## Seminars and Colloquia by Series

### Maximum number of almost similar triangles in the plane

Series
Graph Theory Seminar
Time
Tuesday, April 27, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
Bernard LidickýIowa State University

A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon >0$ and $n \in \mathbb{N}$, Bárány and Füredi asked to determine the maximum number of triangles $h(n,T,\varepsilon)$ being $\varepsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\varepsilon=\varepsilon(T)>0$ such that $h(n,T,\varepsilon)=n^3/24 (1+o(1))$. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof. This is joint work with József Balogh and Felix Christian Clemen.

### Some problems in point-set registration

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 26, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Yuehaw KhooUniversity of Chicago

In this talk, we discuss variants of the rigid registration problem, i.e aligning objects via rigid transformation. In the simplest scenario of point-set registration where the correspondence between points are known, we investigate the robustness of registration to outliers. We also study a convex programming formulation of point-set registration with exact recovery, in the situation where both the correspondence and alignment are unknown. This talk is based on joint works with Ankur Kapoor, Cindy Orozco, and Lexing Ying.

### On the length of the shortest closed geodesic on positively curved 2-spheres.

Series
Geometry Topology Seminar
Time
Monday, April 26, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/579155918
Speaker
Franco Vargas PalleteYale University

Following the approach of Nabutovsky and Rotman for the curve-shortening flow on geodesic nets, we'll show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. On the pinched curvature setting, we prove a bound on the first eigenvalue of the Laplacian and use it to prove a new isoperimetric inequality for pinched 2-spheres sufficiently close to being round. This allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting. This is joint work with Ian Adelstein.

### On Scalable and Fast Langevin-Dynamics-Based Sampling Algorithms

Series
Dissertation Defense
Time
Friday, April 23, 2021 - 15:00 for 1.5 hours (actually 80 minutes)
Location
ONLINE
Speaker
Ruilin LiGeorgia Institute of Technology

Langevin dynamics-based sampling algorithms are arguably among the most widely-used Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big data applications, and (ii) Tight convergence analysis of MCMC algorithms, with explicit dependence on various characteristics of the target distribution, in a non-asymptotic manner.

This thesis continues the previous efforts in these two lines and consists of three parts. In the first part, we study stochastic gradient MCMC methods for large-scale applications. We propose a non-uniform subsampling of gradients scheme to approximately match the transition kernel of a base MCMC base with full gradient, aiming for better sample quality. The demonstration is based on underdamped Langevin dynamics.

In the second part, we consider an analog of Nesterov's accelerated algorithm in optimization for sampling. We derive a  dynamics termed Hessian-Free-High-Resolution (HFHR) dynamics, from a high-resolution ordinary differential equation description of Nesterov's accelerated algorithm. We then quantify the acceleration of HFHR over underdamped Langevin dynamics at both continuous dynamics level and discrete algorithm level.

In the third part, we study a broad family of bounded, contractive-SDE-based sampling algorithms via mean-square analysis. We show how to extend the applicability of classical mean-square analysis from finite time to infinite time. Iteration complexity in the 2-Wasserstein distance is also characterized and when applied to the Langevin Monte Carlo algorithm, we obtain an improved iteration complexity bound.

### Contact structures on hyperbolic L-spaces

Series
Dissertation Defense
Time
Friday, April 23, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Hyunki MinGeorgia Tech

Ever since Eliashberg distinguished overtwisted from tight contact structures in dimension 3, there has been an ongoing project to determine which closed, oriented 3–manifolds support a tight contact structure, and on those that do, whether we can classify them. This thesis studies tight contact structures on an infinite family of hyperbolic L-spaces, which come from surgeries on the Whitehead link. We also present partial results on symplectic fillability on those manifolds.

### Normal form and existence time for the Kirchhoff equation

Series
CDSNS Colloquium
Time
Friday, April 23, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Emanuele HausUniversity of Roma Tre

In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with Pietro Baldi.

Computing the first step of quasilinear normal form, we erase from the equation all the cubic terms giving nonzero contribution to the energy estimates; thus we deduce that, for small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ coming from the linear theory).

In the second step of normal form, there remain some resonant terms (which cannot be erased) that give a non-trivial contribution to the energy estimates; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$.

### Learning Gaussian mixtures with algebraic structure

Series
Stochastics Seminar
Time
Thursday, April 22, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/129119189
Speaker
Victor-Emmanuel BrunelENSAE/CREST

We will consider a model of mixtures of Gaussian distributions, called Multi-Reference Alignment, which has been motivated by imaging techniques in chemistry. In that model, the centers are all related with each other by the action of a (known) group of isometries. In other words, each observation is a noisy version of an isometric transformation of some fixed vector, where the isometric transformation is taken at random from some group of isometries and is not observed. Our goal is to learn that fixed vector, whose orbit by the action of the group determines the set of centers of the mixture. First, we will discuss the asymptotic performances of the maximum-likelihood estimator, exhibiting two scenarios that yield different rates. We will then move on to a non-asymptotic, minimax approach of the problem.

### A modern take on Huber regression

Series
School of Mathematics Colloquium
Time
Thursday, April 22, 2021 - 12:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Po-Ling LohUniversity of Cambridge

Please Note: Note the unusual time: 12:00pm.

In the first part of the talk, we discuss the use of a penalized Huber M-estimator for high-dimensional linear regression. We explain how a fairly straightforward analysis yields high-probability error bounds that hold even when the additive errors are heavy-tailed. However, the parameter governing the shape of the Huber loss must be chosen in relation to the scale of the error distribution. We discuss how to use an adaptive technique, based on Lepski's method, to overcome the difficulties traditionally faced by applying Huber M-estimation in a context where both location and scale are unknown.

In the second part of the talk, we turn to a more complicated setting where both the covariates and responses may be heavy-tailed and/or adversarially contaminated. We show how to modify the Huber regression estimator by first applying an appropriate "filtering" procedure to the data based on the covariates. We prove that in low-dimensional settings, this filtered Huber regression estimator achieves near-optimal error rates. We further show that the commonly used least trimmed squares and least absolute deviation estimators may similarly be made robust to contaminated covariates via the same covariate filtering step. This is based on joint work with Ankit Pensia and Varun Jog.

### An Alexander method for infinite-type surfaces

Series
Geometry Topology Student Seminar
Time
Wednesday, April 21, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Roberta Shapiro

Given a surface S, the Alexander method is a combinatorial tool used to determine whether two homeomorphisms are isotopic. This statement was formalized in A Primer on Mapping Class Groups in the case that S is of finite type. We extend the Alexander method to include infinite-type surfaces, which are surfaces with infinitely generated fundamental groups.

In this talk, we will introduce a technique useful in proofs dealing with infinite-type surfaces. Then, we provide a "proof by example" of an infinite-type analogue of the Alexander method.

### An analytical study of intermittency through Riemann’s non-differentiable functions

Series
Analysis Seminar
Time
Wednesday, April 21, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE — see abstract for the Zoom link
Speaker
Victor Vilaça Da RochaGeorgia Tech

Intermittency is a property observed in the study of turbulence. Two of the most popular ways to measure it are based on the concept of flatness, one with structure functions in the physical space and the other one with high-pass filters in the frequency space. Experimental and numerical simulations suggest that the two approaches do not always give the same results. In this talk, we prove they are not analytically equivalent. For that, we first adapt them to a rigorous mathematical language, and we test them with generalizations of Riemann’s non-differentiable function. This work is motivated by the discovery of Riemann’s non-differentiable function as a trajectory of polygonal vortex filaments.

The seminar will be held on Zoom.  Here is the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09