Quantitative geometry of arithmetic locally symmetric spaces

par Lam Pham

jeudi 23 avr. 2026, 14:00 → 15:00 Europe/Paris

Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour in mixed-characteristic remains less well understood. Building on recent advancements by Bachmann, Elmanto, Morrow, and Bouis, in a joint work with Bouis, we demonstrate a purity result over deeply-ramified bases. I will also discuss an application of this r

Arithmetic locally symmetric spaces (in the broad sense) occur in several areas of mathematics. Archimedean, non-archimedean, and adelic constructions can be studied using arithmetic and algebraic groups. There is a remarkable interplay of geometry, algebra, and arithmetic, and these objects connect several interesting areas (construction of expander graphs and high-dimensional expanders, Diophantine geometry, geometric analysis, geometric group theory, etc.). It is generally a difficult problem to understand this geometry quantitatively, in infinite families. I will recall several important constructions in this unified framework and present some recent work on conjectures of Margulis' (geometry) and Lehmer's (arithmetic) regarding lower bounds on systoles of arithmetic locally symmetric spaces (length of the shortest closed geodesic). Part of this work is joint with F. Thilmany and M. Fraczyk.

sult in p-adic Hodge theory.