Georgia Institute of Technology College of Sciences

The original concept ofdimension for posets was formulatedby Dushnik and Miller in 1941 and hasbeen studied extensively in the literature.Over the years, a number of variant formsof dimension have been proposed withvarying degrees of interest and application.However, in the recent past, two variantshave received extensive attention. Theyare Boolean dimension and local dimension.This is the first of two talks on these twoconcepts, with the second talk givenby Heather Smith. In this talk, wewill introduce the two parameters and providemotivation for their study. We will alsogive some concrete examples andprove some basic inequalities.This is joint work with a GeorgiaTech team in which my colleaguesare Fidel Barrera-Cruz, Tom Prag,Heather Smith and Libby Taylor.

This talk will focus on tree automata, which are tools to analyze existential monadic second order properties of rooted trees. A tree automaton A consists of a finite set \Sigma of colours, and a map \Gamma: \mathbb{N}^\Sigma \rightarrow \Sigma. Given a rooted tree T and a colouring \omega: V(T) \rightarrow \Sigma, we call \omega compatible with automaton A if for every v \in V(T), we have \omega(v) = \Gamma(\vec{n}), where \vec{n} = (n_\sigma: \sigma \in \Sigma) and n_\sigma is the number of children of v with colour \sigma. Under the Galton-Watson branching process set-up, if p_\sigma denotes the probability that a node is coloured \sigma, then \vec{p} = (p_\sigma: \sigma \in \Sigma) is obtained as a fixed point of a system of equations. But this system need not have a unique fixed point. Our question attempts to answer whether a fixed point of such a system simply arises out of analytic reasons, or if it admits of a probabilistic interpretation. I shall formally defined interpretation, and provide a nearly complete description of necessary and sufficient conditions for a fixed point to not admit an interpretation, in which case it is called rogue.Joint work with Tobias Johnson and Fiona Skerman.

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.