Let $G$ be a finite $p$-group. The stable module category of $G$ is defined as the quotient of the category of $G$-representations over a field $k$ of characteristic $p$ by those morphisms which factor through a projective. It can also be modeled as the category of module spectra over the Tate construction $k^{tG}$.

It is a classical theorem of Dade that the Picard group of the stable module category contains no "exotic" objects when $G$ is abelian. This translates into a statement about the $E_\infty$-ring spectrum $k^{tG}$. We will discuss a general approach to studying the Picard groups of structured ring spectra using descent theory and describe a new proof of Dade's theorem based on Rognes's theory of Galois extensions of ring spectra and Galois descent.

Motivated by the study of Adams spectral sequence, I will explain how techniques from tensor triangulated geometry gives some new insight on some features of the stable category of modules over sub-Hopf algebras of the modulo 2 Steenrod algebra (and also recover old computations, in particular cases).

In particular, this setting explains the importance of some non-bounded localizations of A(1)-modules which appeared in the work of Bruner. Finally, I will show how this allows the study of some local detection of Picard group for these categories.

In characteristic zero, a functorial derived version of the Hochschild--Kostant--Rosenberg theorem connecting Hochschild homology the de Rham complex follows from identification of the rational cochain algebra of the circle with a trivial square-zero extension algebra. The positive characteristic analogue of this result is false, however. Rather, the mod-p cochain algebra of the circle has a natural interpretation in terms of the p-typical Witt vectors, reminiscent of computations due to Hesselholt and Hesselholt--Madsen of topological cyclic homology in this setting.

J.P. May proved that traces in stable monoidal homotopy categories are additive in cofiber sequences. It is then natural to ask about the interaction of traces and homotopy colimits in general. In this talk we will give an answer for homotopy colimits indexed over EI-categories. The proof involves generalizing the notion of traces from the realm of categories to derivators. It will be sketched if time permits.

It is a classical problem to determine which complex topological vector bundles on $P^n$ admit an algebraic structure. We will discuss ``obstructions" to equipping a bundle with an algebraic structure coming from $A^1$-homotopy theory (though I won't presuppose any knowledge of $A^1$-homotopy theory). The main goal of the talk is to analyze the question of whether some rank 2 bundles on $P^{n}$ constructed by Elmer Rees using homotopy theoretic techniques admit an algebraic structure. This talk is based on joint work with Jean Fasel and Mike Hopkins.

An $(\infty, n)$-category should have morphisms at all levels, but those morphisms should be weakly invertible above level $n$. While the case of $n=1$ is fairly well-understood, with several different models known to be equivalent, but there are many different ways to generalize each of them for higher $n$. Making these different models precise and giving explicit comparisons is currently being done by several different authors. In this talk we'll look at several of the approaches to $(\infty, n)$-categories and known comparisons between them.

I'll be speaking in some joint work with Andrew Blumberg where we describe and classifying the algebra underlying the kinds of equivariant commutativity

When one shines white light through a prism it splits up into the rainbow. When one looks at the stable homotopy groups of spheres through the Adams spectral sequence one sees $v_n$-periodicities. I'll explain what this means and the related computational procedure.

I plan on this being an approachable talk for graduate students. Only in the last part of the talk will I mention related results in motivic and equivariant homotopy theory.

The Toledo number is a numerical invariant associated to representations of fundamental groups of surfaces into Lie groups of Hermitian type. Maximal representations are those representations for which the Toledo number is maximal. They form connected components of the representation variety. In the case when the Lie group is $SL(2,R)= Sp(2,R)$ they correspond precisely to holonomy representations of hyperbolic structures. Maximal representations into the symplectic group $Sp(2n,R)$ generalize this situation with a lot of new features appearing. I will describe some of these new features and explain how maximal representations arise as homonym representations of projective structures on iterated sphere bundles over surfaces.

Abstract: The talk will start with mapping degree sets and
simplicial volumes. We then discuss recent results on virtual
representation volumes and on virtual dominations of 3-manifolds, as well
as their relations. Time permitted, we may end with the high dimensional
applications of representation volumes.
This is joint work with P. Derbez, Y. Liu and H. Sun.

Suppose one has determined the Floer theory algebra of a finite non-empty collection of Lagrangians in a Calabi-Yau manifold. I will explain that, if the resulting algebra satisfies a finiteness condition called homological smoothness, then the collection automatically split-generates the Fukaya category. In addition, the Hochschild invariants of the algebra (and hence of the whole Fukaya category) are automatically isomorphic to the quantum cohomology ring. This result immediately extends to the setting of monotone/non-Calabi-Yau symplectic manifolds, under an additional hypothesis on the rank of the algebra’s 0th Hochschild cohomology. The proofs make large use of joint work with Perutz and Sheridan, which in turn is part of a further story about recovering Gromov-Witten invariants from the Fukaya category.

A link is called an L-space link if all sufficiently large surgeries along it are L-spaces. It is well known that the Heegaard Floer homology of L-space knots have rank 0 or 1 at each Alexander grading. However, for L-space links with many components the homology usually has bigger ranks and a rich structure. I will describe the homology for algebraic and cable links, following joint works with Jen Hom and Andras Nemethi. In particular, for algebraic links I will construct explicit topological spaces with homology isomorphic to link Floer homology.

A pseudo-Riemannian manifold is a manifold where each tangent space is endowed with a quadratic form that is non-degenerate, but not necessarily positive definite. A typical example is the hyperbolic space $H(p,q)$, which is a pseudo-Riemannian manifold of signature $(p,q)$ and constant negative sectional curvature. It is homogeneous, as it admits a transitive isometric action of the Lie group $SO(p,q+1)$. A long standing question is to determine for which values of $(p,q)$ one can find a discrete subgroup of $SO(p,q+1)$ acting properly discontinuously and cocompactly on $H(p,q)$. In this talk I will show that there is no such action when $p$ is odd and $q >0$. The proof relies on a computation of the volume of the corresponding quotient manifold. The proof also implies that, when $p$ is even, this volume is essentially rational. I will discuss in more details the case of $H(2,1)$ (the 3-dimensional anti-de Sitter space), for which compact quotients exist and have been described by work of Kulkarni-Raymond and Kassel.

The Homfly skein relations from knot theory can be used to associate an algebra to each (topological) surface. The Hall algebra construction associates an algebra to each smooth (algebraic) curve over a finite field. Using work of Burban and Schiffmann, we show that the skein algebra of the torus is isomorphic to the Hall algebra of an elliptic curve. If time permits we discuss a third (categorical) construction of the same algebra. (Joint with Morton and Licata.)