BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Georg Biedermann (Universidad del Norte) DTSTART:20221021T150000Z DTEND:20221021T161500Z DTSTAMP:20260423T005718Z UID:CIRGET/75 DESCRIPTION:Title: Calculus in homotopy theory\nby Georg Biedermann (Universidad del Nort e) as part of CRM - Séminaire du CIRGET / Géométrie et Topologie\n\nLec ture held in PK-5115.\n\nAbstract\n(joint with M. Anel\, E. Finster and A. Joyal)\nIn classical calculus one studies smooth functions via their Tayl or series. Its $n$-th homogeneous layer is governed by a single coefficien t: the $n$-th derivative. As part of his effort to relate algebraic K-theo ry to topological cyclic homology Goodwillie during the 90s introduced "Go odwillie calculus" to homotopy theory. A homotopy invariant functor is vie wed as an analogue of a smooth function and resolved into a tower whose $n $-th homogeneous layer is governed by a single coefficient: a spectrum (in the sense of homotopy theory) with $\\Sigma_n$-action. Goodwillie calculu s is now a central tool in homotopy theory.\nAround the same time (and inf luenced by Goodwillie) Michael Weiss constructed "orthogonal calculus": sp ace-valued functors from the category of finite dimensional Euclidean vect or spaces with morphism given by Stiefel manifolds are resolved into an or thogonal tower whose $n$-th homogeneous layer is governed by a spectrum wi th an action by $O(n)$. Weiss' theory has found many applications in diffe rential topology.\nPeople have wondered for a long time whether both theor ies have a common description. We can give one. In fact\, it turns out tha t the theory of $\\infty$-topoi is the perfect language. For any left exac t localization $L$ of an $\\infty$-topos we construct a tower $(P_n)_{n\\g e 0}$ of left exact localizations such that $P_0=L$. The pointed objects o f the layers form stable $\\infty$-categories. The tower is analogous to t he completion tower of a commutative ring with respect to an ideal. It spe cializes to Goodwillie's and Weiss' tower.\n\nI am going to tell you a bit about all these towers.\n LOCATION:https://researchseminars.org/talk/CIRGET/75/ END:VEVENT END:VCALENDAR