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.