Past Events
The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can consider a complementary perspective - given a smooth projective variety whose nonrationality is known, how "irrational" is it? I will survey…
In this talk, I will continue to describe aspects of geometric optics, one of the main themes introduced earlier. I also hope to describe a bit more about how this relates to some interesting properties of hyperbolic PDE. In particular, I hope to motivate a bit more about why you may…
Automorphy lifting theorems establish situations in which Galois representations over \bar{Q_p} are automorphic if their residual representation has an automorphic lift. In 2018, Allen et. al. proved the first automorphy lifting theorem for n-dimensional Galois representations over a CM field…
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…