C3

to follow

In this talk, I will give an account of the measure of large values where |ζ(1/2 + it)| > exp(V), with t ∈ [T,2T] and V ∼ αloglogT. This is the range that influences the moments of the Riemann zeta function. I will present previous results on upper bounds by Arguin and Bailey, and new lower bounds in a soon to be completed paper, joint with Louis-Pierre Arguin, and explain why, with current machinery, the lower bound is essentially optimal. Time permitting, I will also discuss adaptations to other families of L-functions, such as the central values of primitive characters with a large common modulus.

The Deligne-Beilinson conjecture predicts that the special values of many L-functions are related to the ranks of certain Ext groups in the category of mixed Hodge structures. In this talk, we present Skinner’s constructions of certain extensions that are extracted from the cohomology of the modular curve using CM points and the Eisenstein series. Through an explicit analytic calculation, which is performed in the adelic setting using (g,K)-cohomology and Tate’s zeta integrals, we obtain a formula relating the non-triviality of these extensions to the well-known non-vanishing at s=1 of the L-functions associated to Hecke characters of imaginary quadratic fields. These constructions have natural analogs in the category of p-adic Galois representations which are useful for Euler systems.

For families of Calabi-Yau threefolds, we derive an explicit formula to count the number of points over $\mathbb{F}_{q}$ in terms of the periods of the holomorphic three-form, illustrated by the one-parameter mirror quintic and the 5-parameter Hulek-Verrill family. The formula holds for conifold singularities and naturally incorporates p-adic zeta values, the Yukawa coupling and modularity in the local zeta function. I will give a brief introduction on the physics motivation and how this framework links arithmetic, geometric and physics.

A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction. No prior knowledge of stacks is needed for this talk!

TBC

Einstein’s theory of general relativity reshaped our understanding of the universe. Instead of thinking of gravity as a force, Einstein showed it is the bending and warping of space and time caused by mass and energy. This radical idea not only explained how planets orbit stars, but also opened the door to astonishing predictions. In this seminar we will explore some of its most fascinating consequences from the expansion of the universe, to gravitational waves, and the existence of black holes.