Georgia Institute of Technology College of Sciences

In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and the location of each hole for B(t). We discuss the theoretical background of this work, the algorithm we used to conduct simulations, and include an extensive discussion of our simulation results. Our results support some of the prior conjectures and further introduce several interesting open problems.

This defense will be conducted on bluejeans, at https://bluejeans.com/611615950.

Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon.  I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

I will present MCMC algorithms as optimization over the KL-divergence in the space of probabilities. By incorporating a momentum variable, I will discuss an algorithm which performs accelerated gradient descent over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained. I will then discuss how MCMC algorithms compare against variational inference methods in parameterizing the gradient flows in the space of probabilities and how it applies to sequential decision making problems.

We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.

   For qualitative properties, we aim to establish sufficient conditions for such solutions to be radially symmetric. The proof  is based on variational argument, using the fact that a uniformly-rotating solution can be formally thought of as  a critical point of an energy functional. It turns out that if positive vorticity is rotating with angular velocity, not in (0,1/2), then the corresponding energy functional has a unique critical point, while radial ones are always critical points. We apply similar ideas to more general active scalar equations (gSQG) and vortex sheet equation. We also prove that for rotating vortex sheets, there exist  non-radial rotating vortex sheets, bifurcating from radial ones. This work is based on the joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao. 

    It is well-known that there are non-radial rotating patches with angular velocity in (0,1/2). Using the variational argument, we derive some quantitative estimates for their angular velocities and the difference from the radial ones.

Link: https://bluejeans.com/974226566

  

A conjecture by Alon, Krivelevich, and Sudakov in 1999 states that for any graph $H$, there is a constant $c_H > 0$ such that if $G$ is $H$-free of maximum degree $\Delta$, then $\chi(G) \leq c_H \Delta / \log\Delta$. It follows from work by Davies et al. in 2020 that this conjecture holds for $H$ bipartite (specifically $H = K_{t, t}$), with the constant $c_H = (t+o(1))$. We improve this constant to $1 + o(1)$ so it does not depend on $H$, and extend our result to the DP-coloring (also known as correspondence coloring) case introduced by Dvořák and Postle. That is, we show for every bipartite graph $B$, if $G$ is $B$-free with maximum degree $\Delta$, then $\chi_{DP}(G) \leq (1+o(1))\Delta/\log(\Delta)$.

This is joint work with Anton Bernshteyn and Abhishek Dhawan.

The talk will present joint work with Almaz Butaev (Calgary) in which we consider local versions of uniform domains and characterize them as extension domains for the nonhomogeneous ("localized") BMO space defined by Goldberg, denoted bmo. As part of this characterization, we show these domains are the same as the $(\epsilon,\delta)$ domains used in Jones' extension theorem for Sobolev spaces, and also that they satisfy a local quasihyperbolically uniform condition. All the above terms will be defined in the talk. The Zoom link for the seminar is here: https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.

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.

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.

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

 

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.

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.