Georgia Institute of Technology College of Sciences

The fundamental EKR theorem states that, when n≥2r, no pairwise intersecting family of r-subsets of {1,2,...,n} is larger than the family of all r-subsets that each contain some fixed x (star at x), and that a star is strictly largest when n>2r. We will discuss conjectures and theorems relating to a generalization to graphs, in which only independent sets of a graph are allowed. In joint work with Kamat, we give a new proof of EKR that is injective, and also provide results on a special class of trees called spiders.

The theme of this talk is walks in a random environment of "signposts" altered by the walker. I'll focus on three related examples: 1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres. 2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss. 3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk “organizes” its environment of signposts. The stationary environment is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco and Boyao Li.

Researchers here at Georgia Tech initiated a "Ramsey Theory" on binary trees and used the resulting tools to show that the local dimension of a poset is not bounded in terms of the tree-width of its cover graph. Subsequently, in collaboration with colleagues in Germany and Poland, we extended these Ramsey theoretic tools to solve a problem posed by Seymour. In particular, we showed that there is an infinite sequence of graphs with bounded tree-chromatic number and unbounded path-chromatic number. An interesting detail is that our research showed that a family conjectured by Seymour to have this property did not. However, the insights gained in this work pointed out how an appropriate modification worked as intended. The Atlanta team consists of Fidel Barrera-Cruz, Heather Smith, Libby Taylor and Tom Trotter The European colleagues are Stefan Felsner, Tamas Meszaros, and Piotr Micek.

In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe two techniques that can be used to shed some light on the nature of a sequence using only some known initial terms. While these methods are, on the face of it, experimental, they often lead to rigorous proofs. As we talk about these two techniques -- automated conjecturing of generating functions, and the method of differential approximation -- we'll exhibit their usefulness through a variety of combinatorial topics, including matchings, permutation classes, and inversion sequences.