Combinatorics

Understanding how network structure gives rise to spatiotemporal dynamics and computation is a central challenge in computational neuroscience and artificial intelligence. Despite increasingly detailed connectomic data in neuroscience and large-scale datasets in machine learning, establishing principled links between connectivity, dynamics, and function in nonlinear neural systems remains difficult. In this talk, I will present a mathematical framework that directly relates network architecture to emergent dynamical patterns and computational capabilities in analytically tractable models. Our approach focuses on networks of coupled oscillators, which are widely used to model interacting neural populations and have recently gained interest as computational substrates in artificial neural networks. With this approach, we can show how key structural features of these networks — including connectivity patterns and transmission delays — determine the emergence and stability of spatiotemporal activity, enabling analytical predictions of collective phenomena such as traveling waves. When applied to empirically derived brain networks, the framework provides a rigorous connection between large-scale anatomy, distance-dependent delays, and wave dynamics observed at mesoscopic and whole-brain scales. Building on these results, we introduce a new class of neural networks that leverage structured spatiotemporal dynamics for computation while remaining exactly solvable. Together, these results outline a general strategy for linking network structure, emergent dynamics, and computation, with implications for understanding neural activity and for developing interpretable dynamical models for neural computation.

Drinfeld modules are the analogues of elliptic curves in positive characteristic. They are essential objects in number theory for studying function fields. They do not have points, in the traditional sense—we're going to count them anyway! The first methods achieving this were inspired by classical elliptic curve results; we will instead explore an algorithm based on so-called Anderson motives that achieves greater generality. Joint work with Xavier Caruso.

Dave Morris (University of Lethbridge, Canada)

We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

In this talk, I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

I will give an overview of a few places where combinatorial structures have an interesting role to play in quantum field theory and which I have been involved in to varying degrees, from the Connes-Kreimer Hopf algebra and other renormalization Hopf algebras, to the combinatorics of Dyson-Schwinger equations and the graph theory of Feynman integrals.

For other events in this series see the quanTA events website.