Idisplays

In joint work, very much in progress, with Johannes Anschütz, Juan Esteban Rodriguez Camargo and Peter Scholze, we define an analogue of prismatic cohomology for rigid analytic varieties. Our construction is formulated using analytic stacks and takes inspiration both from Scholze's theory of diamonds and the work of Bhatt-Lurie and Drinfeld on the prismatization of p-adic formal schemes. It also furnishes a potential formulation of the geometrization of the p-adic local Langlands correspondence. In this talk, I would like to survey various aspects of this work.

We investigate the isomorphism problem for Cayley graphs of graph products. We show that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs. Additionally, we identify conditions under which the Cayley graphs of two graph products are isomorphic, even when the underlying groups are not. This leads to interesting examples of non-isomorphic finitely generated groups with isomorphic Cayley graphs.

I will talk about my recent result regarding the separability of products of subgroups in virtually special cubulated groups. My talk will also contain lots of background on cube complexes, cubulated groups and (virtual) specialness, which has been an important topic in geometric group theory over the last 20 years, particularly with the connection to 3-manifold theory.

Abstract: In this talk we will discuss the structure of $\mathbb E_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group completion. This is joint work with Maxime Ramzi.

Literature on model theory of Henselian valued fields usually establishes, or relies on transfer principles between the theory of a valued field, and that of its value group Gamma and residue field k. Recent contributions often use an alternative approach, which is to study transfer principles with an intermediate reduct of the valued field: the *leading-term structure* RV, the expansion of the Abelian group sitting in the pure short exact sequence (PSES, for short): 1->k*->RV->Gamma->0 The cleanest, most natural and most general framework to study this structure is that developed in Section 4 of the very influential (and recent) article by Aschenbrenner-Chernikov-Gehret-Ziegler: the setting of PSES of *Abelian structures*, with an arbitrary expansion on their term on the left and the right (such as the order on Gamma and addition on k). The aforementioned article establishes transfer principles for *quantifier elimination* and *distality* between the middle term (RV), and the two other terms (Gamma, k). Thanks to this very general setting, those results carry over for free to natural expansions of valued field (by a derivation, an automorphism...). In this talk, we present our contribution to this work, where we establish similar transfer principles for *forking and dividing *in the same setting of expansions of PSES of Abelian structures.

If we complete the rational numbers $\mathbb{Q}$ with respect to the standard absolute value, we obtain the reals $\mathbb{R}$, wich is a connected topological space. The completion $\mathbb{Q}_p$ of $\mathbb{Q}$ with respect to the $p$-adic valuation for a prime $p$, however, is totally disconnected. So it seems that the concept of a path connecting two points in the $p$-adic numbers $\mathbb{Q}_p$ (or $\mathbb{Q}_p^n$) does not make sense. In fact, the space $\mathbb{Q}_p$ itself is not quite suitable for doing geometry. Instead one can consider the affine line $\mathbb{A}_{\mathbb{Q}_p}^1$ as an adic space. It contains $\mathbb{Q}_p$ as so called \emph{classical points} but has many more points. In this talk we will convince ourselves that $\mathbb{A}_{\mathbb{Q}_p}^1$ is indeed path connected if we use the right notion of a path.