Schedule

Titles and abstracts

Lukas Brantner: Canonical lifts of ordinary Calabi–Yau varieties

Classical Serre–Tate theory shows that ordinary abelian varieties and K3 surfaces in characteristic p admit canonical lifts to characteristic zero. In this talk, I will explain how derived deformation theory can be used to generalise this result to Calabi–Yau varieties. If time permits, I will also present some intriguing questions that arise from our canonical lifts, and discuss a generalisation of the BTT theorem to characteristic p. This is joint work with Taelman.

Joana Cirici: Configuration spaces of algebraic varieties I

I will use the theory of weights in étale cohomology to give a simple and conceptual proof of a theorem of Kriz stating that a rational model for the ordered configuration space of a smooth complex projective variety is given by the second page of the Leray spectral sequence for the rational constant sheaf relative to the obvious inclusion, along with its only non-trivial differential. Our proof builds on Totaro's study of this spectral sequence, combined with a basic observation related to the formality of filtered dg-algebras. An advantage of this proof is that it allows for a generalization to study the p-adic homotopy type of configuration spaces for certain algebraic varieties defined over finite fields. This is joint work in progress with Geoffroy Horel.

Johannes Ebert: Tautological classes and higher signatures 

For a bundle $\pi:E \to X$ of closed smooth oriented odd-dimensional manifolds, it has been known for some time that the tautological class $\kappa_{L_m}(E)$ associated to the Hirzebruch $L$-class vanishes. In this talk, we consider a slightly more general situation: given in addition a discrete group $\Gamma$, a map $f: E \to B \Gamma$ and a cohomology class $u \in H^* (B\Gamma;\mathbb{Q})$, we define $\kappa_{L_m,u} (E,f):= \pi_! (L_m (T_v E) \cup f^* u)$ which is a family version of Novikov's higher signatures. One might ask whether $\kappa_{L_m,u} (E) =0$ for all bundles of odd-dimensional manifolds. Surprisingly, the answer depends heavily on $\Gamma$. For instance, if $\Gamma_g$ is the fundamental group of a genus $g$ surface and $u \in H^2 (\Gamma_g)$ is a generator, we prove that $\kappa_{L_m,u}(E,f) =0$ for all bundles when $g \geq 2$. In contrast, when $g=1$, one can construct many examples with $\kappa_{L_m,u}(E,f) \neq 0$.

Fabian Hebestreit: Homology manifolds and euclidean bundles

It is a curious fact of life in geometric topology, that the classification of closed manifolds by surgery theory becomes easier as one passes from smooth to piecewise linear and finally to topological manifolds. It was long conjectured that an even cleaner statement should be expected in the somewhat arcane world of homology manifolds of the title, which ought to fill the role of some "missing manifolds". This was finally proven by Bryant, Ferry, Mio and Weinberger in the 90's, building on an earlier theorem of Ferry and Pedersen that any homology manifold admits a euclidean normal bundle. In the talk I will try to explain this result, and further that its existence is incompatible with the result of Ferry and Pedersen. The latter is therefore incorrect and/or the proof the former incomplete. (joint with M.Land, M.Weiss & C.Winges.)

Geoffroy Horel: Configuration spaces of algebraic varieties II

I will first review tame homotopy theory. This is an old construction that describes the "easy part" of integral homotopy theory namely the part that is insensitive to Steenrod operations. I will then explain how using Galois action on étale cochains of algebraic varieties, we can get some restrictions on the tame homotopy type of some algebraic varieties. In particular, we can show that a tame version of the Kriz model is a model for the tame homotopy type of the configuration spaces of some algebraic varieties. I will also describe some other related applications of these ideas. This is joint work in progress with Joana Cirici. 

Alexander Kupers: Mapping class groups of exotic tori

We will explain how the mapping class group of the connected sum of a d-torus and an exotic sphere depends on the exotic sphere, and use this to give examples of such manifolds that admit no non-trivial action of SL_d(Z). This is joint work with Mauricio Bustamante, Manuel Krannich, and Bena Tshishiku.

Paolo Salvatore: Disc embeddings, deloopings and operad actions

A classical theorem by Morlet identifies the $(n+1)$-fold delooping of the group of diffeomorphisms of the $n$-disc relative to the boundary. We extend the result to spaces of embeddings of $m$-discs into $n$-discs, together with their versions modulo immersions and with frames. In the framed case we construct an action of the framed $(m+1)$-disc operad that combines an $O(m+1)$-action defined by Hatcher and an $E_{m+1}$ action defined by Budney. This is joint work with Victor Turchin. 

Christine Vespa: Polynomial functors associated with beaded open Jacobi diagrams

The Kontsevich integral is a very powerful invariant of knots, taking values is the space of Jacobi diagrams. Using an extension of the Kontsevich integral to tangles in handlebodies, Habiro and Massuyeau construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. The category A has a subcategory equivalent to the linearization of the opposite of the category of finitely generated free groups, denoted by gr^{op}. By restriction to this subcategory, morphisms in the linear category A give rise to interesting contravariant functors on the category gr, encoding part of the composition structure of the category A. Katada studies the functor given by the morphisms in the category A from 0. In particular, she obtains a family of polynomial functors on gr^{op} which are outer functors, in the sense that inner automorphisms act trivially. In this talk, I will explain these results and give extensions of Katada’s results concerning the functors given by the morphisms in the category A from any integer k. These functors give rise to families of polynomial functors on gr^{op}  which are no more outer functors. Our approach is based on an equivalence of categories given by Powell. Through this equivalence the previous polynomial functors correspond to functors given by beaded open Jacobi diagrams.

Thomas Willwacher: Cyclic operads through modules

We describe a way of computing mapping spaces of cyclic operads through their modules. Precisely, to a cyclic operad P, one may associate the pair (P^{nc}, P^{mod}) consisting of P as a non-cyclic operad and P as a right P^{nc}-module. This functor from cyclic operads to pairs is homotopically fully faithful. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.