## Seminars and Colloquia by Series

### Bias-Variance Tradeoffs in Joint Spectral Embeddings

Series
Stochastics Seminar
Time
Thursday, November 5, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/974631214
Speaker
Daniel SussmanBoston University

We consider the ramifications of utilizing biased latent position estimates in subsequent statistical analysis in exchange for sizable variance reductions in finite networks. We establish an explicit bias-variance tradeoff for latent position estimates produced by the omnibus embedding in the presence of heterogeneous network data. We reveal an analytic bias expression, derive a uniform concentration bound on the residual term, and prove a central limit theorem characterizing the distributional properties of these estimates.

Link to the BlueJeans meeting https://bluejeans.com/974631214

### Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Series
Geometry Topology Student Seminar
Time
Wednesday, November 4, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
online: https://bluejeans.com/872588027
Speaker
Surena HozooriGeorgia Tech

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss some uniqueness results for the underlying (bi-) contact structure for an Anosov flow, and/or a characterization of Anosovity based on Reeb flows.

### Post-grazing dynamics of a vibro-impacting energy generator

Series
SIAM Student Seminar
Time
Tuesday, November 3, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Online at https://bluejeans.com/893955256
Speaker
Larissa SerdukovaMathematics &amp; Statistics Department, University of Reading

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

### Forbidden traces in hypergraphs

Series
Graph Theory Seminar
Time
Tuesday, November 3, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
Ruth LuoUniversity of California, San Diego

Let $F$ be a graph. We say a hypergraph $H$ is a trace of $F$ if there exists a bijection $\phi$ from the edges of $F$ to the hyperedges of $H$ such that for all $xy \in E(F)$, $\phi(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in an $r$-uniform hypergraph that does not contain a trace of $F$. We also obtain better bounds in the case $F = K_{2,t}$. This is joint work with Zoltán Füredi and Sam Spiro.

### Theoretical guarantees of machine learning methods for statistical sampling and PDEs in high dimensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Yulong LuUniversity of Massachusetts Amherst

Neural network-based machine learning methods, inlcuding the most notably deep learning have achieved extraordinary successes in numerious  fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.

In this talk, I will demonstrate the power of  neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss a generalization error bound of the Deep Ritz Method for solving high dimensional elliptic problems. For both problems,  our theoretical results show that neural networks-based methods  can overcome the curse of dimensionality.

### What is tropical convexity?

Series
Time
Monday, November 2, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/759112674
Speaker
Cvetelina HillGeorgia Tech

We say that a set is convex if for any two points in the set, the straight line segment connecting them is also contained in the set.  For example, a triangle, a square, a cube, a ball are all convex sets. We typically speak of convex sets in Euclidean space with the ordinary addition and multiplication operations. What happens if we replace addition with taking the minimum between two elements, and multiplication with ordinary addition? These are the tropical arithmetic operations and using these we can define tropical convexity. What does it mean for a set to be tropically convex? What does a tropical triangle look like? In this talk we will answer these questions and explore how ordinary and tropical convexity interact.

### Knots and Links in overtwisted contact structures

Series
Geometry Topology Seminar
Time
Monday, November 2, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
on line
Speaker
Rima ChatterjeeLSU

Please Note: Knots/links associated to overtwisted contact structures have been less explored. There are two types of knots/links in overtwisted contact manifolds, namely loose and non-loose. In this talk, I will start with an overview of these knots and then discuss some of my recent work involving these knots and links. Specifically, I will talk about a coarse classification result of loose, null-homologous Legendrian and transverse links . Next relating them with open book decompositions, I will show that coarse equivalence class of loose null-homologous Legendrian links has support genus zero. I will end with some interesting open questions.

### $k$-planar crossing numbers and the midrange crossing constant

Series
ACO Student Seminar
Time
Friday, October 30, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Dr. Zhiyu WangMath, Georgia Tech

The crossing number of a graph is the minimum number of crossings it can be drawn in a plane. Let $\kappa(n, m)$ be the minimum crossing number of graphs with $n$ vertices and (at least) $m$ edges. Erd\H{o}s and Guy conjectured and Pach, Spencer and T\'oth proved that for any $m = m(n)$ satisfying $n \ll m \ll n^2$, the quatity $\ds\lim_{n \to \infty} \frac{\kappa(n,m) n^2}{m^3}$ exists and is positive. The $k$-planar crossing number of a graph is the minimum crossing number obtained when we partition the edges of the graph into $k$ subgraphs and draw them in $k$ planes. Using designs and a probabilistic algorithm, the guaranteed factor of improvement $\alpha_k$ between the $k$-planar and regular crossing number is $\frac{1}{k^2} (1 + o(1))$, while if we restrict our attention to biplanar graphs, this constant is $\beta_k = \frac{1}{k^2}$ exactly. The lower bound proofs require the existence of a midrange crossing constant. Motivated by this, we show that the midrange crossing constant exists for all graph classes (including bipartite graphs) that satisfy certain mild conditions. The regular midrange crossing constant was shown to be is at most $\frac{8}{9\pi^2}$; we present a probabilistic construction that also shows this bound.

### Explorations in high-dimensional convexity

Series
Research Horizons Seminar
Time
Friday, October 30, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Galyna LivshytsGeorgia Tech

We will discuss a few beautiful questions in high-dimensional convexity, and path their connections to areas such as Analysis, Probability Theory and Differential Geometry. I shall mention some of my recent results too, in particular a new inequality about convex sets in high dimensions. I will describe its relations to one of the difficult problems in the area.

### An Introduction to Gabor Analysis

Series
School of Mathematics Colloquium
Time
Thursday, October 29, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://us02web.zoom.us/j/89107379948
Speaker
Kasso OkoudjouTufts University

In 1946, Dennis Gabor claimed that any Lebesgue square-integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian.  Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have lead to an impressive body of work in harmonic analysis. For example, Gabor analysis, which originated from Gabor's claim, is concerned with both the theory and the applications of the approximation properties of sets of time and frequency shifts of a given function. It re-emerged with the advent of wavelets at the end of the last century and is now at the intersection of many fields of mathematics, applied mathematics, engineering, and science. In this talk, I will introduce the fundamentals of the theory highlighting some applications and open problems.