| Résume | Après-midi de Topologie - 30 janvier 2025
Organisée par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Muriel Livernet (IMJ-PRG), Najib Idrissi (IMJ-PRG).
Victor Roca i Lucio (CNRS / UPCité) : Higher Lie theory in positive characteristic:
Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L-infinity-algebras, giving in general infinity-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.
Robin Sroka (Universität Münster) : Scissors automorphism groups and their homology
Two polytopes in Euclidean n-space are called scissors congruent if one can be cut into finitely many polytopic pieces that can be rearranged by Euclidean isometries to form the other. A generalized version of Hilbert's third problem asks for a classification of Euclidean n-polytopes up to scissors congruence. In this talk, we consider the complementary question and study the scissors automorphism group -- it encodes all transformations realizing the scissors congruence relation between distinct polytopes. This leads to a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory. By varying the notion of polytope, scissors automorphism groups recover many important examples of groups appearing in dynamics and geometric group theory including Brin--Thompson groups and groups of rectangular exchange transformations. Combined with recently developed computational tools for scissors congruence K-theory, we recover and extend calculations of their homology. This talk is based on joint work with Kupers--Lemann--Malkiewich--Miller.
Maria Yakerson (CNRS / Sorbonne Université) : An alternative to spherical Witt vectors
Witt vectors of a ring form a “bridge” between characteristic p and mixed characteristic: for example, Witt vectors of a finite field Fp is the ring of p-adic integers Zp. Spherical Witt vectors of a ring is a lift of classical Witt vectors to the world of higher algebra, much like sphere spectrum is a lift of the ring of integers. In this talk we will discuss a straightforward construction of spherical Witt vectors of a ring, in the case when the ring is a perfect Fp-algebra. Time permitting, we will further investigate the category of modules over spherical Witt vectors, and explain a universal property of spherical Witt vectors as an E1-ring. This is joint work with Thomas Nikolaus. |
