Seminars and Colloquia by Series

Half-integral Erdős-Pósa property for non-null S–T paths (Meike Hatzel)

Series
Graph Theory Seminar
Time
Tuesday, October 22, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Meike HatzelInstitute for Basic Science (IBS)

For a group Γ, a Γ-labelled graph is an undirected graph G where every orientation of an edge is assigned an element of Γ so that opposite orientations of the same edge are assigned inverse elements. A path in G is non-null if the product of the labels along the path is not the neutral element of Γ. We prove that for every finite group Γ, non-null S–T paths in Γ-labelled graphs exhibit the half- integral Erdős-Pósa property. More precisely, there is a function f , depending on Γ, such that for every Γ-labelled graph G, subsets of vertices S and T , and integer k, one of the following objects exists:
• a family F consisting of k non-null S–T paths in G such that every vertex of G participates in at most two paths of F; or
• a set X consisting of at most f (k) vertices that meets every non-null S–T path in G.
This in particular proves that in undirected graphs S–T paths of odd length have the half-integral Erdős-Pósa property.
This is joint work with Vera Chekan, Colin Geniet, Marek Sokołowski, Michał T. Seweryn, Michał Pilipczuk, and Marcin Witkowski.

Schrödinger Equation with Coulomb Potential

Series
PDE Seminar
Time
Tuesday, October 22, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ebru ToprakYale University

I will begin by presenting our recent results on the spherically symmetric Coulomb waves. Specifically, we study the evolution operator of H= -\Delta+q/|x| where q>0. Utilizing a distorted Fourier transform adapted to H, we explicitly compute the evolution kernel. A detailed analysis of this kernel reveals that e^itH satisfies an L^1 \to L^{\infty} dispersive estimate with the natural decay rate t^{-3/2}. This work was conducted in collaboration with Adam Black, Bruno Vergara, and Jiahua Zhou. Following this, I will discuss our ongoing research on the nonlinear Schrödinger equation, where we apply the distorted Fourier transform developed for the Coulomb Hamiltonian. This work is being carried out in collaboration with Mengyi Xie.

Small symplectic fillings of Seifert fibered spaces

Series
Geometry Topology Seminar
Time
Monday, October 21, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bülent TosunThe University of Alabama

It is an important and rather difficult problem in low dimensional topology to determine which rational homology 3-spheres bound smooth rational homology 4-balls. This is largely open even in the case of Brieskorn spheres—a special class of Seifert fibered spaces. In this talk, we will focus on symplectic version of this question, and (almost) determine when a small Seifert fibered space admits a symplectic rational homology ball filling. For some small Seifert fibered spaces, we provide explicit and new examples of such fillings, and for most others we provide strong restrictions. In the talk, we will review these concepts and provide further context; give some details of the techniques involved and finally mention a few applications. This will report on recent joint work with J. Etnyre and B. Özbağcı.

Categorifying the Four Color Theorem

Series
Geometry Topology Seminar
Time
Monday, October 21, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Scott BaldridgeLSU

The four color theorem states that each bridgeless trivalent planar graph has a proper 4-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with m colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

In this talk, I show how to categorify the m-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the ``Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded m-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the $E_1$-page of a spectral sequence whose $E_\infty$-page has a basis that can be identified with proper m-face colorings, that is, each successive page of the sequence provides better approximations of m-face colorings than the last. Since it can be shown that the $E_1$-page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)

Density estimation for Gaussian mixture models

Series
Algebra Seminar
Time
Monday, October 21, 2024 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Julia LindbergUT Austin

Density estimation for Gaussian mixture models is a classical problem in statistics that has applications in a variety of disciplines. Two solution techniques are commonly used for this problem: the method of moments and maximum likelihood estimation. This talk will discuss both methods by focusing on the underlying geometry of each problem.

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

Series
CDSNS Colloquium
Time
Friday, October 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Mark van den BoschLeiden University

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

 

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

 

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

Series
CDSNS Colloquium
Time
Friday, October 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Mark van den BoschLeiden University

Please Note: Talk is in-person; zoom link if needed: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

 

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

Regularity for Semialgebraic Hypergraphs and Applications

Series
Combinatorics Seminar
Time
Friday, October 18, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans Hung-Hsun YuPrinceton University

A semialgebraic hypergraph is a hypergraph whose edges can be described by a system of polynomial inequalities. Semialgebraic hypergraphs appear in many problems in discrete geometry. There has been growing interest in semialgebraic hypergraphs since the discovery that they satisfy strong regularity lemmas, where between most parts, the hypergraph is either complete or empty. In this talk, I will talk about an optimal regularity lemma along these lines and several applications. Based on joint work with Jonathan Tidor.

An interesting variational problem related to the Cwikel-Lieb-Rozenblum inequality and its solution

Series
Math Physics Seminar
Time
Friday, October 18, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Clough 280
Speaker
Tobias ReidGeorgia Tech

The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical estimate on the number of bound states for Schrödinger operators. In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our new proof highlights a natural but overlooked connection of the CLR inequality with bounds for maximal Fourier multipliers from harmonic analysis and leads to a variational problem that can be reformulated in terms of a variant of Hadamard’s three-lines lemma. The solution of this variational problem relies on some interesting complex analysis techniques. (Based on joint work with T. Carvalho-Corso, D. Hundertmark, P. Kunstmann, S. Vugalter)

Non-linear mean-field systems in flocking and sampling

Series
Stochastics Seminar
Time
Thursday, October 17, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sayan BanerjeeUNC

Mean-field particle systems are well-understood by now. Typical results involve obtaining a McKean-Vlasov equation for the fluid limit that provides a good approximation for the particle system over compact time intervals. However, when the driving vector field lacks a gradient structure or in the absence of convexity or functional inequalities, the long-time behavior of such systems is far from clear. In this talk, I will discuss two such systems, one arising in the context of flocking and the other in the context of sampling (Stein Variational Gradient Descent), where there is no uniform-in-time control on the discrepancy between the limit and prelimit dynamics. We will explore methods involving Lyapunov functions and weak convergence which shed light on their long-time behavior in the absence of such uniform control.

 

Based on joint works with Amarjit Budhiraja, Dilshad Imon (UNC, Chapel Hill), Krishnakumar Balasubramanian (UC Davis) and Promit Ghosal (UChicago).

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