| Résume | Après-midi de Topologie - 22 mai 2025
Organisée par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Muriel Livernet (IMJ-PRG), Najib Idrissi (IMJ-PRG).
Juan-Ramón Gómez-García : Turaev’s coproduct and parabolic restriction
Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the fundamental representation from GL_{m+n} to GL_m \times \GL_n. The restriction functor is, however, not braided hence it cannot be extended to the skein category of an arbitrary surface. So there was a priori no reason for Turaev's coproduct to be well-defined.. In this talk, I will explain how to construct a universal version of parabolic restriction on framed surfaces, using skein theory with defects. Precisely, parabolic restriction yields a morphism (bimodule) between the GL_t-skein category and (GL_t \times GL_t)-skein category of S. This construction depends on the choice of the framing, is compatible with gluing surfaces and recovers the Turaev’s coproduct when applied to links, justifying why this is well-defined.
Hyungseop Kim : Some formal gluing diagrams for continuous K-theory
The study of descent properties of K-theory plays an important role in understanding its values and behaviour in many geometric contexts. In this talk, I will explain a construction of certain diagrams arising from formal gluing situations for which continuous K-theory, and more generally all localising invariants, satisfy descent, from the perspective of dualisable categories. I will also discuss how this encompasses both Clausen–Scholze’s gluing result for analytic adic spaces and an adelic descent for dualisable categories.
Maria Yakerson : Fun facts about p-perfection
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. |
