Chicago Chromatic Homotopy Day

Some questions about Wilson spaces

Wilson spaces are (p-local) spaces with even cells and even homotopy, such as CP^\infty and BU.  In this talk, I will review some basic facts about Wilson spaces and then discuss some phenomena that arise when studying their homology with respect to chromatic ring spectra.  The goal of the talk will be to formulate some open ended questions (and unfortunately, will not contain any answers).  

Looking for higher height examples in geometry and physics

In the vast zoo of cohomology theories, only a handful have known descriptions in terms of geometric cocycles: ordinary cohomology, K-theory, and cobordism. The Stolz-Teichner program posits the existence of geometric cocycles for topological modular forms coming from 2-dimensional quantum field theories. Applying similar ideas to other examples from physics leads to some intriguing parallel structures in chromatic homotopy theory and geometry. This talk will review some of these ideas and speculate on some promising examples, extending prior work of Alvarez and Singer. This is joint with Natalia Pacheco-Tallaj. 

On some relationships between homotopy theory and geometric representation theory

Homotopy theory plays two dual roles in mathematics: it can be viewed either as providing good “coefficient systems” with respect to which geometric objects like spaces can be linearized; or it can be viewed as providing a refinement of the theory of algebra, with respect to which one can do algebraic geometry. The former is classical. The latter can be interpreted in the sense of either spectral algebraic geometry, or the theory of even stacks (Hahn-Raksit-Wilson, and joint work of mine with Hahn-Raksit-Yuan), which constructs algebro-geometric objects from homotopy theory (like the moduli stack of formal groups from the sphere spectrum). Matching up these two “dual” roles of homotopy theory is, in some sense, the goal of various mirror symmetry-esque statements. 

In this talk, I will sketch some applications of this perspective to geometric representation theory: namely, if k is an E_∞-ring, the construction of even stacks (and “categorifications” thereof) relates interesting ring spectra like C^*(BG; k) or categories like Shv(Gr_G/G; k) for a connected complex reductive group G to algebro-geometric objects over the even stacks of k constructed from the Langlands dual group G^. When k is an ordinary commutative ring, this leads to (a slight variant of) the derived geometric Satake equivalence; and when k is the sphere spectrum and G = *, this is the relationship between the category of spectra and quasi coherent sheaves on the moduli stack of formal groups. This perspective leads to many questions which I find interesting, and I want to explain some of them.

Higher semiadditive power operations

Power operations for E_∞-rings are a classical theme in homotopy theory, In this talk I will discuss the interaction of this theory with higher semiadditivity both in the chromatic and the categorical setting. In particular, I will explain how this context gives rise naturally to the notion of a beta-ring and clarifies its relation to the more familiar one of a lambda-ring. I will conclude by discussing the Rezk logarithm and its formal inverse the "Ganter exponential" from this perspective.