Seminars and Colloquia Schedule

Welschinger Signs and the Wronski Map (New conjectured reality)

Series
Algebra Seminar
Time
Monday, March 25, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Frank SottileTexas A&M University

There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30 am in Skiles 005.

A general real rational plane curve C of degree d has 3(d-2) flexes and (d-1)(d-2)/2 complex double points. Those double points lying in RP^2 are either nodes or solitary points. The Welschinger sign of C is (-1)^s, where s is the number of solitary points. When all flexes of C are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to C we may associate the local degree of the Wronskii map, which is also 1 or -1. My talk will discuss work with Brazelton and McKean towards a possible conjecture that these two signs associated to C agree, and the challenges to gathering evidence for this.

Function approximation with one-bit Bernstein polynomials and one-bit neural networks

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 25, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Weilin LiCity College of New York
The celebrated universal approximation theorems for neural networks typically state that every sufficiently nice function can be arbitrarily well approximated by a neural network with carefully chosen real parameters. With the emergence of large neural networks and a desire to use them on low power devices, there has been increased interest in neural network quantization (i.e., the act of replacing its real parameters with ones from a much smaller finite set). In this talk, we ask whether it is even possible to quantize neural networks without sacrificing their approximation power, especially in the extreme one-bit {+1,-1} case? We present several naive quantization strategies that yield universal approximation theorems by quantized neural networks, and discuss their advantages/disadvantages. From there, we offer an alternative approach based on Bernstein polynomials and show that {+1,-1} linear combinations of multivariate Bernstein polynomials can efficiently approximate smooth functions. This strategy can be implemented by means of a one-bit neural network and computed from point samples/queries. Joint work with Sinan Gunturk.

 

Gradient Elastic Surfaces and the Elimination of Fracture Singularities in 3D Bodies

Series
PDE Seminar
Time
Tuesday, March 26, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Casey Rodriguez University of North Carolina at Chapel Hill

In this talk, we give an overview of recent work in gradient elasticity.  We first give a friendly introduction to gradient elasticity—a mathematical framework for understanding three-dimensional bodies that do not dissipate a form of energy during deformation. Compared to classical elasticity theory, gradient elasticity incorporates higher spatial derivatives that encode certain microstructural information and become significant at small spatial scales. We then discuss a recently introduced theory of three-dimensional Green-elastic bodies containing gradient elastic material boundary surfaces. We then indicate how the resulting model successfully eliminates pathological singularities inherent in classical linear elastic fracture mechanics, presenting a new and geometric alternative theory of fracture.

Matroids on graphs (Daniel Bernstein, Tulane)

Series
Graph Theory Seminar
Time
Tuesday, March 26, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel BernsteinTulane University

Many problems in rigidity theory and matrix completion boil down to finding a nice combinatorial description of some matroid supported on the edge set of a complete (bipartite) graph. In this talk, I will give many such examples. My goal is to convince you that a general theory of matroids supported on graphs is needed and to give you a sense of what that could look like.

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].

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.

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.

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.

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.

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.

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.

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.