## Seminars and Colloquia by Series

### Hypertrees

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

A finite connected acyclic graph is called a tree. Both properties - connectivity and being acyclic - make very good sense in higher dimensions as well. This has led Gil Kalai (1983) to define the notion of a $d$-dimensional hypertree for $d > 1$. The study of hypertrees is an exciting area of research, and I will try to give you a taste of the many open questions and what we know (and do not know) about them. No specific prior background is assumed.

The talk is based on several papers. The list of coauthors on these papers includes Roy Meshulam, Mishael Rosenthal, Yuval Peled, Lior Aronshtam, Tomsz Luczak, Amir Dahari, Ilan Newman and Yuri Rabinovich.

### A Higher-Dimensional Sandpile Map (note the unusual time/day)

Series
Combinatorics Seminar
Time
Wednesday, September 30, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Alex McdonoughBrown University

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

### A Higher-Dimensional Sandpile Map

Series
Algebra Seminar
Time
Wednesday, September 30, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Alex McdonoughBrown University

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

### Exotic behavior of manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, September 30, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Anubhav MukherjeeGeorgia Tech

Poincare Conjecture, undoubtedly, is the most influential and challenging problem in the world of Geometry and Topology. Over a century, it has left it’s mark on developing the rich theory around it. In this talk I will give a brief history of the development of Topology and then I will focus on the Exotic behavior of manifolds. In the last part of the talk, I will concentrate more on the theory of 4-manifolds.

### Number of Hamiltonian cycles in planar triangulations

Series
Graph Theory Working Seminar
Time
Tuesday, September 29, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
Xiaonan LiuGeorgia Institute of Technology

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices, then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/\log_2 n)$ separating 4-cycles then $G$ has $\Omega(n^2)$ Hamiltonian cycles, and if $\delta(G) \ge 5$ then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture.

### Balian-Low theorems for subspaces

Series
Analysis Seminar
Time
Tuesday, September 29, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
online seminar
Speaker
The Balian-Low theorem is a classical result in time-frequency analysis that describes a trade off between the basis properties of a Gabor system and the smoothness and decay of the Gabor window.
In particular a Gabor system with well localized window cannot be a Riesz basis for the space of finite energy signals.
We explore a few generalizations of this fact in the setting of Riesz bases for subspaces of L^2 and we show that the Gabor space being invariant under additional time-frequency shifts is incompatible with two different notions of smoothness and decay for the Gabor window.

### Off the rails: Train tracks on surfaces

Series
Time
Monday, September 28, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Marissa LovingGeorgia Tech

Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at their computer's screen a bit at times.

### SL3 Skein Algebras of Surfaces by Vijay Higgins

Series
Geometry Topology Seminar
Time
Monday, September 28, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Virtual
Speaker
Vijay HigginsUC Santa Barbara

The SL2 skein algebra of a surface is built from diagrams of curves on the surface. To multiply two diagrams, we draw one diagram on top of the other and then resolve the crossings with the Kauffman bracket. If we replace SL2 with another quantum group, we replace curves by embedded graphs on the surface. Recently, Thang Le showed that the SL2 skein algebra has a nice decomposition into simpler algebras whenever the surface has an ideal triangulation. This triangular decomposition is a powerful tool and should help us to study other skein algebras if we are able to show that the necessary ingredients exist. In this talk, I will explain what these ingredients are and how to find them for the SL3 skein algebra of trivalent webs on a surface.

8.3.3

### Minimal problems in 3D reconstruction

Series
ACO Student Seminar
Time
Friday, September 25, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/264244877/0166
Speaker
Timothy DuffMath, Georgia Tech

I describe my ongoing work using tools from computational and combinatorial algebraic geometry to classify minimal problems and identify which can be solved efficiently. I will not assume any background in algebraic geometry or computer vision.

Structure-from-motion algorithms reconstruct a 3D scene from many images, often by matching features (such as point and lines) between the images. Matchings lead to constraints, resulting in a nonlinear system of polynomial equations that recovers the 3D geometry. Since many matches are outliers, these methods are used in an iterative framework for robust estimation called RANSAC (RAndom SAmpling And Consensus), whose efficiency hinges on using a small number of correspondences in each iteration. As a result, there is a big focus on constructing polynomial solvers for these "minimal problems" that run as fast as possible. Our work classifies these problems in cases of practical interest (calibrated cameras, complete and partial visibility.) Moreover, we identify candidates for practical use, as quantified by "algebraic complexity measures" (degree, Galois group.)

joint w/ Anton Leykin, Kathlen Kohn, Tomas Pajdla arxiv1903.10008 arxiv2003.05015+ Viktor Korotynskiy, TP, and Margaret Regan (ongoing.)

### Taming the randomness of chaotic systems

Series
Research Horizons Seminar
Time
Friday, September 25, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Alex BlumenthalGeorgia Tech
All around us in the physical world are systems which evolve in chaotic, seemingly random ways: fire, smoke, turbulent fluids, the flow of gas around us. Over the last ~60 years, mathematicians have made tremendous progress in understanding these processes and how chaotic behavior can emerge and, remarkably, the extent to which chaotic systems emulate probabilistic randomness. This talk is a brief introduction to these ideas, with an emphasis on examples and pretty pictures.