A Quadratic Panorama

• Jean Fasel (Université Grenoble-Alpes)

Room: Amphithéâtre Yvonne Choquet-Bruhat In this talk, I will first survey well-known connections between algebraic K-theory and algebraic cycles, including the coniveau and Atiyah–Hirzebruch spectral sequences, as well as the classical Riemann–Roch theorem. I will then explain the motivations behind the current trend of enhancing these results using quadratic (or more exactly symmetric bilinear) forms, and highlight a few results in this direction.

Structured Flow Categories and Twisted Spectra

• Alice Hedenlund

Room: Amphithéâtre Yvonne Choquet-Bruhat In 1994, Cohen-Jones-Segal proposed a program to understand the homotopy theory underlying Floer theory. They proposed that a Floer problem should give rise to a “Floer homotopy type”, refining the associated Floer homology. Moreover, they discussed how such a Floer homotopy type might be constructed via the use of flow categories and in particular sketch how to obtain a spectrum from a framed flow category. More recently, Abouzaid-Blumberg show that framed flow categories can be arranged into a stable infinity-category and show that this is equivalent to the infinity-category of spectra. Far from all flow categories associated with Floer data are frameable, though. That some version of twisted stable homotopy theory is needed to deal with non-framed Floer homotopy theory has been known for a while. Twisted spectra were introduced by Douglas in his PhD thesis and recently recast in the infinity-categorical setting by Hedenlund-Moulinos. In this talk, we explain how these are indeed related to flow categories by exhibiting an equivalence between twisted spectra and flow categories structured by certain maps to U/O. This is joint work with Trygve Poppe Oldervoll.