Seminars and Colloquia by Series

Advancements in Persistence Solutions for Functional Perturbed Uniformly Hyperbolic Trajectories: Insights into Relativistic Charged Particle Motion

Series
Math Physics Seminar
Time
Wednesday, April 3, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Joan Gimeno – Universitat de Barcelona –

Please Note: Available online at: https://gatech.zoom.us/j/98258240051

We develop a method to construct solutions of some (retarded or advanced) equations. A prime example could be the motion of point charges interacting via the fully relativistic Lienard-Wiechert potentials (as suggested by J.A. Wheeler and R.P. Feynman in the 1940's). These are retarded equations, but the delay depends implicitly on the trajectory. We assume that the equations considered are formally close to an ODE and that the ODE admits hyperbolic solutions (that is, perturbations transversal to trajectory grow exponentially either in the past or in the future) and we show that there are solutions of the functional equation close to the hyperbolic solutions of the ODE. The method of proof does not require to formulate the delayed problem as an evolution for a class of initial data. The main result is formulated in an "a-posteriori" format and allows to show that solutions obtained by non-rigorous approximations are close (in some precise sense) to true solutions. In the electrodynamics (or gravitational) case, this allows to compare the hyperbolic solutions of several post-newtonian approximations or numerical approximations with the solutions of the Lienard-Weichert interaction. This is a joint work with R. de la Llave and J. Yang.

Spectrahedral Geometry of Graph Sparsifiers (Catherine Babecki, Caltech)

Series
Graph Theory Seminar
Time
Tuesday, April 2, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Catherine Babecki – California Institute of Technology –
We propose an approach to graph sparsification based on the idea of preserving the smallest k eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph G that have the same first k eigenvalues (and eigenvectors) as G is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of k. Various families of graphs illustrate our construction.

Ribbon disks for the square knot

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Alex Zupan – University of Nebraska - Lincoln

A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4. In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy. Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4. Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy. This is joint work with Jeffrey Meier.

A Staircase Proof for Contact Non-Squeezing

Series
Geometry Topology Seminar
Time
Monday, April 1, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
Georgia Tech
Speaker
Lisa Traynor – Bryn Mawr College

Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.

Accelerating Molecular Discovery with Machine Learning: A Geometric, Sampling and Optimization Perspective

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 1, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Yuanqi Du – Cornell University

Please Note: Speaker will present in person. Bio: Yuanqi Du is a PhD student at the Department of Computer Science, Cornell University studying AI and its intersection with Scientific Discovery advised by Prof. Carla P. Gomes. His research interests include Geometric Deep Learning, Probabilistic Machine Learning, Sampling, Optimization, and AI for Science (with a focus on molecular discovery). Aside from his research, he is passionate about education and community building. He leads the organization of a series of events such as the Learning on Graphs conference and AI for Science, Probabilistic Machine Learning workshops at ML conferences and an educational initiative (AI for Science101) to bridge the AI and Science community.

Recent advancements in machine learning have paved the way for groundbreaking opportunities in the realm of molecular discovery. At the forefront of this evolution are improved computational tools with proper inductive biases and efficient optimization. In this talk, I will delve into our efforts around these themes from a geometry, sampling and optimization perspective. I will first introduce how to encode symmetries in the design of neural networks and the balance of expressiveness and computational efficiency. Next, I will discuss how generative models enable a wide range of design and optimization tasks in molecular discovery. In the third part, I will talk about how the advancements in stochastic optimal control, sampling and optimal transport can be applied to find transition states in chemical reactions.

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 Meléndez – University 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 Cai – Georgia 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 Vig – University 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 Nguyen – Princeton 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 Brarda – Emory 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.

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