| 14/4 | 15/4 | 16/4 | |
| 9h-9h30 | Registration | ||
| 9h30 - 10h30 | Oscar Randal-Williams | Thomas Willwacher | Alexis Aumonier |
| 10h30 - 11h | coffee | coffee | coffee |
| 11h - 12h | Joana Cirici | Alex Takeda | Victor Turchin |
| 12h - 14h | lunch | lunch | lunch |
| 14h - 15h | Azélie Picot | Greg Arone | Manuel Krannich |
| 15h15 - 16h15 | Florian Naef | Andrea Bianchi | Open problems |
| 16h15 - 17h | coffee | coffee | coffee |
We will discuss a method for calculating Ext between polynomial functors from groups to abelian groups. It is based on the close relationship between the homotopy theories of spaces and of simplicial groups. After reviewing the general result, I will tell about joint work in progress with Minkyu Kim and Christine Vespa: We consider the functor \(S^n\), that sends a group G to the n-th symmetric power of ab(G). Ext into this functor is trivial rationally, but has a lot of torsion. It turns out, perhaps surprisingly, that one can express Ext into S^n in terms of the homology of Braid groups.
I will introduce and explain how to study a moduli space parameterising algebraic hypersurfaces equipped with a continuous map to a background space. On the surface we will see a scanning map, but the nuts and bolts will come from Goodwillie calculus. I shall describe the method for general moduli of maps, and apply it to the specific case of hypersurfaces.
String topology, introduced by Chas and Sullivan, is the study of the homology of mapping spaces of the form \(M^X\), where \(M\) is a closed oriented d-dimensional manifold and \(X\) is a space. Fixing \(M\) and letting \(X\) vary, the homology groups \(H_*(M^X)\) carry additional algebraic structure coming from (contravariant) functoriality in \(X\) and from Poincare' duality of \(M\); the most famous example is the Chas-Sullivan product. We introduce a symmetric monoidal infty-category GrCob of "graph cobordisms between spaces"; we define compatible local coefficient systems \(\xi_d\) on the morphism spaces of GrCob, and use the twisted homology of the morphism space GrCob(Y,X) to define higher string operations \(H_*(M^X) \to H_*(M^Y)\). We assemble all such operations into a "graph field theory" associated with \(M\), i.e. a contravariant symmetric monoidal functor out of the linearisation of GrCob given by \(\xi_d\). We recover some basic operations, including the Chas-Sullivan product, as special cases.The construction of the graph field theory can in fact be carried out for any oriented Poincare' duality space \(M\), and is natural in \(M\) with respect to orientation-preserving equivalences; in particular all string operations we obtain are automatically homotopy invariant. The main technical input to the construction is a recent result by Barkan-Steinebrunner, giving a universal property for the category of graph cobordisms between finite sets in terms of commutative Frobenius algebras.
I will explain how the theory of weights in homotopy theory produces algebraic models for complements of arrangements in smooth varieties, using Galois actions in étale cohomology. This applies in particular to configuration spaces. Similar ideas lead to equivariant formality results for varieties with algebraic group actions and, via the Grothendieck–Teichmüller group, for the little disks operad equipped with its \(O(n)\)-action. This is joint work with Pedro Boavida and Geoffroy Horel.
The h-principle studies spaces of geometric structures. Typical examples include immersions, submersions, functions with controlled singularities, embeddings, symplectic structures, foliations, complex structures, and metrics with various curvature constraints
A key idea in h-principle, dating back to Gromov, is that geometric structures of a given type form a (pre)sheaf, which should be compared to its homotopy sheafification (the homotopy sheaf of formal structures). The main metaquestion in h-principle is then the following: What makes a geometry flexible (i.e. what properties of a geometry imply that its presheaf is in fact a homotopy sheaf)?
The purpose of this talk is to discuss how the wrinkling ideas of Eliashberg and Mishachev have led to interesting developments in h-principle in the last 10-20 years. I will touch on various key results (e.g. the classification of overtwisted contact structures due to Borman-Eliashberg-Murphy), as well as some recent work of myself joint with A. Fokma and L. Toussaint.
How much of a closed smooth manifold \(M\) is captured by the homotopy types of its framed configuration spaces? This talk serves to make this question precise and to explain recent results in this direction in the case where \(M\) is an exotic sphere, obtained in joint work with A. Kupers and F. Mezher.
Cobordism categories conveniently package closed manifolds of a given dimension as objects and moduli spaces of their cobordisms as morphisms. On the other hand, Poincaré duality spaces are spaces whose homology satisfies Poincaré duality. We can again define a cobordism category for Poincaré duality spaces. By equipping objects and morphisms with a map to a base space \(X\), we get a cobordism category over \(X\). Taking classifying space then yields a functor from spaces into spaces. I will discuss the first Goodwillie derivative of this functor in the Poincaré case.
Given a closed manifold, its cohomology carries the structure of a Frobenius algebra. I will explain how such a structure can be lifted to the cochain-level using its configuration spaces of points. Moreover, the construction is invertible, that is, it gives an equivalence between (strict) Frobenius algebras and Disk-presheaves in commutative algebras. This is a report on an ongoing joint work with Thomas Willwacher.
The Madsen--Weiss theorem in dimension 2 and its extension (Galatius--R-W) to higher even dimensions describes the (co)homology of the diffeomorphism group of a manifold \(M^{2n}\) in a stable range of degrees (depending on how many \(S^n \times S^n\) connect-summands \(M\) has; for \(2n=4\) the statement is a little different). Not only does it allow us to understand the homology of diffeomorphism groups of specific manifolds in a stable range, it has become---usually in combination with the Weiss fibre sequence---a basic theoretical tool for proving qualitative and quantitative results about diffeomorphism groups.
Ebert has explained that the most literal odd-dimensional analogue of the Madsen--Weiss theorem cannot be true. I will describe a different point of view on this using L-theory, and explain how this line of thinking together with the modern perspective on Grothendieck--Witt theory leads to a conjectural Madsen--Weiss theorem for the manifolds \(\#^g S^n \times S^{n+1}\) which fits with all known calculational data.
This talk reflects ongoing work with Fabian Hebestreit, Manuel Krannich, and Robin Stoll.
We define manifold operads, a class of topological operads that generalize the Fulton-MacPherson operad. In particular they are self dual on the chain level. We also define cobordism bimodules between manifold operads, and show how to propagate a surgery in higher arity, producing many new examples of operads that are left or right cobordant but not equivalent to the Fulton-MacPherson operad. This is joint work with Xujia Chen and Connor Malin.
In rational homotopy theory, the notions of formality and coformality of a topological space are related to how “simple” the higher structures on their minimal models are. In particular, for simply connected spaces, the notion of coformality can be related to the formality of the based loop algebra, In this talk I will explain a generalization of this latter notion of formality that extends this notion to a pair (X,[X]) of a space with a fundamental class giving local Poincaré duality. This is phrased in terms of algebras over a certain dioperad Y, which we show to be Koszul. Part of this talk is about joint work with C. Emprin.
The goal of the talk is to explain a conjecture about block embedding spaces appearing naturally in connection with recent delooping results for concordance embeddings. (Project in progress with A. Kupers and T. Willwacher.)
We discuss algebraic (graphical) models for general \(P-E_n\) bimodules of configuration space type, where \(P\) is a (colored) operad.
The main examples of such objects are configuration spaces of points on \(M \times \mathbb{R}^m\) with \(P=E_m\) and \(M\) a parallelized manifold, or configuration spaces of points on parallelized manifolds with boundary.
This work is based on joint work in progress with S. Schwarz, extending my earlier results [arxiv:2302.07369].
