Seminars and Colloquia by Series

A Polynomial Method for Counting Colorings of $S$-labeled Graphs

Series
Combinatorics Seminar
Time
Friday, November 17, 2023 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Hemanshu Kaul – Illinois Institute of Technology –

The notion of $S$-labeling, where $S$ is a subset of the symmetric group, is a common generalization of signed $k$-coloring, signed $\mathbb{Z}_k$-coloring, DP (or Correspondence) coloring, group coloring, and coloring of gained graphs that was introduced in 2019 by Jin, Wong, and Zhu.  In this talk we use a well-known theorem of  Alon and F\"{u}redi to present an algebraic technique for bounding the number of colorings of an $S$-labeled graph from below.  While applicable in the broad context of counting colorings of $S$-labeled graphs, we will focus on the case where $S$ is a symmetric group, which corresponds to DP-coloring (or, correspondence coloring) of graphs, and the case where $S$ is a set of linear permutations which is applicable to the coloring of signed graphs, etc.

 

This technique allows us to prove exponential lower bounds on the number of colorings of any $S$-labeling of graphs that satisfy certain sparsity conditions. We apply these to give exponential lower bounds on the number of DP-colorings (and consequently, number of  list colorings, or usual colorings) of families of planar graphs, and on the number of colorings of families of signed (planar) graphs. These lower bounds either improve previously known results, or are first known such results.

This joint work with Samantha Dahlberg and Jeffrey Mudrock.

Controlled SPDEs: Peng’s Maximum Principle and Numerical Methods

Series
SIAM Student Seminar
Time
Friday, November 17, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lukas Wessels – Georgia Tech –

In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First, we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs.

In the second part, we present a numerical algorithm that allows the efficient approximation of optimal controls in the case of stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm.

This talk is based on joint work with Wilhelm Stannat and Alexander Vogler. Talk will also be streamed: https://gatech.zoom.us/j/93808617657?pwd=ME44NWUxbk1NRkhUMzRsK3c0ZGtvQT09

Algebra from Projective Geometry

Series
Algebra Student Seminar
Time
Friday, November 17, 2023 - 10:00 for
Location
Speaker
Griffin Edwards – Georgia Tech

Join us as we define a whole new algebraic structure, starting from the axioms of the projective plane. This seminar will be aimed at students who have never seen this material and will focus on hands-on constructions of classic (and new!) algebraic structures that can arise from a projective plane. The goal of this seminar is to expose you to Desargues's theorem and hopefully even examine non-Desarguesian planes.

"No (Con)way!"

Series
Geometry Topology Student Seminar
Time
Wednesday, November 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Hwang – Georgia Tech –

 This talk is a summary of a summary. We will be going over Jen Hom's 2024 Levi L. Conant Prize Winning Article "Getting a handle on the Conway knot," which discusses Lisa Piccirillo's renowned 2020 paper proving the Conway knot is not slice. In this presentation, we will go over what it means for a knot to be slice, past attempts to classify the Conway knot with knot invariants, and Piccirillo's approach of constructing a knot with the same knot trace as the Conway knot. This talk is designed for all audiences and NO prior knowledge of topology or knot theory is required. Trust me, I'm (k)not a topologist.

On the curved trilinear Hilbert transform

Series
Analysis Seminar
Time
Wednesday, November 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bingyang Hu – Auburn University –

The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator

$$

H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R

$$

is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1

 

The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

 

1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;

 

2). a structural analysis of suitable maximal "joint Fourier coefficients";

 

3). a level set analysis with respect to the time-frequency correlation set. 

 

This is a joint work with my postdoc advisor Victor Lie from Purdue.

 

Onsager's conjecture in 2D

Series
PDE Seminar
Time
Tuesday, November 14, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Razvan-Octavian Radu – Princeton University –

I will begin by describing the ideas involved in the Nash iterative constructions of solutions to the Euler equations. These were introduced by De Lellis and Szekelyhidi (and developed by many authors) in order to tackle the flexible side of the Onsager conjecture. Then, I will describe Isett’s proof of the conjecture in the 3D case, and highlight the simple reason for which the strategy will not work in 2D. Finally, I will describe a construction of non-conservative solutions that works also in 2D (this is joint work with Vikram Giri).

Subgraphs in multipartite graphs

Series
Graph Theory Seminar
Time
Tuesday, November 14, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yi Zhao – Georgia State University –

In 1975 Bollobas, Erdos, and Szemeredi asked the following question: given positive integers $n, t, r$ with $2\le t\le r$, what is the largest minimum degree among all $r$-partite graphs G with parts of size $n$ and which do not contain a copy of $K_t$? The $r=t$ case has attracted a lot of attention and was fully resolved by Haxell and Szabo, and Szabo and Tardos in 2006. In this talk we discuss recent progress on the $r>t$ case and related extremal results on multipartite graphs.

Knot Homology, Fusion Numbers, and Symmetric Unions

Series
Geometry Topology Seminar
Time
Monday, November 13, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 303)
Speaker
Michael Willis – Texas A&M

I will discuss a mixture of results and conjectures related to the Khovanov homology and Knot Floer homology for ribbon knots. We will explore relationships with fusion numbers (a measure of complexity on ribbon disks) and particular families of symmetric unions (ribbon knots given by particular diagrams). This is joint work with Nathan Dunfield, Sherry Gong, Tom Hockenhull, and Marco Marengon.

Effective bounds for Roth's theorem with shifted square common difference

Series
Additional Talks and Lectures
Time
Monday, November 13, 2023 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Ashwin Sah – MIT

Let $S$ be a subset of $1 ,…, N$ avoiding the nontrivial progressions $x, x+y^2-1, x+2(y^2-1)$. We prove that $|S| < N/\log \log \cdots \log(N)$, where we have a fixed constant number of logarithms. This answers a question of Green, and is the first effective polynomial Szemerédi result over the integers where the polynomials involved are not homogeneous of the same degree and the underlying pattern has linear complexity. Joint work with Sarah Peluse and Mehtaab Sawhney.

 

Products of locally conformal symplectic manifolds

Series
Geometry Topology Seminar
Time
Monday, November 13, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 303)
Speaker
Kevin Sackel – UMass Amherst

Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale. In this talk, we define a notion of product for LCS manifolds, in which the underlying manifold of an LCS product is not simply the smooth product of the underlying manifolds, but which nonetheless appears to fill the same role in LCS geometry as the standard symplectic product does in standard symplectic geometry. As a proof of concept, with input from an LCS result of Chantraine and Murphy, we use the LCS product to prove that C^0 small Hamiltonian isotopies have a lower bound on the number of fixed points given by the rank Morse-Novikov homology. This is a natural generalization of the classical symplectic proof of the analogous result by Laudenbach and Sikorav which uses the graph of a Hamiltonian diffeomorphism in the product manifold. These results are joint work in progress with Baptiste Chantraine.

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