Algebra/topology seminar

Speaker: Elden Elmanto

Title: Nonabelian Poincare Duality in A^1-Homotopy Theory for Fun and Profit

Abstract: In recent years, there has been at least two instances when homotopical ideas have been useful in studying geometry, construed broadly. The first is through a homology theory for E_n algebras (factorization homology), as developed by Lurie and Ayala-Francis, which is used to study manifolds and configuration spaces of them. The second is through A^1-homotopy theory, as developed by Morel and Voevodsky, which brings topological thinking into algebraic geometry.

This talk is a progress report on a fusion of these two areas. We will motivate why such an endeavor might be profitable, with reference to classical theorems in topology (Dold-Thom, Atiyah-Bott). We then give a definition of factorization homology in this context, and sketch the first calculation: an A^1-local analogue of nonabelian Poincare duality, inspired by recent work of Gaitsgory-Lurie. This talk will include a brief sampler to the techniques of factorization homology and the A^1 homotopy theory of schemes.