal and the model, and we exploit this information using a kernel-based score. In the second part of the talk, we turn to uncertainty propagation; in many scientific areas, the parameters of deterministic models are uncertain or noisy. A comprehensive model should therefore provide a statistical description of the quantity of interest. Underlying this computational problem is a fundamental question - if two “similar” functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? Through optimal transport theory, a Wasserstein-distance formulation of our problem yields a simple and applicable theory. Group Actions and Dynamics Sharp conditions for equidistribution of translates of curves by a diagonal flow on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$ 4:00pm - Zoom We consider the action of the diagonal subgroup $\{a(t)=(t^{n-1},t^{-1},\ldots,t^{-1})\}\subset G=\mathrm{SL}(n,\mathbb{R})$  on $X=G/\Gamma$, where $\Gamma=\mathrm{SL}(n,\mathbb{Z})$. Let $C$ be a finite piece of an analytic curve on the expanding horophere ($\cong{\mathbb R}^{n-1}$) of $\{a(t)\}_{{t>1}}$ in $G$ . Let $\mu_{C}$ be a smooth probability measure on the trajectory $C[\Gamma]$ on $X$. We provide necessary and sufficient conditions on the smallest affine subspace containing $C$ in terms of Diophantine approximation and algebraic number fields so that the measures $a(t)\mu_{C}$ get equidistributed in $X$ as $t\to\infty$. This result generalizes the speaker’s earlier work showing equidistribution of translates of curves, which are not contained in proper affine subspaces. The result answers a question of Davenport and Schmidt on non-improvability of Dirichlet’s approximation. The case of $n=3$ is a joint work D. Kleinbock, N.  de Saxcé, and P. Yang; and the general case is a joint work with Pengyu Yang. Geometry, Symmetry and Physics Affine Springer fibers, open Hessenberg varieties, and nabla positivity 4:30pm - Zoom Abstract: I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their Borel-Moore homology turn out to be related to reproducing kernels of the Bergeron-Garsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson. Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode.
 Geometry & Topology Tropical Fock-Goncharov coordinates for SL3-webs on surfaces 4:15pm - LOM 214 For a finite-type surface S, we study a preferred basis for the commutative algebra C[R_SL3(S)] of regular functions on the SL3(C)-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface S. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun. Algebra and Number Theory Seminar Flag varieties and representations of p-adic groups 4:30pm - Zoom Deligne--Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called "depth-zero" supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is based on joint work with A. Ivanov and on joint work with M. Oi.
 Applied Mathematics Efficient distribution classification via optimal transport embeddings 2:30pm - https://yale.zoom.us/j/2188028533 Abstract:  Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years.  There are some drawbacks to OT: Computing OT is usually slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.  In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level.  In particular, we’ll show how to embed the space of distributions into an L^2-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. The proposed framework significantly reduces both the computational effort and the required training data in supervised settings. We demonstrate the benefits in pattern recognition tasks in imaging and provide some medical applications. Undergraduate Seminar Putnam Seminar 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - 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 5 4:00pm - https://yale.zoom.us/j/99019019033 (password was emailed by Ivan) This is the fourth lecture in the series. Details can be found here: Algebra and Geometry lecture series (yale.edu)
 Geometric Analysis and Application Uniform estimates for complex Monge-Ampere and fully nonlinear equations 2:00pm - Abstract: Uniform estimates for the complex Monge Ampere equation has had a long history, starting with Yau’s famous resolution of the Calabi conjecture. Subsequent developments has led to many geometric applications and openings of new fields, but have all relied on the methods of pluripotential theory. In this talk, we will discuss a new PDE method of obtaining sharp uniform estimates for complex Monge-Ampere, without using pluripotential theory. This new methods extends more generally to other fully non-linear equations in complex geometry. This is based on joint work with B. Guo and D.H. Phong.