Speaker: Emma Brink – University of Bonn
Abstract: For a compact Lie group $G$, we describe equivariant versions of geometric bordism with stable tangential structures. These define $\mathbb Z$-graded equivariant homology theories which admit a Thom-Pontryagin comparison map to the homology theory represented by associated Thom spectra.
If $G$ has non-abelian connected identity component (e.g. $G=SU(2)$), the geometric bordism theory is not represented by a genuine $G$-spectrum as it fails to admit certain Wirthmüller isomorphisms....
Speaker: Trygve Poppe – Norwegian University of Science and Technology
Abstract: Floer homotopy theory aims to extract stable homotopy invariants from certain geometric moduli problems that appear for instance in symplectic topology. Flow categories were introduced for this purpose by Cohen-Jones-Segal in the 90’s. The recent development of a good homotopy theory of flow categories has made the study of these objects more feasible, and of potential interest outside purely symplectic circles....
Speaker: Eigil Rischel – University of Strathclyde
Abstract: In computer science (and mathematics), we often study various different types of “machine” or dynamical system. There are a number of different ways of fitting such things into a category - there are various notions of morphism between such systems, but also ways of treating systems as morphisms in a higher category to build composite systems.
Combining these different notions, one arrives naturally at the conclusion that there should be a pseudo triple category of systems....
Speaker: Jonathan Sejr Pedersen – University of Toronto
Abstract: We prove that the Madsen-Tillmann spectrum $$MT\theta_n := Th(− \theta^{*}_ {n} \gamma_{2n}\longrightarrow BO(2n)\langle n\rangle)$$ splits into the sum of spectra $\Sigma^{−2n}MO⟨n⟩\oplus \Sigma^{\infty−2n}\mathbb{R}P^{\infty}_{2n}$ after Postnikov trunctation $\tau_{\leq l}$ for $l=\frac{n}{2}−c$. This is achieved by showing the connecting homomorphism $\tau_{\leq l}MO\langle n\rangle \longrightarrow \tau_{\leq l}\Sigma^{\infty+1}\mathbb{R}P^{\infty}_{2n}$ is nullhomotopic in this range by applying Adams spectral sequence arguments.
We discuss a number of applications, most prominently the computation of $H_2(B\mathrm{Diff}(W^{2n}_g,D^{2n});\mathbb{Z})$ which is connected to moduli spaces of high dimensional manifolds....
Speaker: Sonja M. Farr – University of Nevada
Abstract: In 1990 V. Drinfeld introduced a pro-unipotent algebraic group $\mathrm{GRT}$, which is closely related to the absolute Galois group $\mathrm{Gal}(\overline{Q}/Q)$. It was later shown by B. Fresse that this group is isomorphic to the group of homotopy automorphisms of the rationalization of the little disks operad. Deligne’s conjecture – now a theorem with multiple proofs – asserts that the natural operations on the Hochschild cochain complex of any smooth algebraic variety lift to an action of the dg operad of little disks in a precise way....
Speaker: Ben Szczesny – Ohio State University
Abstract: Blumberg and Hill have introduced the notion of $\mathbb{N}_ \infty$-operads as an equivariant extension of $\mathbb{E}_\infty$ that can also encode norm maps – a type of twisted multiplication. It has been shown that the homotopy category of $\mathbb{N}_\infty$-operads is equivalent to a category of combinatorial objects called transfer systems. In this talk we will introduce a new construction that allows us to build operads realizing transfer systems that is both simpler than the currently available methods and, in some respects, more general as it also allows us to construct homotopy incoherent operads reminiscent of $\mathbb{E}_k$-operads....
Full title: A point-free approach to fundamental groups: the Kennison fundamental pro-group of a connected and locally connected locale
Speaker: Jean Paul Schemeil – University of Nottingham
Abstract: The aim of this talk is to introduce the notions of the fundamental pro-sheaf and pro-group of locales, first introduced by John F. Kennison in “What is the fundamental group?” (Journal of Pure and Applied Algebra, 1989). We will review some foundational aspects of locale theory, motivating how locales can be correctly thought as “pointless” topological spaces....
Speaker: Guglielmo Nocera – Institut des Hautes Études Scientifiques (IHÉS)
Abstract: The Borel conjecture states that, for a discrete group $G$, any two closed topological manifolds homotopy equivalent to $BG$ are indeed homeomorphic to each other. The truth of the conjecture can be checked by inspecting whether the so-called assembly maps are equivalences. Informally, these maps relate certain $G$-equivariant homology theories to $K$-theory and $L$-theory of the group ring $\mathbb{Z}[G]$. In the special case of $G=\mathbb{Z}$, they are also related to classical splitting results in $K$- and $L$-theory of Laurent polynomials....
Speaker: Guoqi Yan – University of Notre Dame
Abstract: The equivariant slice spectral sequence is invented by Dugger and is generalized and utilized to a great extent by Hill–Hopkins–Ravenel in their resolution of the Kervaire invariant one problem. Later, because of the work of Hahn–Shi and Meier–Hill–Shi–Zeng, the equivariant slice spectral sequence can be used to compute the homotopy fixed point of Lubin–Tate theories. In this talk, I will talk about equivariant computations around the slice spectral sequence using the generalized Tate diagrams invented by Greenlees–May....
Speaker: Robert Szafarczyk – University of Copenhagen
Abstract: Every discrete ring is a $\mathbb{Z}$-algebra. In homotopy theory we have a deeper base though, the sphere spectrum. Hence, one can ask if a given scheme arises as base change to the integers of some spectral scheme. We will present an obstruction (a necessary condition) to lift existence. In the affine case, there is a relatively simple $1$-algebraic obstruction due to Thomas Nikolaus....