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.
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.
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 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.
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
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.
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.
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters are generically identifiable for 2-tree CFN and K3P mixtures. We also show that the k-cycle phylogenetic network parameter is identifiable under the K2P and K3P models. This is joint work with Benjamin Hollering.
Meeting Link: https://gatech.bluejeans.com/348270750
