Seminars and Colloquia by Series

Statistical problems for Smoluchowski processes

Series
Stochastics Seminar
Time
Tuesday, March 25, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Goldenshluger – University of Haifa –

Suppose that particles are randomly distributed in $R^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. The goal is to infer on particle displacement process from such count data. We discuss probabilistic properties of the Smoluchowski processes and consider related statistical problems for two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In these settings we develop estimators with provable accuracy guarantees.

Domain branching in ferromagnets: elliptic regularity in action

Series
PDE Seminar
Time
Tuesday, March 25, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tobias Reid – Georgia Tech –

The Landau-Lifshitz model of micromagnetics is a powerful continuum theory that describes the occurrence of magnetization patterns in a ferromagnetic body. In this talk I will discuss  domain branching in strongly uniaxial materials resulting from the competition between a short-range attractive interaction (surface energy), a long-range repulsive interaction (stray field energy), and a non-convex constraint coming from the strong uniaxiality. 

 

On a mathematical level, we use modern tools from elliptic regularity theory, convex duality, ideas from statistical physics, and fine geometric constructions to describe the occurrence of domain branching through local energy estimates at the boundary of the sample (where the branching is infinitely fine). Our approach provides a robust framework for other domain branching problems and is the first step to prove self-similarity in a statistical sense.

 

(Joint work with Carlos Román)

Automorphisms of the Smooth Fine Curve Graph

Series
Dissertation Defense
Time
Tuesday, March 25, 2025 - 13:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Katherine Booth – Georgia Tech –

The smooth fine curve graph of a surface provides a combinatorial perspective to study the action of maps on smooth curves in the surface. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves.

Committee:

  • Dan Margalit (advisor)
  • John Etnyre
  • Jen Hom
  • Igor Belegradek 
  • Michael Wolf

Torsor structures on spanning quasi-trees of ribbon graphs

Series
Algebra Seminar
Time
Monday, March 24, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Changxin Ding – Georgia Tech

Previous work of Chan-Church-Grochow and Baker-Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$.

Our proof shows, more generally, that the family of "BBY torsors'' constructed by Backman-Baker-Yuen and later generalized by Ding admit natural generalizations to regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids to a natural combinatorial problem about graphs. 

 Joint work with Matt Baker and Donggyu Kim. 

Information Theory in Scientific Domains

Series
Combinatorics Seminar
Time
Friday, March 14, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bill Kay – Pacific Northwest National Labs –

In this talk, the speaker will present three applications of information theory in applied spaces. No background on information theory, hypergraphs, or RF signals analysis will be assumed. Bill Kay is a pure mathematician in combinatorics by training who now lives in an applied space at Pacific Northwest National Laboratory.

On the Low-Complexity Critical Points of Two-Layer Neural Networks

Series
SIAM Student Seminar
Time
Friday, March 14, 2025 - 11:00 for
Location
Skiles 006
Speaker
Leyang Zhang – Georgia Tech –

Abstract: Critical points significantly affect the behavior of gradient-based dynamics. Numerous works have been done for global minima of neural networks. Thus, the recent work characterizes non-global critical points. With the idea that gradient-based methods of neural networks favor “simple models”, this work focuses on the set of low-complexity critical points, i.e., those representing underparameterized network models. Specifically, we investigate: i) the existence and ii) geometry of such sets, iii) the output functions they represent, iv) saddles in them. The talk will discuss these results based on a simple example. The general theorems will also be included. No specific knowledge in neural networks is required. 

Matrix superconcentration inequalities

Series
Stochastics Seminar
Time
Thursday, March 13, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tatiana Brailovskaya – Duke University –

One way to understand the concentration of the norm of a random matrix X with Gaussian entries is to apply a standard concentration inequality, such as the one for Lipschitz functions of i.i.d. standard Gaussian variables, which yields subgaussian tail bounds on the norm of X. However, as was shown by Tracy and Widom in 1990s, when the entries of X are i.i.d. the norm of X exhibits even sharper concentration. The phenomenon of a function of many i.i.d. variables having strictly smaller tails than those predicted by classical concentration inequalities is sometimes referred to as «superconcentration», a term originally dubbed by Chatterjee. I will discuss novel results that can be interpreted as superconcentration inequalities for the norm of X, where X is a Gaussian random matrix with independent entries and an arbitrary variance profile. We can also view our results as a nonhomogeneous extension of Tracy-Widom-type upper tail estimates for the norm of X.

Theory of valuations and geometric inequalities

Series
School of Mathematics Colloquium
Time
Thursday, March 13, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Semyon Alesker – Tel Aviv University and Kent State University –

Valuations are finitely additive measures on convex compact sets. In the last two decades a number of structures (e.g. product and convolution)  with non-trivial properties were discovered on the space of valuations. One such recently discovered property is an analogue of the classical Hodge-Riemann bilinear relations known in algebraic/Kaehler geometry. In special cases, they lead to new inequalities for convex bodies, to be discussed in the talk. No familiarity with valuations theory and algebraic/Kaehler geometry is assumed.

VC dimension and point configurations in fractals

Series
Analysis Seminar
Time
Wednesday, March 12, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alexander McDonald – Kennesaw State

An important class of problems at the intersection of harmonic analysis and geometric measure theory asks how large the Hausdorff dimension of a set must be to ensure that it contains certain types of geometric point configurations. We apply these tools to study configurations associated to the problem of bounding the VC-dimension of a naturally arising class of indicator functions on fractal sets.

Tight minimum colored degree condition for rainbow connectivity

Series
Graph Theory Seminar
Time
Tuesday, March 11, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Xiaofan Yuan – Arizona State University –

Let $G = (V,E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$.  In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$ the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. This is joint work with Andrzej Czygrinow.

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