Jakub Löwit (ISTA)
Equivariant K-theory and trace maps after perfection abstract
Abstract:
et X be a variety with a G-action. On one hand, we can consider its equivariant algebraic K-theory. On the other hand, we can study the associated family of fixed points -- this has interesting geometry and its global functions model equivariant Hochschild homology. These two sides compare by the so-called Dennis trace map. Although ubiquitous in the study of K-theory of rings, this comparison did not receive much attention in the equivariant setup. In the first part of the talk, I will explain this setup and show some simple examples.In the second half of the talk, I will explain how to control the trace map on singular examples in positive characteristic after inverting the Frobenius. It becomes an equivalence in interesting situations. In particular, this is the case for affine Schubert varieties in the perfect affine Grassmannian of GL_n. In a different direction, same techniques can be used to prove non-vanishing results for negative equivariant K-theory of proper toric varieties.
14:15 • EPF Lausanne, CM 1 517
Artem Semidetnov (UniGE)
The operad associated to a crossed simplicial group abstract
Abstract:
By the Kan–Thurston theorem, any connected homotopy type is realizable as (BG)+ for a discrete group G. Deep results of Barratt–Priddy–Quillen, Galatius, Madsen–Weiss, and others concretely identify the homotopy types of (BG∞)+ for families such as Gn = Σn+1,Bn+1, Aut(Fn),GLn(K). However, these methods do not extend to other natural families, such as the homotopy braids Gn = hBn+1, for which (BhB∞)+ remains unexplored.In this talk I will present an operadic enhancement of the crossed simplicial groups of Loday–Fiedorowicz. To each operadic crossed simplicial group one associates an operad in groupoids, recovering the E∞- and E2-operads in the symmetric and braid cases. The main applications are a generalized bar construction specializing to the classical symmetric and braided variants, and an identification of the associated group-completed monads with Barratt– Priddy–Quillen type spaces—providing a new approach to (BhB∞)+, (BΣ∞)+, and (BB∞)+.
15:30 • Université de Genève, Conseil Général 7-9, Room 1-07