Seminars and Colloquia Schedule

Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, October 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michelle ChuUniversity of Illinois at Chicago

The smallest volume cusped hyperbolic 3-manifolds, the figure-eight knot complement and its sister, contain many immersed but no embedded closed totally geodesic surfaces. In this talk we discuss the existence or lack thereof of codimension-1 closed embedded totally geodesic submanifolds in minimal volume cusped hyperbolic 4-manifolds. This talk is based on joint work with Alan Reid.

A Few Thoughts on Deep Learning-Based Scientific Computing

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Haizhao YangPurdue University

The remarkable success of deep learning in computer science has evinced potentially great applications of deep learning in computational and applied mathematics. Understanding the mathematical principles of deep learning is crucial to validating and advancing deep learning-based scientific computing. We present a few thoughts on the theoretical foundation of this topic and our methodology for designing efficient solutions of high-dimensional and highly nonlinear partial differential equations, mainly focusing on the approximation and optimization of deep neural networks.

Synchronization of coupled pendulum clocks and metronomes

Series
Undergraduate Seminar
Time
Monday, October 26, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Guillermo GoldszteinGeorgia Tech

In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they tend to synchronize in phase, not antiphase. Here, using a simple model of coupled clocks and metronomes, we calculate the regimes where antiphase and in-phase synchronization are stable. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and a three-time scale asymptotic analysis.

Two results on the interaction energy

Series
Analysis Seminar
Time
Tuesday, October 27, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Speaker
Yao YaoGeorgia Tech


For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. 

I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m>=2, where the threshold is sharp.

Maximum diameter of $k$-colorable graphs

Series
Graph Theory Seminar
Time
Tuesday, October 27, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Éva CzabarkaUniversity of South Carolina

Erdős, Pach, Pollack and Tuza conjectured that for fixed integers $r$, $\delta \ge 2$, for any connected graph $G$ with minimum degree $\delta$ and order $n$:

(i) If $G$ is $K_{2r}$-free and $\delta$ is a multiple of $(r-1)(3r+2)$, then, as $n$ tends to infinity, the diameter of $G$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)} \cdot \frac{n}{\delta} + O(1)$.

(ii) If $G$ is $K_{2r+1}$-free and $\delta$ is a multiple of $3r-1$, then, as $n$ tends to infinity, the diameter of $G$ is at most $\frac{3r-1}{r} \cdot \frac{n}{\delta} + O(1)$.

They created examples that show that the above conjecture, if true, is tight.

No more progress has been reported on this conjecture, except that for $r=2$ in (ii), under a stronger hypothesis ($4$-colorable instead of $K_5$-free), Czabarka, Dankelman and Székely showed that for every connected $4$-colorable graph $G$ of order $n$ and minimum degree $\delta \ge 1$, the diameter of $G$ is at most $\frac{5n}{2\delta} - 1$.

For every $r>1$ and $\delta \ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $\delta$ and diameter $\frac{(6r-5)n}{(2r-1)\delta+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $\delta > 2(r-1)(3r+2)(2r-3)$. We also prove positive results under a stronger hypothesis, $k$-colorability, instead of being $K_{k+1}$-free. We show that the diameter of connected $k$-colorable graphs with minimum degree at least $\delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}+O(1)$, while for $k=3$, it is at most $\frac{57n}{23\delta}+O(1)$.

This is joint work with Inne Singgih and László A. Székely.

Patchworking oriented matroids

Series
Algebra Seminar
Time
Wednesday, October 28, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Marcel CelayaTU Berlin

A classical result on oriented matroids due to Folkman and Lawrence in
1978 states that they are in bijection with pseudosphere arrangements up
to cellular homeomorphism. A more recent result, conjectured by Ardila and
Develin in 2007 and proved by Silke Horn in 2016, states that a similar
result holds for tropical oriented matroids and tropical hyperplane
arrangements. In a joint work with Georg Loho and Chi Ho Yuen, we show how
to unify these two results based on a variant of Viro's patchworking
technique, generalized to complete intersections by Sturmfels, for a
certain class of uniform oriented matroids arising from a product of two
simplices.

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.

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.

$k$-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.