Seminars and Colloquia by Series

On two Notions of Flag Positivity

Series
Algebra Seminar
Time
Monday, March 31, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jonathan BoretskyCentre de Recherches Mathématiques, Montreal

Please Note: There will be a preseminar from 10:55 to 11:15 in the morning in Skiles 005.

The totally positive flag variety of rank r, defined by Lusztig, can be described as the set of rank r flags of real linear subspaces which can be represented by a matrix whose minors are all positive. For flag varieties of consecutive rank, this equals the subset of the flag variety with positive Plücker coordinates, yielding a straightforward condition to determine whether a flag is totally positive. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the totally positive Grassmannian equals the subset of the Grassmannian with positive Plücker coordinates. We discuss the "tropicalization" of this result, relating the nonnegative tropical flag variety to the nonnegative Dressian, a space parameterizing the regular subdivisions of flag positroid polytopes into flag positroid polytopes. Many results can be generalized to flag varieties of types B and C. This talk is primarily based on joint work with Chris Eur and Lauren Williams and joint work with Grant Barkley, Chris Eur and Johnny Gao.

Universality for graphs of bounded degeneracy

Series
Combinatorics Seminar
Time
Friday, March 28, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anita LiebenauUNSW Sydney

What is the smallest number of edges that a graph can have if it contains all $D$-degenerate graphs on $n$ vertices as subgraphs? A counting argument shows that this number is at least of order $n^{2−1/D}$, assuming n is large enough. We show that this is tight up to a polylogarithmic factor.

Joint work with Peter Allen and Julia Böttcher.

Randomized Iterative Sketch-and-Project Methods as Efficient Large-Scale Linear Solvers

Series
Stochastics Seminar
Time
Thursday, March 27, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Elizaveta RebrovaPrinceton

Randomized Kaczmarz methods — popular special case of the sketch-and-project optimization framework — solve linear systems through iterative projections onto randomly selected equations, resulting in exponential expected convergence via cheap, local updates. While known to be effective in highly overdetermined problems or under the restricted data access, identifying generic scenarios where these methods are advantageous compared to classical Krylov subspace solvers (e.g., Conjugate Gradient, LSQR, GMRES) remained open. In this talk, I will present our recent results demonstrating that properly designed randomized Kaczmarz (sketch-and-project) methods can outperform Krylov methods for both square and rectangular systems complexity-wise. In addition, they are particularly advantageous for approximately low-rank systems common in machine learning (e.g., kernel matrices, signal-plus-noise models) as they quickly capture the large outlying singular values of the linear system. Our approach combines novel spectral analysis of randomly sketched projection matrices with classical numerical analysis techniques, such as including momentum, adaptive regularization, and memoization.

Local-to-global in thin orbits

Series
School of Mathematics Colloquium
Time
Thursday, March 27, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kate StangeUniversity of Colorado, Boulder

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

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 ReidGeorgia 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)

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 GoldenshlugerUniversity 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.

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 BoothGeorgia 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 DingGeorgia 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 KayPacific 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 ZhangGeorgia 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. 

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