Activities This Week

Let $F = (F_1, \ldots, F_m)$ be a collection of (not neccessarily distinct) sets. A (partial) rainbow set for $F$ is a set of the form $R = {x_{i_1}, \ldots, x_{i_k}}$ of distinct elements, where $1 \leq i_1 < \cdots < i_k \leq m$ and $x_{i_j}$ is an element of $F_{i_j}$. We are interested in the following question: given sufficiently many independent sets of size $a$ in a graph belonging to a certain class, there exists a rainbow independent set of size $b$. In this talk, I will present our recent results on this question, mainly regarding $H$-(induced) free graphs and graphs of bounded maximum degree. This is joint work with Ron Aharoni, Joseph Briggs and Jinha Kim.

In the early 1970’s, Hindman proved a beautiful theorem in additive Ramsey theory asserting that for any partition of the set of natural numbers into finitely many cells, there exists some infinite set such that all of its finite sums belong to a single cell.

In this talk, we shall address generalizations of this statement to the realm of the uncountable. Among others, we shall present a new theorem concerning the real line which simultaneously generalizes a recent theorem of Hindman, Leader and Strauss, and a classic theorem of Galvin and Shelah.

This is joint work with David Fernandez-Breton.

In a series of recent papers, Minhyong Kim defined an arithmetic analogue of topological Chern-Simons theory. In this talk, I will introduce this arithmetic Chern-Simons theory and then explain how to compute the arithmetic Chern-Simons invariant for finite, cyclic gauge groups. I will then give some recent applications of these computations.

My work in this talk is based on joint works with Tomer Schlank and Eric Ahlquist.

Let T be a bounded linear operator on a Banach space X; the elements of (I − T)X are called coboundaries. For two commuting operators T and S, elements of (I − T)X ∩ (I − S)X are called joint coboundaries, and those of (I − T)(I − S)X are double coboundaries. By commutativity, double coboundaries are joint ones. Are there any other? Let θ and τ be commuting invertible measure preserving transformations of (Ω, Σ, m), with corresponding unitary operators induced on L2(m). We prove the existence of a joint coboundary g ∈ (I − U)L2 ∩ (I − V )L2 which is not in (I − U)(I − V )L2. For the proof, let E be the spectral measure on T 2 obtained by Stone’s spectral theorem. Joint and double coboundaries are characterized using E, and properties of the maximal spectral type of E, together with a result of Foia³ on multiplicative spectral measures acting on L2, are used to show the existence of the required function.