Seminars and Colloquia by Series

Some results on a simple model of kinetic theory

Series
School of Mathematics Colloquium
Time
Thursday, April 15, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Federico BonettoGeorgia Institute of Technology

In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, together with M. Loss and others, we worked to extend the analysis to more "realistic" versions of the original Kac model. I will give a brief overview of kinetic theory, introduce the Kac model and explain the standard results on it. Finally I will present to new papers with M. Loss and R. Han and with J. Beck.

l^p improving and sparse bounds for discrete averaging operators using the divisor function

Series
Analysis Seminar
Time
Wednesday, April 14, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE. https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Christina GiannitsiGeorgia Tech

We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n) $. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is

$$ \Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'}  \leq C  \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p} $$

as long as $f$ is supported on the interval $[0,N]$.

We will also see that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

The seminar will be held on Zoom, and can be accessed by the link

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

Description:Chromatic index of dense quasirandom graphs

Series
Graph Theory Seminar
Time
Tuesday, April 13, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Songling ShanIllinois State University

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. We show that the conjecture of Glock, Kühn and Osthus is affirmative.

Geometric and Statistical Approaches to Shallow and Deep Clustering

Series
Time
Monday, April 12, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
James MurphyTufts University

We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning.  By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved.  Connections to geometric analysis, stochastic processes, and deep learning are emphasized.  The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points.  Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance.  Extensions to active learning, generative modeling, and computational geometry will be discussed.

Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

Series
Geometry Topology Seminar
Time
Monday, April 12, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
On line
Speaker
David GayUGA

I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well. Putting all this stuff together we can show that there is a surjective homomorphism from (a certain limit of) fundamental groups of spaces of embeddings of 2-spheres in connected sums of S^2XS^2 onto this smooth mapping class group of S^4. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded S^2's in S^4 (Montesinos twins).

Abelian Livshits Theorem

Series
CDSNS Colloquium
Time
Friday, April 9, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Andrey GogolevThe Ohio State University

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

The classical Livshits theorem characterizes coboundaries over a transitive Anosov flow as precisely those functions which integrate to zero over all periodic orbits of the flow. I will present a variant of this theorem which uses a weaker assumption and gives a weaker conclusion that the function is an ``abelian coboundary.” Such weaker version corresponds to studying the cohomological equation on infinite abelian covers e.g., for geodesic flows on abelian covers of hyperbolic surfaces. I will also discuss a connection to the marked length spectrum rigidity of Riemannian metrics. Joint work with Federico Rodriguez Hertz.

Obstructions to embeddings in 4-manifolds

Series
Geometry Topology Seminar
Time
Friday, April 9, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
On line
Speaker
Anubhav MukherjeeGeorgia Tech

Please Note: Note special day and time.

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk)  in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Mathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining

Series
Dissertation Defense
Time
Friday, April 9, 2021 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Online
Speaker
Yuchen (Roy) HeSchool of Math, Georgia Institute of Technology

Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this presentation, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns’ properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we introduce suitable mathematical representations using techniques ranging from group theory to deep neural networks.

Join Zoom Meeting

https://zoom.us/j/97642529845?pwd=aS9aTGloQnBGVVNQMHd6d0I4eGFNQT09

Meeting ID: 976 4252 9845

Passcode: 42PzXb

 

On a conjectural symmetric version of the Ehrhard inequality

Series
Stochastics Seminar
Time
Thursday, April 8, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Galyna LivshytsGeorgiaTech

We will discuss a conjectured sharp version of an Ehrhard-type inequality for symmetric convex sets, its connections to other questions, and partial progress towards it. We also discuss some new estimates for non-gaussian measures.

Erdős covering systems

Series
School of Mathematics Colloquium
Time
Thursday, April 8, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Rob MorrisNational Institute for Pure and Applied Mathematics, Rio de Janeiro, Brazil

A covering system of the integers is a finite collection of arithmetic progressions whose union is the integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. 

In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.

Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

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