Seminars and Colloquia by Series

q-Chromatic Polynomials

Series
Algebra Seminar
Time
Monday, April 1, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrés R. Vindas MeléndezUniversity of California, Berkeley
We introduce and study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely,
\[\chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) },\] where $\lambda \in \mathbb{Z}^V$ is a fixed linear form.
Via work of Chapoton (2016) on $q$-Ehrhart polynomials, $\chi_G^\lambda(q,n)$ turns out to be a polynomial in the $q$-integer $[n]_q$, with coefficients that are rational functions in $q$.
Additionally, we prove structural results for $\chi_G^\lambda(q,n)$ and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes.
We offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of $P$-partitions for graphs.
This is joint work with Esme Bajo and Matthias Beck.

Topics in Toric and Tropical Geometry: Positivity and Completion

Series
Dissertation Defense
Time
Monday, April 1, 2024 - 11:30 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
May CaiGeorgia Institute of Technology

This defense will also be on zoom at: https://gatech.zoom.us/j/99428720697

In this defense we describe three topics in tropical and toric positivity and completion. In the first part, we describe the finite completability of a partial point to a log-linear statistical model: a toric variety restricted to the probability simplex. We show when a generic point in some projection of a log-linear model has finite preimage, and the exact number of preimages in such a case. In the second part, we describe the tropical variety of symmetric tropical rank 2 matrices. We give a description of the tropical variety as a coarsening of the simplicial complex of a type of bicolored trees, and show that the tropical variety is shellable. Finally, we discuss two tropical notions of positivity, and give results on the positive part of certain tropical determinantal varieties.

Committee:

Josephine Yu, Georgia Institute of Technology (Advisor)
Matt Baker, Georgia Institute of Technology
Greg Blekherman, Georgia Institute of Technology,
Kaie Kubjas, Aalto University
Anton Leykin, Georgia Institute of Technology

Thesis draft:
Link

Silent geodesics and cancellations in the wave trace

Series
CDSNS Colloquium
Time
Friday, March 29, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Amir VigUniversity of Michigan

Can you hear the shape of a drum? A classical inverse problem in mathematical physics is to determine the shape of a membrane from the resonant frequencies at which it vibrates. This problem is very much still open for smooth, strictly convex planar domains and one tool in that is often used in this context is the wave trace, which contains information on the asymptotic distribution of eigenvalues of the Laplacian on a Riemannian manifold. It is well known that the singular support of the wave trace is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. In a recent work together with Vadim Kaloshin and Illya Koval, we construct large families of domains for which there are multiple geodesics of a given length, having different Maslov indices, which interfere destructively and cancel arbitrarily many orders in the wave trace. This shows that there are potential obstacles in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned.

Erdős–Hajnal and VC-dimension (Tung Nguyen, Princeton)

Series
Combinatorics Seminar
Time
Friday, March 29, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Tung NguyenPrinceton University

A hereditary class $\mathcal C$ of graphs is said to have the Erdős–Hajnal property if every $n$-vertex graph in $\mathcal C$ has a clique or stable set of size at least $n^c$. We discuss a proof of a conjecture of Chernikov–Starchenko–Thomas and Fox–Pach–Suk that for every $d\ge1$, the class of graphs of VC-dimension at most $d$ has the Erdős–Hajnal property. Joint work with Alex Scott and Paul Seymour.

Efficient hybrid spatial-temporal operator learning

Series
SIAM Student Seminar
Time
Friday, March 29, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Francesco BrardaEmory University

Recent advancements in operator-type neural networks, such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet), have shown promising results in approximating the solutions of spatial-temporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new operator learning framework to address these issues. The proposed paradigm leverages the traditional wisdom from numerical PDE theory and techniques to refine the pipeline of existing operator neural networks. Specifically, the proposed architecture initiates the training for a single or a few epochs for the operator-type neural networks in consideration, concluding with the freezing of the model parameters. The latter are then fed into an error correction scheme: a single parametrized linear spectral layer trained with a convex loss function defined through a reliable functional-type a posteriori error estimator.This design allows the operator neural networks to effectively tackle low-frequency errors, while the added linear layer addresses high-frequency errors. Numerical experiments on a commonly used benchmark of 2D Navier-Stokes equations demonstrate improvements in both computational time and accuracy, compared to existing FNO variants and traditional numerical approaches.

Improving Predictions by Combining Models

Series
Stochastics Seminar
Time
Thursday, March 28, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jason KlusowskiPrinceton University

When performing regression analysis, researchers often face the challenge of selecting the best single model from a range of possibilities. Traditionally, this selection is based on criteria evaluating model goodness-of-fit and complexity, such as Akaike's AIC and Schwartz's BIC, or on the model's performance in predicting new data, assessed through cross-validation techniques. In this talk, I will show that a linear combination of a large number of these possible models can have better predictive accuracy than the best single model among them. Algorithms and theoretical guarantees will be discussed, which involve interesting connections to constrained optimization and shrinkage in statistics.

Galois groups in Enumerative Geometry and Applications

Series
School of Mathematics Colloquium
Time
Thursday, March 28, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Frank SottileTexas A&M University

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.

Local canonical heights and tropical theta functions

Series
Number Theory
Time
Wednesday, March 27, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Farbod ShokriehUniversity of Washington, Seattle

I will describe some connections between arithmetic geometry of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical invariants. Our formula extends a classical computation of local height functions due to Tate (involving Bernoulli polynomials).
Based on ongoing work with Robin de Jong.

Clifford Algebra: A Marvelous Machine Offered By the Devil

Series
Geometry Topology Student Seminar
Time
Wednesday, March 27, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden WangGeorgia Tech

Clifford algebra was first developed to describe Maxwell's equations, but the subject has found applications in quantum mechanics, computer graphics, robotics, and even machine learning, way beyond its original purpose. In topology and geometry, Clifford algebra appears in the proofs of the celebrated Atiyah-Singer Index Theorem and Bott Periodicity; it is fundamental to the understanding of spin structures on Riemannian manifolds. Despite its algebraic nature, it somehow gives us the power to understand and manipulate geometry. What a marvelous machine offered by the devil! In this talk, we will investigate the unreasonable effectiveness of Clifford algebra by exploring its algebraic structure and constructing the Pin and Spin groups. If time permits, we will prove that Spin(p,q) is a double cover of SO(p,q), complementing the belt trick talk of Sean Eli.

Matrix generalization of the cubic Szegő equation

Series
Analysis Seminar
Time
Wednesday, March 27, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruoci SunGeorgia Tech

This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation enjoys a Lax pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136arXiv:2310.13693].

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