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.

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