Seminars and Colloquia by Series

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 Klusowski – Princeton 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 Sottile – Texas 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 Shokrieh – University 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 Wang – Georgia 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 Sun – Georgia 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:2309.12136, arXiv:2310.13693].

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 Bernstein – Tulane 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.

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.

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 Li – City 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.

 

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 Sottile – Texas A&M University

Please Note: 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.

Enhanced diffusion for time-periodic alternating shear flows

Series
CDSNS Colloquium
Time
Friday, March 15, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Kyle Liss – Duke –

The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions relaxing to equilibrium at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced diffusion. In this talk, I will discuss a joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of a divergence-free velocity field on the two-dimensional torus that results in optimal enhanced diffusion.  The flow consists of time-periodic, alternating piece-wise linear shear flows. The proof is based on the probabilistic representation formula for the advection-diffusion equation, a discrete time approximation, and ideas from hyperbolic dynamics.

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