Program

The Kummer surface is a K3 surface constructed as a generically 2:1 quotient of an abelian surface. However, the same Kummer surface can also arise as 2:1 quotient of other K3 surfaces, and the two covering surfaces (the abelian and this new K3) can be related in terms of their second cohomology; the triangular diagram that arises is known as a Shioda-Inose structure. We will discuss the conditions that allow for this construction, and some possible generalizations.

Since the beginning, mankind has felt the need to see “to infinity and beyond” in order to get a better understanding of its own world. This necessity turned out to be a great boost that led humanity to new discoveries. All these breakthroughs are the result of a realization of sophisticated instruments: had the telescope not been invented, the Earth would still be the center of the Universe. In this speech, we try to show why in modern mathematics we have the need to see “to infinity and beyond” and why the primordial theory of Infinity-Category is supposed to be the natural tool to deal with infinite.

Since their introduction in 1971 thanks to the seminal work of James Serrin, overdetermined problems have been an important field of study within the PDEs and geometric analysis community. I will present the result of Serrin together with the two main tools needed in order to prove it: maximum principles and the method of moving planes. I will then introduce the basic features of the fractional laplacian and present a result for an overdetermined problem involving this nonlocal operator.

A way to study integer solutions to diophantine equations is by looking at the reduction of the equation modulo prime numbers. It follows from this idea that it is possible to study rational points on a variety by looking at their image in the p-adic points. During this talk, I will give an overview of this strategy and explain the role that cohomological obstructions play.

In my seminar I will do an introduction to the concept of essential dimension: roughly speaking, the essential dimension is a measure of how many independent parameters we need to describe some algebraic object. The concept of essential dimension was introduced by Buhler and Reichstein in 1995 and it is linked to an algebraic version of Hilbert’s 13th problem. For a finite group G, the essential dimension measures how much one can compress a faithful representation of G. When G is the symmetric group S_n, the essential dimension tells us how many independent parameters we need to write a generic polynomial of degree n on a field k of characteristic zero; equivalently, the essential dimension of S_n computes the number of parameters needed to write a generating polynomial for separable field extensions of degree n. This is still an open problem for n ≥ 8. Suprisingly, the analogue problem for inseparable field extensions has been solved explicitely.

The notions of context, judgement, and deduction all pertain to logic and are in fact the basic blocks which most traditions in the subject are built on. While mathematicians rightly feel they have a grasp on what each of these pieces are, and how to deal with them, the perspective which one comes from decisively skews the way they interpret and use each: for example, a string of symbols as “♥ ⊢ ♣” would be interpreted as a judgement by someone studying dependent type theory, so that ♣ is a type in context ♥, or as a consecution if the person is a proof theorist, with ♣ being something we can derive from ♥. Still, there is no reason as to why these two perspectives should not be supported by the same theoretical system, and this is the topic of the work we will be presenting. In fact, such an effort has already been attempted and proved fruitful: we just take it to the extreme, and see how that brings remarkable results.

The L^p-Calderón-Zygmund inequality (CZ(p)) is a global second order regularity estimate for the solution of the Poisson equation. While CZ(p) always holds on R^n, in a Riemannian setting some geometric assumptions are needed. In this talk, we will present some existence results, show the relation of CZ(p) to other regularity estimates and discuss its role in proving the equivalence of different notions of Sobolev spaces. Finally, we will sketch how to use singular metric spaces in order to produce some new smooth counterexamples.

L-functions play a crucial role in the formulation of several (and often open) conjectures in Number Theory. We will introduce these objects and try to motivate their importance, with a particular emphasis on the p-adic side of the story.

The study of derived categories is a central theme in modern algebraic geometry. Fourier-Mukai functors appear as the natural maps between them. But are they the only functors? Some results point in this direction, but unfortunately things are not so easy. In the talk we will see why it is necessary to move to a higher level (in our case, the realm of differential graded categories) and what happens there.