Miguel Barata (Utrecht University)

⤷Title: A dendroidal approach to Goodwillie-Weiss embedding calculus

⤷Abstract: Manifold calculus is a technique developed by T. Goodwillie and M. Weiss for approximating the homotopy type of the space of smooth embeddings \(\text{Emb}(M,N)\) via the configuration spaces of the manifolds in question. Together with the convergence results by Goodwillie-Klein, embedding calculus has had a meaningful impact in both homotopy theory and differential topology, for instance for proving finiteness results for the homotopy groups of diffeomorphism groups. In this talk I want to explain the Boavida-Weiss approach to manifold calculus in terms of operads and operadic right modules. Moreover, we will describe how this strategy can be carried out using trees and the language of dendroidal spaces. Finally, we will provide an explicit description of the constituents of the embedding tower using the dendroidal formalism.

Severin Barmeier (University of Cologne)

⤷Title: Strict deformation quantization via homotopy transfer

⤷Abstract: Kontsevich’s Formality Theorem relies on an \(L_\infty\)-quasi-isomorphism between two DG Lie algebras: on one side the Hochschild complex of the algebra of smooth functions on \(\mathbb{R}^d\) equipped with the Gerstenhaber bracket and on the other side the graded vector space of multivector fields equipped with the Schouten-Nijenhuis bracket. I will explain how in the setting of polynomial functions on \(\mathbb{R}^d\), the complex of polynomial multivector fields naturally inherits an \(L_\infty\)-structure by homotopy transfer, the binary bracket coinciding with the Schouten-Nijenhuis bracket. I will explain how this \(L_\infty\)-structure gives rise to explicit “combinatorial” star products which can be used to quantize polynomial Poisson structures. These star products turn out to admit a graphical calculus, similar to the Kontsevich star product, but without weights, allowing one to prove convergence results for these star products and to construct strict deformation quantizations.
This talk is based on joint work with Zhengfang Wang and with Philipp Schmitt.

Thomas Blom (MPIM Bonn)

⤷Title: Koszul duality and envelopes of operads

⤷Abstract: Over the years, several definitions of a spectral version of the Lie operad have been introduced. While these are all expected to be equivalent as operads in spectra, there exists no such proof in the literature. In this talk I will describe how to prove that several of these spectral Lie operads are equivalent. The key ingredient will be a reformulation of operadic Koszul duality in terms of symmetric monoidal structures on certain presheaf categories.
This is joint work with Connor Malin and Niall Taggart.

Žan Grad (KU Leuven)

⤷Title: Yang–Mills theory for multiplicative Ehresmann connections

⤷Abstract: Classical Yang–Mills theory is a theoretical cornerstone of elementary particle physics. Roughly speaking, it is an application of calculus of variations to the theory of principal bundles, seeking to minimize the norm of the curvature of a principal bundle connection. In this talk, I will present a broad generalization of this framework to Lie groupoids and Lie algebroids, where principal bundle connections are replaced with (infinitesimal) multiplicative Ehresmann connections. This extends the classical theory to the non-integrable and non-transitive setting, capturing the dynamics of gauge fields in both longitudinal and transversal directions relative to the (singular) orbit foliation. Time permitting, we will look at an example from the theory of bundle gerbes, and discuss the relation to other known generalizations.

Fabian Hebestreit (University of Bielefeld)

⤷Title: Idempotent ideals and derived localisations

⤷Abstract: By now classical work of Thomason classifies the thick subcategories of the category of perfect complexes over a commutative ring \(R\), or equivalently the subcategories of the derived category \(D(R)\) that are closed under colimits and generated by perfect complexes, in terms of the spectrum of \(R\). Recently, a slightly more general class of subcategories has come into view, namely those whose inclusion admits a right adjoint, that preserves colimits. These make up what is often called the smashing spectrum of \(R\). Besides the examples in Thomason’s work, Faltings’ theory of almost rings can be interpreted as providing examples associated to idempotent ideals \(I\) in \(R\). I will explain a simple new proof of this result that fundamentally makes use of the fact that such subcategories correspond in a one-to-one fashion to derived localisations of \(R\), which can be explicitly described in the case at hand. As an application of this description we answer a question of Efimov by showing that the Frobenius does not always induce an isomorphism on the smashing spectrum of a ring in positive characteristic, contrary to the case of the ordinary spectrum.
This is joint work with P. Scholze.

Annika Kraasch-Tarnowsky (MPIM Bonn)

⤷Title: Computing Differentiable Stack Cohomology — a journey via multiplicative Ehresmann connections

⤷Abstract: The concept of a “differentiable stack” is a higher geometric notion similar to algebraic or topological stacks, where examples include orbifolds, quotients of Lie group actions and many moduli spaces. While understanding differentiable stacks by unraveling their definition as categories is possible, there is also another way to approach their study: It can be seen that the 2-category of differentiable stacks is equivalent to a 2-category of Lie groupoids. This makes it possible to transfer notions, methods and results between both concepts, expanding the mathematical toolbox applicable to either. For example, we have that the differentiable stack cohomology associated to an action groupoid agrees with the equivariant cohomology of that action. In particular, results on computing equivariant cohomology, such as the existence of infinitesimal models, can be interpreted as results on computing the cohomology of the associated differentiable stacks. It has been a continuing effort over the past 20 years to generalize the well-known Cartan model for equivariant cohomology to a model computing the differentiable stack cohomology of a general proper Lie groupoid. In this talk, we will explore the problem and its challenges and present a recent advancement yielding a model in the case of a proper and regular Lie groupoids. We will furthermore explore the key tools in the construction of this model, including multiplicative Ehresmann connections. These are a certain type of connection on a Lie groupoid, which through a construction using jet groupoids induce a useful notion of invariance on differential forms on the nerve of the groupoid.

Maxime Wybouw (Radboud University)

⤷Title: Homotopy transfer and minimal models

⤷Abstract: A classical theorem by Kadeishvili states that the information of an associative dg algebra over a field can be fully transferred to a minimal \(A_\infty\) structure on its homology. In this talk, I will discuss this result and some generalizations. One main tool is to work with diagrams of chain complexes indexed by finite sets and injections. This for example allows transferring commutative or derived homotopy structures.