Tuesday, September 12 at 1:00pm to 3:00pm

Kaprielian Hall (KAP), 414

3620 South Vermont Avenue, Los Angeles, CA 90089

**Elden Elmanto, Northwestern University**

Abstract:

A familiar notion in topology is that of a spectrum: sequences of spaces {Xi}i ∈ ℕ equipped with ``bonding maps" ϵi: Xi → Ω Xi. If each of the bonding maps are weak homotopy equivalences, then one can think of a spectrum as a gadget that encodes the notion of an infinite loop space in the sense that for all n ∈ ℕ, we have a weak homotopy equivalence Ωn Xn ≃ X0. A classical question in topology asks: How can one tell if a space is an infinite loop space? A solution to the above question should isolate key structural features of a pointed space X0 such that we can produce these Xi's and their bonding maps. One solution was due to Graeme Segal. The punchline of his solution reads: ``Infinite loop spaces are obtained from spaces by formally adjoining wrong way maps along finite covering maps."

In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we give a solution to the analogous problem in motivic homotopy theory. Our punchline reads: ``motivic infinite loop spaces are obtained from motivic spaces by formally adjoining wrong way maps along finite syntomic morphisms equipped with trivializations of of their relative stable normal bundle." I will make this punchline precise, and explain how to obtain this theorem from recent work on framed motives by the ``St. Petersburg school." Time permitting, I will explain how our theorem gives a concrete interpretation of the motivic sphere spectrum as the motivic homotopy type of a certain explicit ind-smooth scheme. Only rudiments of motivic homotopy theory will be assumed. In particular, I will explain our take on framed motives from scratch.