Department of Mathematics

Title: Making LaTeX PDFs ADA Compliant: Setup, Workflows, and Verification Tools

Abstract: Upcoming legal requirements mandate that all instructional materials provided to students be fully accessible and ADA compliant. Historically, LaTeX—despite being the standard tool for producing high-quality mathematical and technical documents—has been unable to generate PDFs that meet modern accessibility standards in a reliable and verifiable way.

This talk focuses on the Phase 3 release of the LaTeX tagging project, officially released on November 1, 2025. Developed by the core LaTeX development team, this release represents a fundamental rewrite of key components of the LaTeX engine to support the semantic document structure and metadata required for accessible, standards-compliant PDFs. As a result, LaTeX can now natively produce properly tagged PDFs suitable for screen readers and accessibility validation tools.

The presentation will demonstrate where and how to obtain a sufficiently recent LaTeX engine—since most commonly used distribution channels do not yet provide a version that supports Phase 3 tagging—and will outline the specific changes required to update existing LaTeX documents for accessibility compliance. Particular attention will be given to commonly used document features and packages, including which are currently supported and which require modification or replacement.

There will also be reference materials, including a demonstration LaTeX source file and curated resources that will assist with updating previous documents. The session will conclude with a discussion of tools and workflows for verifying that compiled PDFs are genuinely ADA compliant, allowing faculty to confidently assess their materials before distribution to students.

Here is the talk recording and the speaker’s site with more details.

Title: Toward a “GPT” Moment for Scientific Computing

Abstract: Foundation models such as ChatGPT have reshaped AI by learning reusable representations that transfer across tasks. This talk asks whether a similar shift is possible in scientific computing: moving beyond solvers for a single partial differential equation (PDE) toward foundation models for families of PDE-governed systems. A central obstacle is that high-fidelity PDE data are expensive—often requiring hours to millions of CPU-hours per simulation—making purely data-driven scaling impractical. I present a physics-first roadmap that replaces data scale with physical structure, using governing equations as supervision.

I will first focus on the single-PDE setting and show how physics-informed neural networks (PINNs) can be made reliable by diagnosing and addressing key training pathologies, leading to substantial accuracy improvements and successful simulations of challenging problems including 3D turbulence. I will then extend physics supervision from learning individual PDE solutions to learning solution operators for parametric PDE families. I will introduce the framework of physics-informed DeepONet and improve its scalability with continuous vision transformers. Finally, I will discuss how these advances motivate a longer-term direction toward unified models that can generalize across heterogeneous PDEs. Together, these results provide practical and theoretical steps toward PDE foundation models, with implications for accelerated simulation,design and control in computational science and engineering.

Title: Intersec(K)tions

Abstract: What do we associate to a cycle? A classical fact is that if $X$ is a nice complex manifold or variety, and $Y\subset X$ can be (locally) given by one equation, then we can attach to $Y$ a line bundle on $X$. The situation is a lot murkier if $Y$ has higher codimension, that is, roughly speaking, you need more than one equation to describe it—what to attach to it is a lot less clear. I would like to argue that one can indeed attach to $Y$ a certain object (a sheaf of spaces or homotopy types) computed in terms of the $K$-Theory of $X$.

$K$-Theory usually gets a bad rap because of its (deserved?) reputation for being abstruse, so, on my way to show what to attach to $Y$, I will illustrate some simple ideas on how to get to the $K$-Theory space’s lower homotopy types.

One conjectures, and this is in fact a theorem in lower codimensions, that the proposed assignment can account for the intersection of cycles. This is work, partly in progress, in collaboration with Niranjan Ramachandran (UMD) and Maxime Ramzi (U Münster). No previous knowledge of $K$-Theory, Sheaf Theory, or Algebraic Geometry is assumed.

FSU/UF Topology & Geometry Meeting

Title: Exploiting Low-Rank and Related Structures in Data Science Compute

Abstract: Most machine learning and AI models combine large data, intense computations and special hardware. At the intersections mathematical problems emerge, which enable systematic solutions. In this talk, I discuss three novel methods, with open-source software, that exploit certain problem properties, such as kronecker form or low-rankness. In particular, when tensor data is a combination of continuous and discrete information (hybrid), standard decompositions either don’t exploit this structure fully or become computationally very expensive. Through careful preprocessing, I decouple large factor subproblems into a sequence of smaller ones or apply effective iterative solvers. Second, for high throughput streaming data, it is typically important to process this information rapidly. For large systems, recomputing the best low-rank approximation is prohibitive. Therefore, I develop an updating method that is closely related to updating the singular value decomposition, however is more effective at tracking a low-rank subspace. Third, nonlinear optimization is a mathematical foundation for training AI models. When only gradients are available, it is often effective to approximate 2nd order curvature information for the iterates. Thus, I propose a parametric family of 2nd derivative Hessian estimates, which applies to deterministic or stochastic optimization. I conclude with a set of future directions.

Title: Algebraic foundations of machine learning

Abstract: This talk explores how algebraic geometry offers powerful tools for understanding models in statistics and machine learning. I will explain how viewing statistical models as algebraic varieties can address practical problems such as maximum likelihood estimation. Building on this perspective, I will focus on modern machine learning models and how their algebraic and semi-algebraic structure gives rise to polynomial invariants and complexity measures, such as Euclidean Distance degree. These tools provide theoretical guarantees for neural network verification, optimization, and robustness, supporting safety-critical applications.

Title: A complex scaling method for junctions of semi-infinite interfaces

Abstract: Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to deriving explicit error bounds for these methods. Many of these methods also require a large computational domain and so require a large number of discretization points to accurately solve the problem.

In this talk, I present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integrals equations can be analytically continued into the complex plane, where they can be safely truncated. I demonstrate this procedure for a junction of two dielectric waveguides. For this problem, I show that the fundamental solutions and integral equation densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with provable exponential accuracy. I will also demonstrate how this method can be applied to a number of other scattering problems, including junctions of periodic gratings and corrugated waveguides.

Title: A Universal Description of Stochastic Oscillators

Abstract: Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. In joint work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we have introduced a nonlinear transformation of stochastic oscillators via a complex-valued function, Q, that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The Q function is the eigenfunction of Kolmogorov’s backward operator (also called the stochastic Koopman operator) with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of both weakly and strongly coupled stochastic oscillators.

Joint work with:
Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona)
Boris Gutkin (Ecole Normale Supérieure, Paris)
Benjamin Lindner (Humboldt University, Berlin)
Max Kreider (Penn State University)

Title: Random commuting matrices

Abstract: The study of the eigenvalue distribution of random matrices is a well-established field, dating back to the 1920’s. It became popular with the work of Wigner and Dyson in the 1950’s and 60’s, and today is a major field in both probability and theoretical physics. Typically a random matrix is generated by choosing the entries identically and independently distributed.

What can one say about random d-tuples of commuting matrices? What does it even mean, since one can no longer choose entries independently?

We will describe one approach to defining a random d-tuple of commuting matrices. We shall show that in the Hermitian case, the description of their eigenvalue distribution parallels to some extent the single matrix theory, though there is a qualitative change when d ≥ 5. In the non-self adjoint case the eigenvalue distribution is quite unlike the single matrix case.

Erdős Colloquium

Number Theory Conference

Title: More unlikely intersections on elliptic surfaces
(w/ G. Urzua and F. Voloch)

Abstract: In arithmetic geometry, one encounters situations where an intersection is “unlikely”, for example on dimension grounds, and where one wants to give the unlikely thing infinitely many chances to happen (say by varying some parameter). A statement that the intersection happens only finitely many times is then interesting (and a bound is even better). Families of elliptic curves host many examples of this type of problem. We discuss one such situation where the main result is a beautiful bound of topological nature, but whose proof, somewhat surprisingly, requires us to briefly leave the world of algebraic geometry for a real-analytic context. Time permitting, I’ll also mention a connection to a famous homomorphism of Manin and point out some inaccuracies in the literature about it.