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K-theory and Motivic Homotopy Theory Seminar - Elden Elmanto

Elden Elmanto
October 5, 2017
3:00 pm - 3:55 pm
Enarson Classroom Bldg 340

Title: Infinite Loop Spaces in Algebraic Geometry

SpeakerElden Elmanto (Northwestern University)

Abstract: A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception.

In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry.

Time permitting, I will also provide

  1. an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and,
  2. a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

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