Talk

We will present recent developments in the theory of overconvergent F-isocrystals, the p-adic analogue of ell-adic lisse sheaves. For the most part of the talk, we will explain a new result on the algebraic monodromy groups of these objects. At the end, we will mention an application of the theorem to the finiteness of separable p-torsion points of an abelian variety.

Given a topological group G, we can think of the space ofhomomorphisms $\hom(\mathbb{Z}^n, G)$ as the space of
$n$-tuples of elements of $G$ that commute pairwise. These spaces are more subtle than one might think, and even basic
invariants such as the number of connected components can lead to surprising results. Fixing $G$ and varying $n$ we can
construct what is known as the classifying space for commutativity in $G$. I will survey what is known about these
classifying spaces, whose study is still young.

I will discuss modularity of a Calabi-Yau threefold $X$ with Hodge numbers of $H^3(X)$ equal to (1,1,1,1).
The restriction of the Galois representation on $H^3(X)$ decomposes over $Q(\sqrt{2})$ into the direct sum of
the Galois representation for a Hilbert modular form of weight [4,2] and its conjugate.

The Calabi-Yau threefold $X$ is defined as a resolution of singularities of a double covering of $P^3$ branched
along a union of eight planes. The proof is based on a careful study of geometry of $X$, which allows us to find a

(joint work with Clara Aldana, Emanuel Carneiro, Carlos Andres Chirre Chavez and Julian Mejia Cordero)

At the simplest level, an upper bound sieve of Selberg type is a choice of $\rho(d),$ $d\le D$, with $\rho(1)=1$, such that $$S = \sum_{n\leq N}\left(\sum_{d\mid n}\mu(d) \rho(d)\right)^2$$

The first part of this lecture series is meant to illustrate the dictum that all suÿciently beautiful mathematical objects are connected. The two objects we choose to illustrate this are the icosahedron, the most subtle of the Platonic solids, and the Rogers-Ramanujan identities, often considered the most beautiful formulas in all of mathematics.

In this talk, I will present models of Lubin-Tate theories at $p=2$ and all heights. These models come with explicit formulas for some finite subgroups of the Morava stabilizer groups on the coefficient rings. The construction utilizes equivariant formal group laws and are based on techniques of Hill-Hopkins-Ravenel. I will also talk about implications of the theorem, such as periodicity and differentials in spectral sequences. This is joint work with Beaudry, Hill and Shi.

To a Poisson manifold $(M,\pi)$ one can naturally associated a cohomology called Poisson cohomology. Although Poisson cohomology is important for questions such as linearization and deformations of poisson structures, it is in general quite difficult to compute. In this talk we look at the Poisson structure on the dual of a Lie algebra $(g, [~,~])$. We look at the relation of Poisson cohomology with the linearization problem and outline the general ideas behind the calculations for the case of $\mathrm{sl}_2(\mathbb{C})$. This is based on joint work with Ioan Marcut.