Talk

On a Riemann Surface $\Sigma$, the moduli space of polystable $\mathrm{SL}_n(\mathbb{C}$)-Higgs bundles can be identified with the space of reductive representations $\pi _1 (\Sigma) \to \mathrm{SL}_n(\mathbb{C})$. In this talk, we discuss a proof of this so called non-abelian Hodge correspondence. Our goal is to understand how
to construct a Higgs bundle from a given representation and how this construction relates to the theory of harmonic maps.

I will talk about lower bounds for the discrete negative 2k-th moment of the derivative of the Riemann zeta function for all
fractional k > 0. The bounds are in line with a conjecture of Gonek and Hejhal. This is a joint work with Winston Heap and
Junxian Li. 

I will explain how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams. I will also explain, with very little detail, how this is used in the construction of some very well-behaved poly-time computable knot polynomials.

I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

In the general study of regulators, we present three inequalities. We first bound from below the regulators of number fields, following previous works of Silverman and Friedman. We then bound from below the regulators of Mordell-Weil groups of abelian varieties defined over a number field, assuming a conjecture of Lang and Silverman. Finally we explain how to prove an unconditional statement for elliptic curves of rank at least 4. This third inequality is joint work with Pascal Autissier and Marc Hindry.

Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. Its modern formulation is due to the works of Hinich, Keller-Lefevre and Positselski. It is a certain correspondence between categories of differential graded (dg) algebras and conilpotent dg coalgebras; there is also a module-comodule level version of it. In this talk I explain what happens if one drops the condition of conilpotency on the coalgebra side; the consequences turn out to be quite dramatic.

The relation between representations of the fundamental group of a Riemann surface into the unitary group, flat connections on Hermitian vector bundles, and stable holomorphic bundles on the surface, goes back to the celebrated theorem of Narasimhan and Seshadri. The case of representations into a non-compact reductive Lie group required the introduction of new holomorphic objects on the Riemann surface, called Higgs bundles.