# Lior Yanovsky: Higher semiadditivity (a.k.a. "ambidexterity") and chromatic homotopy theory

Time: Thu 2019-12-05 10.15 - 12.00

Lecturer: Lior Yanovsky, Max-Planck-Institut für Mathematik, Bonn

Abstract

According to the general philosophy of "brave new algebra", the stable $$\infty$$-category of spectra serves as a higher homotopical version of the usual category of abelian groups. In ordinary algebra one has the local-to-global paradigm of studying objects one prime at a time and then assembling the local information together. The chromatic picture affords and analogues paradigm for the oo-category of spectra. For each classical prime p, the Morava K-theories $$\mathrm{K}(n)$$ for $$0\leq n \leq \infty$$ interpolate between char 0 and char p. Thus, the categories of spectra localized with respect to these "intermediate" primes $$\mathrm{K}(n)$$ exhibit some intermediate characteristic behavior. A particularly remarkable property of the $$\mathrm{K}(n)$$-local categories is that of "higher semiadditivity" (introduced and proved by Hopkins and Lurie) which allows integration of morphisms along homotopically finite spaces. In this talk, I will give an exposition of the theory of higher semiadditivity and describe a joint work with Shachar Carmeli and Tomer Schlank establishing higher semiadditivity for the telescopic localizations $$Sp_T(n)$$ and applications to the construction of Galois extensions of the telescopic spheres $$S_T(n)$$.

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