Posts

Speaker: Ahina Nandy – Osnabrück University Title: An interpolation between the algebraic (co)bordism spectrum $MGL$, and the special linear algebraic (co)bordism spectrum $MSL$ Abstract: Just like the universal complex oriented cohomology theory, complex cobordism ($MU$), we have a similar notion of universal oriented cohomology theory in the realm of stable $A^1$-homotopy theory. Algebraic cobordism ($MGL$) would be our analog of $MU$. Similar to $SU$-(co)bordism $MSU$, there is a notion of “universal” special linear oriented theory in motivic world....

Speaker: Violeta Borges Marques – University of Antwerp Abstract: The goal of this talk is to introduce enriched versions of Segal categories, and in particular, Segal dg-categories. I will start by reviewing the classical notion of Segal category and then I will introduce templicial objects following Lowen-Mertens as replacements for simplicial objects in a non-cartesian context, allowing me to arrive at the definition of Segal dg-category. After this, I will outline a strategy to obtain an adequate model structure for these objects, taking this chance to overview the category of Necklaces, and finally I will show the results we have so far been able to prove....

Speaker: Catrin Mair – Technical University of Darmstadt Abstract: The $\infty$-category $Cond(Ani)$ of condensed anima combines the homotopy theoretic direction of anima with the topological space direction of condensed sets. Hence, it is natural to ask for its role in homotopy theory. For instance, one can assign to every condensed anima a pro-homotopy type by which we can recover the “shape” of a sufficiently nice topological space. In my talk, I will focus on explaining how to define a refinement of the étale homotopy type of a scheme as an object in $Cond(Ani)$....

Speaker: Qi Zhu – University of Bonn Abstract: We compare notions generalizing features of topology, namely condensed mathematics and cohesive resp. fractured structures on topoi. After introducing/recalling the theory of condensed mathematics and cohesive resp. fractured topoi, we discuss their interrelations. Once we have seen that cohesion is not sensible on the $\infty$-topos of condensed anima $\mathbf{Cond(An)}$ we provide a fractured structure on $\mathbf{Cond(An)}$. We apply this to compare sheaf cohomology and condensed cohomology and show that for the corporeal objects of the fractured structure which are also topological spaces the cohomologies agree....

Speaker: Itamar Mor – Queen Mary University of London Abstract: Using the proetale site of the Morava stabiliser group, I give a construction of the $K(n)$-local $E_n$ Adams spectral sequence as a HFPSS. A modified version gives a spectral sequence computing the Picard and Brauer groups of $K(n)$-local spectra. body {text-align: justify}

Speaker: Arun Debray – Purdue University Abstract: The Atiyah-Bott-Shapiro map $\widehat{A}\colon MSpin \longrightarrow KO$ is the preeminent example of an orientation in stable homotopy theory. In the first part of this talk, I will describe the data of $\widehat{A}$: what are $MSpin$ and $KO$, and what is the map between them? In the second half of the talk, I will describe how $\widehat{A}$ and some of its variants provide a mathematical model for the classification of topological phases of matter with free fermion Hamiltonians....

Speaker: Jack Davies – University of Bonn Abstract: Topological $K$-theory has a natural extension to a $G$-equivariant cohomology theory for compact Lie groups $G$—one just simply exchanges vector bundles with $G$-equivariant vector bundles in the usual construction of $K$-theory. There is a reinterpretation of this equivariant cohomology theory though, in terms of algebraic geometry. Although this second perspective could be taken as an insightful curiosity for $K$-theory, for equivariant elliptic cohomology this perspective becomes our (only) definition....

Speaker: Kaya Arro – University of California Riverside Abstract: Representation stability is a regularity property enjoyed by the homology of many naturally arising objects indexed by the category of finite sets and injections. It turns out, any such FI-object in a presentable stable $\infty$-category admits a Taylor tower of approximations by representation stable FI-objects. From this Taylor tower, one obtains Taylor coefficients as well as natural transformations between these coefficients, and, assuming the vanishing of a Tate construction (satisfied e....

Speaker: Maxime Ramzi – University of Copenhagen Abstract: In this talk, I will give a general introduction to the theory of dualizable presentable categories, as is currently being developped by Efimov and others for K-theoretic purposes. The goal is to talk about generalities, with an end goal to prove that the category of dualizable categories is itself presentable. body {text-align: justify}

Speaker: Andrei Konovalov – University of Duisburg-Essen Abstract: I will discuss the problem of constructing a natural rational structure on periodic cyclic homology of dg-algebras and dg-categories. The promising candidate is A. Blanc’s topological K-theory. I will show that it, indeed, provides a rational structure in a number of cases and I will discuss its structural properties and possible applications. body {text-align: justify}