April 18, 2023

**Title:** *The Segal conjecture and the field with one element (MI talk) *

**Speaker:** Christian Carrick

Finite sets behave in many ways as if they were vector spaces over a "field with one element," F_1. Via this analogy, many combinatorial statements may be read as statements about linear algebra over F_1. This analogy is more useful than one might expect, and we will discuss F_1-linear vector bundles over a space. This will allow us to recast the Segal conjecture in algebraic topology as an F_1-linear version of the classical Atiyah-Segal completion theorem on the K theory of classifying spaces. We finish by discussing a new proof of the Z/2 Segal conjecture using equivariant bordism theory.