Log prismatic cohomology, motivic spectra and comparison theorems

• Alberto Merici (Università degli Studi di Milano La Statale)

Room: Aula Magna We prove that Nygaard completed log prismatic and log syntomic cohomology are represented by motivic ring spectra in the category of logarithmic motives of Binda-Park-Østvær. Using a descent technique inspired by the work of Nizioł on log K-theory, we construct comparison isomorphisms of motivic ring spectra classifying classical cohomology theories of logarithmic schemes (de Rham, Hodge, crystalline...). As an application, we immediately obtain Gysin maps that are compatible with the comparison isomorphisms, and we precisely identify their cofibers. This is joint work with F. Binda, T. Lundemo, and D. Park.

Combing a hedgehog (algebraically speaking)

• Marc Levine (Universität Duisburg-Essen)

Room: Aula Magna A classical result from differential topology tells us that an even-dimensional sphere does not admit a nowhere-vanishing vector field, in other words, the tangent bundle does not admit a nowhere zero section. Umberto Zannier noted that for the algebraic 2-sphere over a field $k$, i.e., the variety defined by $x^2+y^2+z^2=1$ in affine space over $k$, the tangent bundle does have a nowhere zero section for $k$ the field $\mathbb{Q}_p$ of p-adic rationals, if $p$ is odd, and asked: what about $p=2$ (for $k$= the real numbers, the topological result shows there is no nowhere zero vector field in the algebraic case as well). With Alexey Ananyevskiy, we applied some modern tools from $\mathbb{A}^1$ homotopy theory to show that the tangent bundle of the algebraic two-sphere over $\mathbb{Q}_2$ does admit a nowhere zero section. We went further and gave a nearly complete answer to the analogous question for a smooth affine quadric over an arbitrary field (of characteristic not 2). We will discuss the tools and methods that go into our proof. Joint work with Alexey Ananyevskiy.

A p-adic analogue of a theorem of Narasimhan and Seshadri

• Fabrizio Andreatta (Università degli Studi di Milano La Statale)

Room: Aula Magna Given a compact Riemann surface, a classical theorem of Narasimhan and Seshadri characterizes vector bundles arising from unitary representations of the fundamental group as polystable vector bundles of degree 0. Given a projective curve with a good reduction over a local field of mixed characteristic 0-p, works of Faltings and Deninger Werner associate semistable Higgs bundles of degree 0 to representations of the fundamental group. We explain some progress towards the question of characterizing which vector bundles arise in this way.

Finite approximations as a tool for studying triangulated categories

• Amnon Neeman (Australian National University & Università degli Studi di Milano La Statale)

Room: Aula Magna A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric. And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories. And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner, and Heller about the non-existence of bounded t-structures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.