Seminars and Colloquia by Series

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 Anderson – Georgia 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
Graduate Student Colloquium
Time
Friday, October 28, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Christina Giannitsi – Georgia Tech

 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)

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

Series
Time
Friday, October 28, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Maciej Capinski – AGH University of Science and Technology

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

In this talk we will discuss a shooting method designed for solving two point boundary value problems in a setting where a system has integrals of motion. We will show how it can be applied to obtain certain families of orbits in the circular restricted three body problem. These include transverse ejection/collisions from one primary body to the other, families of periodic orbits, orbits passing through collision, and orbits connecting fixed points to ejections or collisions.

This is joint work with Shane Kepley and Jason Mireles James.

An Introductory Proof of the Compactness Theorem and Some Applications

Series
Algebra Student Seminar
Time
Friday, October 28, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Griffin Edwards – Georgia Institute Of Technology

We will introduce the foundations of model theory, by defining languages, models, and theories. Then we will look at a couple proofs of the compactness theorem, state Gödel's completeness theorem, and prove that any planar graph is four colorable. Expect a lot of examples, and I hope everyone comes away understanding the foundations of this wonderful theory.

Persistence of periodic orbits in functional perturbations of an ODE

Series
Math Physics Seminar
Time
Thursday, October 27, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Joan Gimeno – Universitat de Barcelona & Georgia Institute of Technology –

With very minor assumptions, I show that periodic orbits in
an ODE can persist under (singular) perturbations of including a delay
term.  These perturbations change the phase space from finite to
infinite dimensions. The results apply to electrodynamics and give new
approaches to handle state-dependent, small, nested, and distributed
delays.

I will discuss and explain some motivations, the new methods, sketches
of the proofs, and possible applications. I will end the talk giving
some ideas of work in progress and possible future works.

Ballistic Annihilation

Series
Stochastics Seminar
Time
Thursday, October 27, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matthew Junge – Baruch College, CUNY –

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

Linear and nonlinear stability of shear flows and vortices

Series
Colloquia
Time
Thursday, October 27, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online： Zoom link: https://gatech.zoom.us/j/96410391996?pwd=VkQvcUdoREtsbUJPNVFTbzdKaC9TQT09
Speaker
Alexandru Ionescu – Princeton University –

I will talk about some recent work on the stability problem of shear flows and vortices as solutions of the Euler equations in 2D.  Our results include nonlinear stability theorems for monotonic shear  flows and point vortices, as well as linear stability theorems for more general flows. This is joint work with Hao Jia.

The cohomological dimension of the terms of the Johnson filtration

Series
Geometry Topology Student Seminar
Time
Wednesday, October 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Minahan – Georgia Tech –

Abstract: How big is a group?  One possible notion of the size of the group is the cohomological dimension, which is the largest n for which a group G can have non—trivial cohomology in degree n, possibly with twisted coefficients.  Following the work of Bestvina, Bux and Margalit, we compute the cohomological dimension of the terms Johnson filtration of a closed surface.  No background is required for this talk.

Normalizable frames

Series
Analysis Seminar
Time
Wednesday, October 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pu-Ting Yu – Georgia Tech –

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a Bessel sequence or a frame for $H$ which does not contain any zero elements. We say that $\{x_n\}$ is a normalizable Bessel sequence or normalizable frame if the normalized sequence $\{x_n/||x_n||\}$ remains a Bessel sequence or frame. In this talk, we will present characterizations of normalizable and non-normalizable frames . In particular, we prove that normalizable frames can only have two formulations.  Perturbation theorems tailored for normalizable frames will be also presented. Finally, we will talk about some open questions related to the normalizable frames.

Dyadic Matroids with Spanning Cliques

Series
Graph Theory Seminar
Time
Tuesday, October 25, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin Grace – Vanderbilt University –

The Matroid Minors Project of Geelen, Gerards, and Whittle describes the structure of minor-closed classes of matroids representable by a matrix over a fixed finite field. To use these results to study specific classes, it is important to study the matroids in the class containing spanning cliques. A spanning clique of a matroid M is a complete-graphic restriction of M with the same rank as M.

 

In this talk, we will describe the structure of dyadic matroids with spanning cliques. The dyadic matroids are those matroids that can be represented by a real matrix each of whose nonzero subdeterminants is a power of 2, up to a sign. A subclass of the dyadic matroids is the signed-graphic matroids. In the class of signed-graphic matroids, the entries of the matrix are determined by a signed graph. Our result is that dyadic matroids with spanning cliques are signed-graphic matroids and a few exceptional cases.

 

The main results in this talk will come from joint work with Ben Clark, James Oxley, and Stefan van Zwam. This talk will include a brief introduction to matroids.

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