## Seminars and Colloquia by Series

### Statistical Tensor Learning in 2020s: Methodology, Theory, and Applications

Series
Stochastics Seminar
Time
Thursday, October 20, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anru ZhangDuke University

The analysis of tensor data, i.e., arrays with multiple directions, has become an active research topic in the era of big data. Datasets in the form of tensors arise from a wide range of scientific applications. Tensor methods also provide unique perspectives to many high-dimensional problems, where the observations are not necessarily tensors. Problems in high-dimensional tensors generally possess distinct characteristics that pose great challenges to the data science community.

In this talk, we discuss several recent advances in statistical tensor learning and their applications in computational imaging, social network, and generative model. We also illustrate how we develop statistically optimal methods and computationally efficient algorithms that interact with the modern theories of computation, high-dimensional statistics, and non-convex optimization.

### No Seminar - Fall Break

Series
Geometry Topology Seminar
Time
Monday, October 17, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

### Examples of constructions of higher dimensional hyperbolic tori with controlled splitting

Series
Joint School of Mathematics and CDSNS Colloquium
Time
Friday, October 14, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Online via Zoom; &quot;viewing party&quot; in Skiles 006
Speaker
Jean-Pierre MarcoSorbonne Universite

In this talk I will generalize a simple trick to produce splitting for the separatrices of (the time-one map of) a simple pendulum, to hyperbolic tori of any dimension $m\geq 2$. The examples will be constructed in the Gevrey class, and the splitting is bounded from below by a term of the form $\exp (-c(1/\eps)^a)$, where $a=\frac{1}{2(\alpha-1)(m-2)}$. This will be compared to usual upper bounds in the same setting.

### TBA by Ruth Luo

Series
Combinatorics Seminar
Time
Friday, October 14, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Ruth LuoUniversity of South Carolina

### Minimum degree conditions ensuring the existence of long cycles in hypergraphs

Series
Combinatorics Seminar
Time
Friday, October 14, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Ruth LuoUniversity of South Carolina

Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. Moreover, every graph with minimum degree $k \geq 2$ contains a cycle of length at least $k+1$, and this can be further improved if the graph is 2-connected. In this talk, we prove analogs of these theorems for hypergraphs. That is, we give sharp minimum degree conditions that imply the existence of long Berge cycles in uniform hypergraphs. This is joint work with Alexandr Kostochka and Grace McCourt.

### Efficient and Near-Optimal Online Portfolio Selection

Series
Stochastics Seminar
Time
Friday, October 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitrii M. OstrovskiiUniversity of Southern California

In the problem of online portfolio selection as formulated by Cover (1991), the trader repeatedly distributes her capital over $d$ assets in each of $T > 1$ rounds, with the goal of maximizing the total return. Cover proposed an algorithm called Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with

an $O(d\log(T))$ regret in terms of the logarithmic return. Without imposing any restrictions on the market, this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing the expectation over certain log-concave density in R^d, so in a practical implementation this expectation has to be approximated via sampling, which is computationally challenging. In particular, the fastest known implementation, proposed by Kalai and Vempala in 2002, runs in $O( d^4 (T+d)^{14} )$ per round, which rules out any practical application scenario. Proposing a practical algorithm with a near-optimal regret is a long-standing open problem. We propose an algorithm for online portfolio selection with a near-optimal regret guarantee of $O( d \log(T+d) )$ and the runtime of only $O( d^2 (T+d) )$ per round. In a nutshell, our algorithm is a variant of the follow-the-regularized-leader scheme, with a time-dependent regularizer given by the volumetric barrier for the sum of observed losses. Thus, our result gives a fresh perspective on the concept of volumetric barrier, initially proposed in the context of cutting-plane methods and interior-point methods, correspondingly by Vaidya (1989) and Nesterov and Nemirovski (1994). Our side contribution, of independent interest, is deriving the volumetrically regularized portfolio as a variational approximation of the universal portfolio: namely, we show that it minimizes Gibbs's free energy functional, with accuracy of order $O( d \log(T+d) )$. This is a joint work with Remi Jezequel and Pierre Gaillard.

### Parallel computations to study complex dynamics in neuroscience and other chaotic nonlinear systems

Series
Time
Friday, October 14, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Krishna PusuluriGSU

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

We will begin with a brief overview of several parallel and hybrid computing approaches including CUDA, OpenAcc, OpenMP, and OpenMPI, followed by a demonstration of how we can leverage these technologies to study complex dynamics arising from diverse nonlinear systems. First, we discuss multistable rhythms in oscillatory 4-cell central pattern generators (CPGs) of inhibitory coupled  neurons. We show how network topology and intrinsic properties of the cells affect dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. We then discuss symbolic methods and parametric sweeps to analyze isolated neuron dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits and animal CPGs. We also demonstrate how such symbolic methods can help identify the universal principles governing both simple and complex dynamics, and chaotic structure in various Lorenz-like systems, their key self-similar organizing structures in 2D parameter space, as well as detailed computational reconstructions of 3D bifurcation surfaces.

### What is a Coxeter group, and why is a Coxeter group?

Series
Algebra Student Seminar
Time
Friday, October 14, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tong JinGeorgia Institute of Technology

A Coxeter group is a (not necessarily finite) group given by certain types of generators and relations. Examples of finite Coxeter groups include dihedral groups, symmetric groups, and reflection groups. They play an important role in various areas. In this talk, I will discuss why I am interested in Coxeter groups from a combinatorial perspective - the geometric concepts associated with the finite Coxeter groups form the language of Coxeter matroids, which are generalizations of ordinary matroids. In particular, finite Coxeter groups are related to Coxeter matroids in the same way as symmetric groups are related to ordinary matroids. The main reference for this talk is Chapter 5 of Borovik-Gelfand-White's book Coxeter Matroids. I will only assume basic group theory, but not familiarity with matroids.

### Spectral Properties of Periodic Elastic Beam Hamiltonians on Hexagonal Lattices

Series
Math Physics Seminar
Time
Thursday, October 13, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Burak HatinogluSchool of Mathematics, Georgia Tech

Elastic beam Hamiltonians on single-layer graphs are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. In this talk, I will first consider spectral properties of this Hamiltonian with periodic potentials on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a joint work with Mahmood Ettehad (University of Minnesota),https://arxiv.org/pdf/2110.05466.pdf.

### Learning to Solve Hard Minimal Problems

Series
Colloquia
Time
Thursday, October 13, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton LeykinGeorgia Tech

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others. The talk is suitable for both professors and students.)