# Past Events

An “abstract polyhedron” means, roughly, a graph that “might be the edges and vertices of a polyhedron”. When can we promote “might be” to “is”? This question is answered by a beautiful theorem about circle packings on the sphere. I will explain the proof of this theorem, as well as some…

Abstract: The set of all n by n orthogonal (or unitary) matrices form a compact topological group. This means that orthogonal matrices have a uniform distribution called Haar measure. A natural question is, what do typical orthogonal or unitary matrices “look like”? We will answer two version of…

I will talk about a new algebra of operations on polynomials which has the property T_iT_j=T_{j+1}T_i for j>i and a family of polynomials dual to them called forest polynomials. This family of operations plays the exact role for quasisymmetric polynomials and forest polynomials as the…

If Δ is a contractible compact d-manifold, then its boundary Σ will be a homology (d-1)-sphere, but the boundary need not be simply connected and Δ need not be homeomorphic to the d-disk. In joint work with Randal-Williams, we show that the topological group consisting of homeomorphisms of…

In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size $T^{3/2-\varepsilon}$. We will also explain how this result relates…

Abstract

Degree-*d* multivariate polynomials over small finite fields are of central importance in theoretical computer science. And yet they retain many mysteries; for example, their Fourier spectra are very poorly understood. We will discuss the so-called "Fourier growth" of such functions…

In the 1970's dihedral representations of knot groups were used to define twisted signature-type invariants which generalize the older invariants of Levine and Tristram. The most prominent examples are the Casson-Gordon invariants, which provide obstructions to being topologically slice as well…

Abstract: This is a talk about concavity and convexity of trace functionals. In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigner, Yanase and Dyson in 1963. This result, together with its many extensions, has found plenty of…

*Abstract:* Take an irrational rotation of the two-sphere; it only has the north and south poles as its periodic points. However, Franks proved that for any area-preserving diffeomorphism of the two-sphere, if it has more than two fixed points, then it must have infinitely many periodic…