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.

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.

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: 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.