Student Homotopy Seminar 

Title: Goerss-Hopkins obstruction theory and synthetic spectra

Abstract: Given an $E_*E$-comodule for some homology theory $E$, when is it actually the $E$-homology of the space or spectrum? If the comodule is actually a ring, when is it the $E$-homology of an $E_\infty$ ring spectrum? Goerss-Hopkins obstruction theory is a family of methods for answering questions like these. Piotr Pstrągowski and I have shown that these obstruction theories are a feature of any $\infty$-category equipped with a certain kind of periodicity; the classical Goerss-Hopkins obstruction theories about spectra can be obtained from Piotr's $\infty$-categories of "synthetic spectra". I'll also mention some interesting questions about the convergence of these obstruction theories, which is closely related to the existence of vanishing lines in Adams spectral sequences.

Factorization homology and nonablian Poincaré duality.

The Hochschild homology of the ring $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ has been known and calculated several ways. This talk uses those calculations to calculate cyclic, negative cyclic, and periodic homology of the ring. The calculations are relatively straightforward for $k=\mathbb{Q}$, but we see interesting torsion phenomena over $k=\mathbb{Z}$.

Title: Lifting Modules

Abstract: Let $B$ be a subring of $A$. Any $A$-module is also a $B$-module (simply forget the action of elements in $A$ that are not in $B$). However, there is no guarantee that any $B$-module can be given an action of $A$ that agrees with the existing $B$-module structure. If a $B$-module can be given a compatible action of $A$, we say it lifts to an $A$-module.

We'll talk about the lifting problem in the case where $A$ is the mod 2 Steenrod algebra and give a criterion for determining when a certain class of modules lift to modules over $A$. A key tool for these results is Margolis homology, so we'll discuss the definition and uses of this invariant.

Title: The Picard Group of the Stable Module Category for Quaternion Groups

Abstract: One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thévenaz using the theory of support varieties. However, I provide new, homotopical proofs of their results for the quaternion group of order 8, and for generalized quaternion groups, using the descent ideas and techniques of Mathew and Mathew-Stojanoska. Notably, these computations provide conceptual insight into the classical work of Carlson-Thévenaz.

Title: Multiplicative Motivic Infinite Loop Space Theory

Abstract: From a spectrum $E$ one can extract its infinite loop space $\Omega^\infty E = X$. The space $X$ comes with a rich structure. For example, since $X$ is a loop space, we know $\pi_0 X$ comes with a group structure. Better yet, since $X$ is a double loop space, we know $\pi_0 X$ is in fact an abelian group. How much structure does this space $X$ possess? In 1974 Segal gave the following answer to this question: the structure of an infinite loop space is exactly the structure of a grouplike $E_\infty$ monoid. In fact, this identification respects multiplicative structures. In this talk, I'd like to discuss the analogue of this story in the setting of motivic homotopy theory. No knowledge of motivic homotopy theory will be assumed.

Title: Approximating homotopy with algebra

Abstract: I'll talk about some machinery you can use to custom-build "algebra to homotopy" spectral sequences and obstruction theories, along the lines of Künneth spectral sequences or obstruction theories built from power operations for computing maps between ring spectra.

Title: In search of new model structures on simplicial sets

Abstract: In the way Kan complexes and quasi-categories model up-to-homotopy groupoids and categories, can we find model structures on simplicial sets which give up-to-homotopy versions of more general objects? We investigate this question, with the particular motivating example of 2-Segal sets. Cisinski's work on model structures in presheaf categories provides abstract blueprints for these new model structures, but turning these blueprints into a satisfying description is a nontrivial task. As a first step, we describe the minimal model structure on simplicial sets arising from Cisinski's theory.

Title: The $H\underline{\mathbb{F}}_2$ homology of $C_2$ Eilenberg-MacLane Spaces

Abstract: While a number of equivariant Steenrod algebras are well understood, the $RO(G)$-graded homology of $G$-equivariant Eilenberg-MacLane spaces has remained more mysterious. I will discuss current computations using a twisted version of the classical bar spectral sequence where the input is a $C_2$ - equivariant Eilenberg-MacLane space. Extending a non-equivariant argument of Ravenel and Wilson, I will outline how to compute $RO(C_2)$-graded homology in this case.

Title: Rigidity of the $K(1)$-local stable homotopy category

Abstract: In some cases, it is sufficient to work in the homotopy category $Ho(C)$ associated to a model category $C$, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question has been investigated during the last decade, and some examples have been studied, but there are still a lot of open questions regarding this subject. Starting with the stable homotopy category $Ho(Sp)$, that is the homotopy category of spectra, it has been proved to be rigid by S. Schwede. In this talk, we investigate a new case of rigidity, which is the localisation of spectra with respect to the Morava $K$-theory $K(1)$, at $p=2$.