## Online Seminar: Integral Homotopy Theory

Given a simply connected topological space, its rational and p-adic homotopy types can be understood in terms of the algebras of its cochains. In more detail, rational homotopy theory, due to Sullivan and Quillen, states that associating to a space its cochain algebra with rational coefficients induces a fully faithful embedding from simply connected rational spaces to rational commutative dgas. Later, an analogous statement for the p-adic case was proved by Mandell, with cdgas replaced by $E_\infty$-algebras. However, there was no known way to "glue" the information about rational and p-adic cochains together, in order to reconstruct the integral homotopy type. In his recent paper "Integral models for spaces via the higher Frobenius", Yuan solves this problem by refining the p-adic model, using Nikolaus-Scholze Frobenius map on $E_\infty$-rings as one of the main instruments. In this seminar, we will study the paper by Yuan. We will use the language of modern homotopy theory, the necessary notions from the Nikolaus-Scholze paper will be recalled.

## Dates and location

SS 2020, Tuesdays, 14-16, Online Seminar via Zoom.

 Talk Date Title Speaker 1 5. May Overview Marc Hoyois 2 12. May Tate construction and Tate diagonal Harry Gindi 3 19. May Equivariant stable homotopy theory Marc Hoyois 4 26. May (at 16:00) The $E_\infty$ Frobenius Martin Speirs 5 2. June Borel global algebras Peter Haine 6 9. June Partial algebraic K-theory: $S_\bullet$-construction Maria Yakerson 7 16. June Partial algebraic K-theory: Q-construction Denis-Charles Cisinski 8 23. June The partial algebraic K-theory of $F_p$ Denis Nardin 9 30. June The p-complete Frobenius and the action of $B\Z_{\geqslant 0}$ Milton Lin 10 7. July Integral models for the unstable homotopy category Georgios Raptis 11 14. July E-infinity coalgebras and a generalized Segal conjecture Allen Yuan