## Seminars and Colloquia by Series

### Coloring graphs with forbidden bipartite subgraphs

Series
Graph Theory Seminar
Time
Tuesday, April 6, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
James AndersonGeorgia Institute of Technology

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.

### On the stationary/uniformly rotating solutions of active scalar euquations

Series
Dissertation Defense
Time
Tuesday, April 6, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Jaemin ParkGeorgia tech

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.

### Optimization in the space of probabilities with MCMC: Uncertainty quantification and sequential decision making

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 5, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE https://bluejeans.com/884917410
Speaker
Prof. Yian MaUCSD

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.

### Right-veering open books and the Upsilon invariant

Series
Geometry Topology Seminar
Time
Monday, April 5, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Diana HubbardBrooklyn College, CUNY

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.

### Numerical Estimation of Several Topological Quantities of the First Passage Percolation Model

Series
Dissertation Defense
Time
Monday, April 5, 2021 - 13:00 for 2 hours
Location
ONLINE
Speaker
Yuanzhe MaGeorgia Institute of Technology

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 in Phylogenetics using Algebraic Matroids

Series
Mathematical Biology Seminar
Time
Friday, April 2, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Seth SullivantNorth Carolina State University

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.

### Optimal Ranking Recovery from Pairwise Comparisons

Series
Stochastics Seminar
Time
Thursday, April 1, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/129119189
Speaker
Anderson Y. ZhangUniversity of Pennsylvania

Ranking from pairwise comparisons is a central problem in a wide range of learning and social contexts. Researchers in various disciplines have made significant methodological and theoretical contributions to it. However, many fundamental statistical properties remain unclear especially for the recovery of ranking structure. This talk presents two recent projects towards optimal ranking recovery, under the Bradley-Terry-Luce (BTL) model.

In the first project, we study the problem of top-k ranking. That is, to optimally identify the set of top-k players. We derive the minimax rate and show that it can be achieved by MLE. On the other hand, we show another popular algorithm, the spectral method, is in general suboptimal.

In the second project, we study the problem of full ranking among all players. The minimax rate exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. A divide-and-conquer ranking algorithm is proposed to achieve the minimax rate.

### Symmetry and uniqueness via a variational approach

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

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

### Symplectic rigidity, flexibility, and embedding problems

Series
Geometry Topology Student Seminar
Time
Wednesday, March 31, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Agniva RoyGeorgia Tech

Embedding problems, of an n-manifold into an m-manifold, can be heuristically thought to belong to a spectrum, from rigid, to flexible. Euclidean embeddings define the rigid end of the spectrum, meaning you can only translate or rotate an object into the target. Symplectic embeddings, depending on the object, and target, can show up anywhere on the spectrum, and it is this flexible vs rigid philosophy, and techniques developed to study them, that has lead to a lot of interesting mathematics. In this talk I will make this heuristic clearer, and show some examples and applications of these embedding problems.

### Intersecting families of sets are typically trivial

Series
Graph Theory Seminar
Time
Tuesday, March 30, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
A family of subsets of $[n]$ is intersecting if every pair of its members has a non-trivial intersection. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n \geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Significantly improving their results, we show that the same conclusion holds for $n \geq 2k + 100 \ln k$. Our proof uses the Sapozhenko’s graph container method and the Das-Tran removal lemma.