VT Math Colloquium

January 23

Speaker: Ryan Shifler

Host: Leonardo Mihalcea

Title: Geometry, Symmetry, and Counting Curves: A Combinatorial Perspective

Hilbert’s 15th problem calls for a rigorous foundation for Schubert’s calculus, which provides classical rules for counting geometric figures satisfying incidence conditions, such as the number of lines in 3‑space meeting four given lines. After surveying a few examples of “counting with geometry,” we turn to a modern viewpoint using the cohomology ring of the complex projective plane to illustrate how intersection theory organizes such problems and leads naturally to quantum cohomology, where curve counts are built into the product structure itself. We then introduce Goresky–Kottwitz–MacPherson (GKM) theory, whose combinatorial framework becomes especially transparent in the case of the complex projective plane, allowing geometric information to be encoded on a graph in a way that simplifies computations and highlights structural patterns. Finally, we describe recent work showing how GKM theory can be used to detect the behavior of quantum parameters, providing a powerful bridge between combinatorics and quantum cohomology.

January 30

Speaker: Claus Kadelka (Iowa State)

Host: Michael Robert

Title: Structure and dynamics of gene regulatory networks

Gene regulatory networks (GRNs) play a central role in cellular decision-making. Understanding their structure and how it impacts their dynamics constitutes thus a fundamental biological question. GRNs are frequently modeled as Boolean networks, which are intuitive, simple to describe, and can yield qualitative results even when data are sparse. We assembled the largest repository of expert-curated Boolean GRN models. A meta-analysis of this diverse set of models reveals several surprising structural and dynamical features. GRNs exhibit more canalization, redundancy, and stable dynamics than expected. Moreover, they are enriched for certain recurring network motifs. Altogether, this raises the important question how these features benefit GRNs.

February 13

Speaker: Alex Iosevich (University of Rochester)

Host: Eyvi Palsson

Title: From harmonic analysis to exact signal recovery

We are going to describe how classical techniques from harmonic analysis, such as linear and bilinear restriction theory, Bourgain/Talagrand $\Lambda_p$ inequalities, and related ideas have been used to study signal recovery, both in the classical setting, and in the context of imputation of time series in applied data science.

February 27

Speaker: Jian-Guo Liu (Duke University)

Host: Yingda Chen

Title: Analysis of the adhesion model and reconstruction in cosmology

In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model suppresses multi-streaming by introducing viscosity. We study mass flow in this model by analysis of Lagrangian advection in the zero-viscosity limit. Under mild conditions, we show that a unique limiting Lagrangian semi-flow exists. Limiting particle paths stick together after collision and are characterized uniquely by a differential inclusion. The absolutely continuous part of the mass measure satisfies a Monge-Ampère equation related to convexification of the free-streaming velocity potential. The use of Monge-Ampère equations and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier and Frisch et al (2003). We show that the singular part of the mass measure can differ from the Alexandrov solution to the Monge-Ampère equation, however, when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. In a neighborhood of merging singular structures in our examples, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.

March 20

Speaker: Camil Muscalu (Cornell University)

Host: Eyvi Palsson

Title: Iterated Fourier Series

The plan of the talk is to describe a bridge that connects in a natural way two mathematical worlds at the first glance far away from each other: the KdV equation and the absolute Galois group. The pillars of this bridge are given by the analytical objects from the title. The presentation will be “elementary”, one would not need to know what is the KdV equation or the absolute Galois group.

March 27

Speaker: Xiantao Li (Penn State)

Host: Daniel Appelo

Title: Dilation Methods for Deterministic and Stochastic Dynamics with Quantum Computing Applications

Quantum algorithms are naturally formulated in terms of unitary dynamics, while many models from scientific computing involve dissipation, forcing, or randomness. In this talk I will describe a dilation-based approach for bridging this gap. The main idea is to embed deterministic or stochastic dynamics into a larger system whose evolution has a structure more suitable for quantum computation. I will first discuss deterministic dynamics, where this approach provides a unified framework for several recent constructions, including LCHS-type methods and Schrödingerization. I will then show how the same perspective extends to stochastic differential equations, leading both to trajectory-based representations and to deterministic evolution equations for quadratic statistics. These ideas connect naturally to imaginary-time evolution, Lindblad-type formulations, and recent quantum algorithms for stochastic dynamics and fermionic systems.

April 3

Speaker: Christine Kelley (University of Nebraska-Lincoln)

Title: Expander Graphs and Coding Theory

Host: Gretchen Matthews

One fundamental goal in coding theory is to design explicit algebraic code ensembles that are asymptotically good, meaning that their minimum distance and rate do not go to zero as the block length tends to infinity. In the mid-90s, it was shown that codes whose underlying graph representations had good expansion properties could be used to obtain such ensembles. In this talk, we give an introduction to the area of graph-based codes and review the basics of expander graphs and expander codes. We derive bounds on performance guarantees for four types of expander code frameworks, and end the talk with a discussion of open problems and current uses of expander graphs in coding theory.

April 24

Speaker: Omar Ghattas (Oden Institute & Mechanical Engineering, UT Austin)

Host: Johann Rudi

Title: Real Time Bayesian Inference for Tsunami Early Warning

The Cascadia Subduction Zone (CSZ) extends over 1000 km from northern California to northern Vancouver Island. The CSZ is capable of unleashing a magnitude 9 earthquake with resulting 30 meter high tsunamis. While 43 earthquakes have occurred on the Cascadia fault over the last 10,000 years, it has been eerily silent since 1700, and many consider it overdue for a major earthquake. Plans are underway to deploy a network of seafloor-mounted pressure sensors in the CSZ. The sensors will record acoustic waves in the ocean that are generated when the seafloor is suddenly uplifted during an earthquake. Our goal is to use these observed pressure transients, along with a coupled acoustic-gravity wave propagation forward model, to infer the seafloor motion, and then forward propagate the resulting tsunamis toward coastal regions. Since destructive tsunami waves can arrive onshore in as little as 10 minutes, the inverse solution and subsequent tsunami forecast must be carried out in seconds to be useful for early warning. Not only do we want to predict the mean of the tsunami wave heights, but we also want to equip this digital twin with uncertainty estimates in a fully Bayesian framework. However, a single forward wave propagation simulation takes an hour on 512 A100 GPUs. To infer the billion parameters representing the uncertain spatio-temporal seafloor motion, state-of-the-art inversion algorithms need hundreds of thousands of such forward simulations. We show that by exploiting the time shift-invariance and linearity of the parameter-to-observable map, transforming the inverse operator from the parameter space to the data space, and devising novel parallel Bayesian inference algorithms that map well onto GPUs, we induce a fast offline–online decomposition that allows the seafloor motion inversion and subsequent tsunami forecast to be carried out exactly in a fraction of a second. Finally, we present fast algorithms for the optimal experimental design problem of optimizing the locations of the seafloor pressure sensors to maximize information gain from the data. This work is joint with Stefan Henneking (UT Austin), Sreeram Venkat (UT Austin), and Alice Gabriel (Scripps/UCSD).

Bio: Omar Ghattas is Professor of Mechanical Engineering at The University of Texas at Austin and holds the Cockrell Chair in Engineering. He is also Principal Faculty and Director of the OPTIMUS (OPTimization, Inverse problems, Machine learning, and Uncertainty for complex Systems) Center in the Oden Institute for Computational Engineering & Sciences, and a member of the faculty in the Computational Science, Engineering, and Mathematics graduate program. He holds courtesy appointments in Earth & Planetary Sciences and Biomedical Engineering. Before moving to UT Austin in 2005, he spent 16 years on the faculty of Carnegie Mellon University. His current research focuses on theory and algorithms for large-scale Bayesian inversion, stochastic optimal control/design, and digital twins for complex engineered and natural systems. He is a three-time winner of the ACM Gordon Bell Prize, a recipient of the SIAM Geosciences Career Prize and the SIAM Babuska Prize, and a Fellow of SIAM and USACM. He holds BSE (civil and environmental engineering) and MS and PhD (computational mechanics) degrees from Duke University.

May 1

Speaker: Nathan Kaplan (UC Irvine)

Host: Sarah Arpin

Title: Sublattices and Subrings of ℤⁿ and Random Finite Abelian Groups

How many sublattices of ℤⁿ have index at most X? If we choose such a lattice Λ at random, what is the probability that ℤⁿ/Λ is cyclic? What is the probability that its order is odd? Now let R be a random subring of ℤⁿ. What is the probability that ℤⁿ/R is cyclic? We will see how these questions fit into the study of random groups in number theory and combinatorics. We will discuss connections to Cohen-Lenstra heuristics for class groups of number fields, sandpile groups of random graphs, and cokernels of random matrices over the integers.

May 8

Speaker: Aaron Bertram (Utah)

Host: Leo Herr

Title: Two Variations on a Theme by Cremona

The Cremona transformation is a non-linear birational map between a projective space and its dual. It can be understood in terms of toric data, in which case it corresponds to the self-duality of the generalized tetrahedron. Or it can be embedded as the diagonal in the projective space of square matrices, in which case it corresponds to the cofactor method for inverting a matrix. I want to talk about two variations on this. The first, jointly with Alicia Lamarche, looks at the analogous geometry for the duality between a hypercube and a generalized octahedron. When we embed this in the space of orthogonal matrices, we obtain an interesting answer to the question: What is the best completion of the space SO(2n+1,C) for the standard representation? Hint: it is not a manifold. The second variation is concerned with Cremona in families. Here, the varieties of secants to a variety embedded in projective space give rise to dual "cosecant" varieties that should be understood as families of Cremona-transformed projective spaces. The interesting question is what happens when the points defining the secants come together. When the variety is a curve, this was explained by Michael Thaddeus via moduli of vector bundles (with a section) on the curve. When the variety is a surface, I have a proposal for the cosecant variety inspired by the moduli of objects in the "derived category" of coherent sheaves on the surface. It is somewhat surprising to see this machinery appearing in such a classical problem, but it has been a very productive point of view that I hope to convey.