## Seminars and Colloquia by Series

### Fluctuation results for size of the vacant set for random walks on discrete torus

Series
Stochastics Seminar
Time
Thursday, November 3, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daesung KimGeorgia Tech

We consider a random walk on the $d\ge 3$ dimensional discrete torus starting from vertices chosen independently and uniformly at random. In this talk, we discuss the fluctuation behavior of the size of the range of the random walk trajectories at a time proportional to the size of the torus. The proof relies on a refined analysis of tail estimates for hitting time. We also discuss related results and open problems. This is based on joint work with Partha Dey.

### Long-time dynamics of dispersive equations

Series
Research Horizons Seminar
Time
Wednesday, November 2, 2022 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gong ChenGeorgia Institute of Technology

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.

### Generic Mean Curvature Flow with Cylindrical Singularities

Series
PDE Seminar
Time
Tuesday, November 1, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ao SunUniversity of Chicago

We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University)

### Wild Rose, Narcissus and other Elliptic Flowers: a new class of billiards with surprising properties.

Series
Geometry Topology Seminar
Time
Monday, October 31, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonid BunimovichGeorgia Tech

I'll talk about some 2D billiards, the most visual class of dynamical systems, where orbits (rays) move along straight lines within a billiard table with elastic reflections off the boundary.  Elliptic flowers are built “around" convex polygons, and the boundary of corresponding billiard tables consists of the arcs of ellipses. It will be explained why some classes of such elliptic flowers demonstrate a never expected before dynamics, and why it raises a variety of (seemingly new) questions in geometry (particularly in 3D), in bifurcation theory (particularly about singularities of wave fronts and creation of wave trains), in statistical mechanics,  quantum chaos, and perhaps some more. The talk will be concluded by showing a free movie. Everything (including various definitions of ellipses) will be explained/reminded.

### Extremal Combinatorics, Real Algebraic Geometry and Undecidability

Series
Algebra Seminar
Time
Monday, October 31, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Greg BlekhermanGeorgia Institute of Technology

I will highlight recent interplay between problems in extremal combinatorics and real algebraic geometry. This sheds a new light on undecidability of graph homomorphism density inequalities in extremal combinatorics, trace inequalities in linear algebra, and symmetric polynomial inequalities in real algebraic geometry. All of the necessary notions will be introduced in the talk. Joint work with Jose Acevedo, Sebastian Debus and Cordian Riener.

### The Potts model on expander graphs

Series
Combinatorics Seminar
Time
Friday, October 28, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Corrine YapRutgers University

The Potts model is a distribution on q-colorings of a graph, used to represent spin configurations of a system of particles. Intuitively we expect most configurations to be "solid-like" at low temperatures and "gas-like" at high temperatures. We prove a precise version of this statement for d-regular expander graphs. We also consider the question of whether or not there are efficient algorithms for approximate counting and sampling from the model, and show that such algorithms exist at almost all temperatures. In this talk, I will introduce the different tools we use in our proofs, which come from both statistical physics (polymer models, cluster expansion) and combinatorics (a new container-like result, Karger's randomized min-cut algorithm). This is joint work with Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, and Aditya Potukuchi.

### Identifiability and inference of phylogenetic birth-death models

Series
Mathematical Biology Seminar
Time
Friday, October 28, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Jonathan TerhorstUniversity of Michigan - Department of Statistics

The phylogenetic birth-death process is a probabilistic model of evolution that
is widely used to analyze genetic data. In a striking result, Louca & Pennell
(Nature, 2020) recently showed that this model is statistically unidentifiable,
meaning that an arbitrary number of different evolutionary hypotheses are
consistent with any given data set. This grave finding has called into question
the conclusions of a large number of evolutionary studies which relied on this
model.

In this talk, I will give an introduction to the phylogenetic birth-death
process, and explain Louca and Pennell's unidentifiability result. Then, I will
describe recent positive results that we have obtained, which establish that, by
restricting the evolutionary hypothesis space in certain biologically plausible
ways, statistical identifiability is restored. Finally, I will discuss some
complementary hardness-of-estimation results which show that, even in identifiable
model classes, obtaining reliable inferences from finite amounts of data may be
extremely challenging.

No background in this area is assumed, and the talk will be accessible to a
mathematically mature audience. This is joint work with Brandon Legried.

### Smooth structures on open 4-manifolds V

Series
Time
Friday, October 28, 2022 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.

### Coloring Graphs with Forbidden Almost Bipartite Subgraphs

Series
ACO Student Seminar
Time
Friday, October 28, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
James AndersonGeorgia Tech Math

For graphs with maximum degree $\Delta$, a greedy algorithm shows $\chi(G) \leq \Delta + 1$. Brooks improved this to $\chi(G) \leq \Delta$ when $G$ has no cliques of size $\Delta + 1$, provided $\Delta \geq 3$. If is conjectured that if one forbids other graphs, the bound can be pushed further: for instance, Alon, Krivelevich, and Sudakov conjecture that, for any graph $F$, there is a constant $c(F) > 0$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ for all $F$-free graphs $G$ of maximum degree $\Delta$. The only graphs $F$ for which this conjecture has been verified so far---by Alon, Krivelevich, and Sudakov themselves---are the so-called almost bipartite graphs, i.e., graphs that can be made bipartite by removing at most one vertex. Equivalently, a graph is almost bipartite if it is a subgraph of the complete tripartite graph $K_{1,t,t}$ for some $t \in \N$. The best heretofore known upper bound on $c(F)$ for almost bipartite $F$ is due to Davies, Kang, Pirot, and Sereni, who showed that $c(K_{1,t,t}) \leq t$. We prove that in fact $c(F) \leq 4$ for any almost bipartite graph $F$, thus making the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of this result in the setting of DP-coloring (also known as correspondence coloring), which we give a gentle introduction to. Finally, we consider consequences of this result in the context of sublinear algorithms.

This is joint work with Anton Bernshteyn and Abhishek Dhawan.

### A brief introduction to the circle method and sparse domination

Series
In this talk we will go over the Hardly-Littlewood circle method, and the major and minor arc decomposition. We shall then see a toy-example of the High-Low decomposition, and proceed with defining sparse families and sparse domination. We will conclude by explaining why sparse domination is of interest to us  when studying $L^p$ bounds. This talk aims to be accessible to people without a strong background in the area. Some basic concepts of real and harmonic analysis will be useful (e.g. $L^p$ spaces, Fourier transform,  Holder inequality, the Hardy-Littlewood Maximal function, etc)