Virginia Tech Mathematics Colloquium – Spring 2026

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.

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.

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.

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.

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