Special Year Research Seminar

Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period...

The goal of these lectures is to present the fundamentals of Simpson’s correspondence, generalizing classical Hodge theory, between complex local systems and semistable Higgs bundles with vanishing Chern classes on smooth projective varieties.

A compact hyperkahler manifold is a higher-dimensional analog of a K3 surface; Lagrangian fibrations of hyperkahler manifolds are higher-dimensional versions of elliptic fibrations of K3 surfaces. A result of Voisin shows that these fibrations yield...

On a projective variety, Simpson showed that there is a homeomorphism between the moduli space of semisimple flat bundles and that of polystable Higgs bundles with vanishing Chern classes. Recently, Bakker, Brunebarbe and Tsimerman proved a version...

We give a lower bound on the codimension of a component of the non-abelian Hodge locus within a leaf of the isomonodromy foliation on the relative de Rham moduli space of flat vector bundles on an algebraic curve. The bound follows from a more...

In this talk, I will discuss some effective computations for variations of integral Hodge structures.

Several years ago, with Ren and Javanpeykar-Kühne, I conjectured (in the Shimura setting) that a variation has only finitely many "non-factor"...

I will describe recent work in progress on logarithmic--exponential preparation theorems in analytically generated sharply o-minimal structures. Our results imply the sharp o-minimality of ℝexp

as well as a uniform version of Wilkie’s conjecture on...

Globally valued fields form a generalisation of global fields that fits into the context of first order (continuous) logic. I will describe these structures, and outline how they are connected to various parts of arithmetic geometry: Arakelov...

Let SO(3,R) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure \mu. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO(3,R) with sufficiently small measure, then...

This talk is based on a joint work with Steve Lester.

We review the Gauss circle problem, and Hardy's conjecture regarding the order of magnitude of the remainder term. It is attempted to rigorously formulate the folklore heuristics behind Hardy's...