Georgia Institute of Technology College of Sciences

Combinatorial homotopy theory, or $A$-theory, is a homotopy theory of simplicial complexes which is known to have far-reaching applications. In the graph case, it coincides with a notion of homotopy first introduced by Maurer to study matroid basis graphs. In the language of $A$-theory, Maurer's celebrated homotopy theorem states that matroid basis graphs have trivial fundamental group. We ask whether this result can be strengthened and make progress toward showing that matroid basis graphs are $A$-contractible. We look at this problem through the lens of Malle's "gangster problem," which formulates $A$-contractibility of graphs in terms of gangsters travelling between towns.

Zoom link: https://gatech.zoom.us/j/99884528900

Engel structures are maximally non-integrable rank-two plane fields on four-dimensional manifolds. They are closely related to contact geometry, but their global behavior is still much less understood.

In contact topology, complex tangencies of real hypersurfaces in complex manifolds give a fundamental source of contact structures, often with strong rigidity properties. This motivates the Engel analogue: can a compact four-dimensional submanifold of $\mathbb C^3$ have complex tangencies forming an Engel structure?

In this talk, I will explain how to construct such examples in the case of embeddings $M \times S^1 \subset \mathbb C^3$. The main idea is to start from a standard construction of Engel structures on circle bundles over $3$-manifolds, and then realize these Engel distributions as complex tangencies of a suitable embedding into $\mathbb C^3$.  This gives the first compact examples of submanifolds of $\mathbb C^3$ whose complex tangencies are Engel, answering a question of Yakov Eliashberg. This is joint work with E. Fernández and Á. del Pino.

Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields. They appear in nature: fireflies flashing in sync with each other is one fun situation.

In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneous Kuramoto systems (we will define these notions!). They are determined from a finite graph. In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations. We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems. We will see how computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions.

Note: we assume essentially NO dynamical systems nor algebraic geometry in this talk! This talk should be understandable to a general mathematical audience. The parts of the talk that are new represent joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend.