School of Mathematics

A few years ago, I was introduced to a surprising result of Payan stating that certain highly symmetric "cube-like" graphs cannot have chromatic number exactly $3$. (Specifically, his result applies to graphs on vertex set $(\mathbf{Z}/2\mathbf{Z})^d$ for some d that are translation-invariant: every translation $\vec{v} \mapsto \vec{v}+\vec{x}$ of the vertex set is also an automorphism of the graph.)  Payan's theorem is unexpected because one generally shows a graph $H$ either has chromatic number $\leq k$ or $>k$ --- which is equivalent to $H$ being $k$-colorable or not. In comparison, this theorem states a cube-like graph is either $2$-colorable or non-$3$-colorable, which are difficult conditions to compare.

In this talk, I'll tell you a short new proof of Payan's theorem (joint with Mike Krebs). The proof uses topological ideas and applies beyond cube-like graphs --- including to quadrangulations of $\mathbf{RP}^2$ and to translation-invariant graphs on $(\mathbf{Z}/4\mathbf{Z})^d$.

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss recent work with Peter Cho and Asif Zaman on a closely related problem, namely determining the least prime with a given cycle type.  We develop a new, comparatively elementary approach for thinking about this problem that nevertheless frequently yields the strongest known results.  We obtain particularly strong results in the case that the Galois group is the symmetric group S_n for some n, where determining the cycle type of a prime is equivalent to Chebotarev.

Sponsored by the Minerva Research Foundation

Abstract: Projection theory asks how the size of a set in $R^n$—often measured by Hausdorff dimension—behaves under projections onto lower-dimensional spaces, both for orthogonal projections and for more general nonlinear families. One would like to understand when dimension is typically preserved and to control the (often small) exceptional set of directions or parameters where it is not.

In this lecture series we study how to find structures in an arbitrary set in $R^n$ and how those structures can be used to prove projection estimates.  We will survey discretized and multi-scale ideas initiated by Bourgain and developed further by Shmerkin, Orponen, and others, which yield recent progress on projection problems. 

The study of descriptive set theory in the context of determinacy axioms began nearly 60 years ago. The context for this study is now understood to be the Axiom AD+, which is a refinement of the Axiom of Determinacy (AD).  The objects of this study are the universally Baire sets of reals which form a transfinite  hierarchy which extends the borel sets.

This has led to what is arguably the main duality program of Set Theory, which is the connection between the universally Baire sets,  and generalizations of L, the inner model of the universe of sets constructed by Gödel.