Posts

Speaker: Liam Keenan – University of Minnesota Abstract: One natural invariant associated to a ring spectrum R is its topological Hochschild homology. If $R$ admits a multiplicative filtration or grading, then these both prolong to $THH(R)$ inducing a “filtered” or “graded” cyclotomic structure. In this talk, I’ll try to explain this point of view and survey some applications, highlighting the case of the May filtration and the $THH$ of a twisted group ring....

Speaker: Elizabeth Tatum – University of Bonn Abstract: In the $G$-equivariant setting, one typically uses $RO(G)$-graded cohomology theories. Costenoble-Waner have constructed an extension of this grading, the $RO(\Pi)$-grading, which allows classical results from nonequivariant topology, such as the Thom Isomorphism Theorem and Poincare duality, to be imported to the $G$-equivariant setting. I will discuss the computation of the cohomology of $BC_{2}O(1)$, the classifying space for real $C_{2}$-line bundles, in this extended grading....

Speaker: Ben Szczesny – Ohio State University Abstract: A classic result by Dunn establishes the additivity of the little cube operads with respect to the Boardman-Vogt tensor of operads. Unlike its $\infty$-category counterparts, the Boardman-Vogt tensor does not preserve homotopic properties, marking Dunn’s result as notably distinct. This talk aims to demystify Dunn’s additivity for graduate students and explain the speaker’s work in extending Dunn’s results to various embedding operads, including equivariant and framed little disks....

Speaker: Florian Riedel – University of Copenhagen Abstract: I discuss how we can use deformation theoretic methods to show that Weil-restriction of coalgebras lets us recover the $p$-completed suspension spectrum of a space from its $\mathbb{F}_ p$-homology as its unique lift to an $\mathbb{E}_\infty$-coalgebra in $p$-complete spectra. I will talk about higher deformation theory as developed by Lurie and introduce the notion of formally étale coalgebra. I will also sketch how this relates to work in progress on construct an integral coalgebraic model for the category of spaces....

Speaker: Julie Rasumsen – University of Warwick Abstract: In recent years, several different attempts at developing the theory of enriched $\infty$-categories have been carried out by Gepner-Haugseng, Heine, and Hinich. These different approaches all have their advantages and disadvantages but have been shown to be equivalent. However, the theory is yet to be uniformly formulated using a single language, making it difficult to approach. In this talk, I will give a general overview of these different approaches and explain why intuitively, in particular, the approach by Gepner-Haugseng gives the desired structure....

Speaker: Torgeir Aambø – Norwegian University of Science and Technology (NTNU) Abstract: Chromatic homotopy theory is the “quantum mechanics” of stable homotopy theory, where we split the theory down and study its smallest and most fundamental pieces. It has gotten the name “chromatic” as each piece operates at a certain fundamental frequency, or a certain wavelength. This gives a filtration of the pure white light $Sp$ into colors $Sp_{K(n)}$, and collections of colors $Sp_{E(n)}$....

Speaker: Arne Mertens – University of Antwerp Abstract: I will introduce a candidate model for ($\infty$-)categories weakly enriched in simplicial objects, based on the Joyal model for quasi-categories. The main idea is to replace the category of simplicial sets by a category of certain colax monoidal functors, inspired by a result of Leinster and Bacard’s work on Segal enriched categories. We call them “templicial objects” and define quasi-categories analogously to the classical situation....

Speaker: Klaus Mattis – University of Mainz Abstract: In homotopy theory, one uses the $p$-completion functors to simplify questions about spectra or anima to the much easier case of $p$-complete spectra or anima. A similar thing can be done in motivic homotopy theory: I will define the unstable $p$-completion functor of a general $\infty$-topos and of the category of motivic spaces. Then I will sketch how one can translate classical results about unstable $p$-completion from homotopy theory to the motivic setting....

Speaker: Marco Praderio Bova – Lancaster University Abstract: Fusion systems are categories that, in a sense, represent an abstraction of the $p$-local structure of a finite group. Mackey functors on the other hand are algebraic structures with induction, restriction and conjugation operations satisfying certain properties that seem to appear in a variety of different contexts such as representation theory, group cohomology or algebraic K-theory among others. Mackey functors can be related to a variety of different constructions including fusion systems and, when related to fusion systems, they can be viewed as a pair of a covariant and a contravariant functors....

Speaker: Vignesh Subramanian – Indian Statistical Institute, Kolkata Abstract: Given an $\mathbb{E}_{\infty}$-algebra $A$ over $\mathbb{F}_p$, just as in classical algebra, there exists a homotopically coherent version of Frobenius on $A$, called the Tate valued Frobenius $A \rightarrow A^{tC_p}$. In this talk, we recall the notion Frobenius perfect $\mathbb{F}_p$-algebra and construct a certain version of perfection $A^{\flat}$, we call this construction tilting and give an explicit formula for the computation of homotopy groups of the tilt via power operations....