# Upcoming Talks

The seminar will meet at 4:00pm on Tuesdays in Eckhart room 203 unless otherwise noted. There will also be a pretalk at 2:30pm in the same room.

• Nov 122019

### Jun Hou Fung (Harvard)

#### Strict units of commutative ring spectra

Just as an ordinary commutative ring has a multiplicative group of units, a E_\infty-ring spectrum R also has a spectrum of units gl_1 R, which plays an important role for example in orientation theory and twisted cohomology theories. However, these spectra are typically very large, and to understand twists by Eilenberg-Mac Lane spaces or to isolate those units that come from geometry, it sometimes suffices to study the space of \emph{strict units} of R. Previously, Hopkins and Lurie have computed the strict units of Morava E-theories, but much remains unknown about them in general. In this talk, I will introduce these strict units and illustrate various methods for computing them, and sketch how these calculations relate to other interesting questions in homotopy theory.

• Nov 192019

TBA

• Nov 262019

### Tim Campion (Notre Dame)

#### Duality in homotopy theory

We explore some implications of a fact hiding in plain sight: Namely, the n-sphere has the remarkable property that the “swap” map σ: Sn ∧ Sn → Sn ∧ Sn can be “untwisted”: it is homotopic to (-1)n ∧ 1. This simple fact remains true in equivariant and motivic contexts. One consequence is a structural fact about symmetric monoidal ∞-categories with finite colimits and duals for objects: it turns out that any such category splits as the product of three canonical subcategories (for instance, one of these subcategories is characterized by being stable). As another consequence, we show that for any finite abelian group G, the symmetric monoidal ∞-category of genuine finite G-spectra is obtained from finite G-spaces by stabilizing and freely adjoining duals for objects. This universal property vindicates one motivation sometimes given for studying genuine G-spectra: namely that genuine G-spectra (unlike naive G-spectra or Borel G-spectra) have a good theory of Spanier-Whitehead duality. We take steps toward a similar universal property for nonabelian groups and also in motivic homotopy theory.

• Dec 32019

TBA

• Dec 102019

TBA