## ARITHTMETIC GEOMETRY AND NUMBER THEORY

DEPARTMENT OF MATHEMATICS ⟡ UNIVERSITY OF MILAN

**THE GROUP**

The arithmetic geometry group at the University of Milan covers a broad range of topics including

- p-adic Hodge theory
- algebraic cycles and L-functions
- arithmetic of modular and automorphic forms
- motivic homotopy theory
- K-theory
- logarithmic and rigid analytic geometry
- analytic and algorithmic aspects of number theory

Faculty members

- Fabrizio Andreatta
- Federico Binda
- Luca Barbieri-Viale
- Carlo Mazza
- Giuseppe Molteni
- Amnon Neeman
- Marco Adamo Seveso
- Rodolfo Venerucci
- Alberto Vezzani
- Paul Arne Østvær

Post-docs and PhD students

- Luca Dall’Ava
- Alberto Merici
- Raphaël Ruimy
- Kaixing Cao
- Ruichuan Zhang

**SEMINARS**

We host a regular seminar, focusing in arithmetic geometry, number theory and motivic homotopy theory

9 September 2024 – 14.30 Aula Dottorato

Swann Tubach (ENS Lyon)

**Higher enhancements of mixed Hodge modules**

Let X be a smooth complex algebraic variety. Deligne proved that the isomorphism between singular and de Rham cohomology of X give rise to a mixed Hodge structure, which led to a very fruitful area of research. M. Saito constructed a theory of Mixed Hodge modules, which is a sheafy version of mixed Hodge structures, and in which the complex of mixed Hodge structures computing the cohomology of X is naturally a derived pushforward of an object living on X, as in ell-adic cohomology or singular cohomology. We prove that the constructions of Saito (that is the derived category of mixed Hodge modules together with the 6 sheaf operations) can be lifted to the world of infinity-categories in a very coherent way. This is done using a « change of variables »: adapting a proof of Nori, we can prove that the derived category of (perverse) mixed Hodge modules is also the derived category of constructible Hodge modules. This enhancement has useful applications, such as a Hodge realisation of motives commuting with the 6 operations and an extension to stacks of the formalism of mixed Hodge modules. This also applies to perverse Nori motives.

**PIZZA SEMINARS**

“Pizza seminars”, funded by the ALGANT Consortium, offer a laid-back atmosphere in which students and young researchers can interact with the speaker and with each other…eating pizza!

*There are no upcoming events*