Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

We will discuss the problem of finding the maximum number of 4-point circles given 9 points, where a 4-point circle is a circle on which lie exactly 4 of the 9 given points. This problem is solved for points in $\mathbb{R}^2$, we present a generalization to $\mathbb{C}^2$. Our approach uses techniques from polyhedral geometry and group theory. In particular, we will construct a polytope whose lattice points represent candidate point-circle incidences of 9-point configurations. This method works until the number of circles is greater than 4, at which point solving for the lattice points becomes computationally infeasible. We present then another method which instead recursively constructs these configurations. We reduce computations by leveraging symmetries of $S_9$. Finally, we discuss the original motivation for this work, coming from an interpolation problem in mechanical engineering involving four-bar linkages.

This is a final presentation for the course AM 4999Z.

A basic and naive problem in homotopy theory is to compute the sets $[S^m, S^n]$ of homotopy classes of maps between spheres of different dimensions. I will describe the preliminary results of a machine-based approach to these computations. Historically, there are two separate paradigms for such computations: the EHP sequence, and the unstable Adams spectral sequence. Our approach exploits both, and the interaction between them.

I will not assume any familiarity with either the EHP sequence or the unstable Adams spectral sequence.

Building on her seminal work regarding moduli space volumes for Riemann surfaces, Mirzakhani also calculated expected values for various geometric functions on moduli space. Notably, she examined the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. We will review several of Mirzakhani’s key results, which collectively offer insights into the geometry of random surfaces in high genus. Following this, we will explore some extensions of her findings in the context of translation surfaces.

(Based on work by recent MSc Thesis student, George Shillcock)

Understanding the capacity of pathogens to cause severe disease is of fundamental importance to human health and preserving biodiversity. Many of those pathogens are not only transmitted horizontally between unrelated hosts but also vertically between parents and their progeny. It is widely accepted that vertical transmission leads to the evolution of less virulent pathogens, but this idea stems from research that neglects the evolutionary response of hosts. Here, we use a game-theory model of coevolution between pathogen and host to show that vertical transmission does not always lead to more benign pathogens. We highlight scenarios in which vertical transmission results in pathogens exhibiting more virulence. However, we also predict that more benign outcomes are still possible (a) when generating new horizontal infections inflicts too much damage on hosts, (b) when clearing an infection is too costly for the host, and (c) when vertical transmission is promoted by a greater growth rate of the host population. Though our work offers a new perspective on the role of vertical transmission in pathogen–host systems, it does agree with previous experimental work.

Many matroid invariants take values in polynomial rings. The study of the coefficients and roots of matroid invariants is an important topic in algebraic combinatorics. Examples are the characteristic polynomial, the Tutte polynomial or the Kazhdan-Lusztig polynomial of a matroid. Connections between matroid theory to the intersection theory of toric varieties has played an important role in the proof of several long-standing conjectures in recent years. In this talk I will explain techniques that have been successfully used to investigate properties of such polynomials.