Georgia Institute of Technology College of Sciences

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erdos and Rado in 1960 proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to $c^w$ for some constant $c$. Despite much research, the best bounds until recently were all of the order of $w^{cw}$ for some constant c. In this work, we improve the bounds to about $(\log w)^{w}$.

There are two main ideas that underlie our result. The first is a structure vs pseudo-randomness paradigm, a commonly used paradigm in combinatorics. This allows us to either exploit structure in the given family of sets, or otherwise to assume that it is pseudo-random in a certain way. The second is a duality between families of sets and DNFs (Disjunctive Normal Forms). DNFs are widely studied in theoretical computer science. One of the central results about them is the switching lemma, which shows that DNFs simplify under random restriction. We show that when restricted to pseudo-random DNFs, much milder random restrictions are sufficient to simplify their structure.

Joint work with Ryan Alweiss, Kewen Wu and Jiapeng Zhang.

We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.

Note that the sequence of topological entropies grows at the order of log(n), whereas, in sharp contrast, the order of magnitude of the sequence of metric entropies is only log(n)/n.

Also, we prove the convexity of the admissible rotation set AR, and the fact that the rotation vectors obtained from
periodic admissible trajectories form a dense subset in AR.

One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path satisfying certain combinatorial properties in the arc complex of the 4-punctured sphere, and how we can reconstruct such surfaces from these paths. This extends and generalizes results of Floyd, Hatcher, and Thurston.

Did you know that a wheel or a ball bearing does not need to be round? Convex regions that can roll smoothly come in many remarkable shapes and have practical applications in engineering and science. Moreover, the mathematics used to describe them, known as convex geometry, is a subject that beautifully ties together analysis and geometry. I'll bring some of these objects along and tell the class how to describe them effectively and recount their interesting history.

We will give a brief introduction to Schur polynomials and intersection theory and show how to use Section 10 of the Brändén-Huh paper to obtain log-concavity results for Schur polynomials.

https://arxiv.org/abs/1902.03719
https://arxiv.org/abs/1906.09633

Anosov flows provide beautiful examples of interactions between dynamics, geometry and analysis. In dimension 3 in particular, they are known to have a subtle relation to topology as well. Motivated by a result of Mitsumatsu from 1995, I will discuss their relation to contact and symplectic structures and argue why contact topological methods are natural tools to study the related global phenomena.