June 21st 2024 - Alexander Schmeding (University of Trondheim), "When the Lie group exponential is bad".
Abstract: From finite dimensional Lie theory, it is well known that the Lie group exponential yields a local diffeomorphism from the Lie algebra onto a unit neighborhood of the Lie group. These exponential coordinates are useful tools to understand the interplay between Lie algebra and Lie group. It is well known that this correspondence breaks down in infinite-dimensional Lie theory. Beyond Banach spaces, the Lie group exponential is in general not a local diffeomorphism. For these non-locally exponential Lie groups the Lie group exponential becomes much more restricted. In this talk we will revisit some of the classical examples of the breakdown of the exponential (e.g. the diffeomorphism group). It turns out that in many known examples this defect can be traced to properties of flows of vector fields. Time permitting, we will show some new results for non-local exponentiality of semidirect products of Lie groups. This is joint work with R. Dahmen and K.-H. Neeb.
June 14th 2024 - Leonid Ryvkin (University of Lyon), "Universal central extensions of the Lie algebra of volume-preserving vector fields".
Abstract: The goal of the talk is proving a conjecture of Claude Roger about the universal central extension of the Lie algebra of volume-preserving vector fields. In the beginning we will briefly review the notion of central extensions of Lie algebras and their link to Chevalley-Eilenberg-cohomology. We will then proceed to Rogers conjecture, which lies in the (continuous) infinite-dimensional setting. To solve it we will need a combination of analytical and geometrical methods, and maybe even a bit of representation theory. Based on an ongoing collaboration with Bas Janssens and Cornelia Vizman.
May 28th 2024 - Rosa Marchesini (University of Göttingen), "On the homotopy invariance of Lie algebroid cohomology".
Abstract: Several extensively studied cohomology theories play an important role in differential geometry and mathematical physics. Examples are De Rham cohomology, Chevalley-Eilenberg cohomology, BRST cohomology, and then Poisson, Foliated, and Principal de Rham cohomologies. All of these can be elegantly formalized at once using special smooth vector bundles called Lie algebroids. Lie algebroids also arise naturally as an infinitesimal description of Lie groupoids and are therefore a central subject of study in higher Lie theory. After an accessible introduction to Lie algebroids and their cohomology, we propose a notion of homotopy for Lie algebroids and justify it with examples and applications. This is a joint work with my supervisor Madeleine Jotz. We mention some related research projects currently in progress with Ryszard Nest.
May 24th 2024 - Stefano Ronchi (University of Göttingen), "Cotangent spaces for higher Lie groupoids and applications".
Abstract: In the same way Lie groupoids can encode local symmetries of a manifold, Lie 2-groupoids are a way to encode higher symmetries. After a short crash course on this topic, we present the construction of a cotangent space for Lie 2-groupoids analogous to the well known one for Lie groupoids. This turns out to have a canonical shifted symplectic structure (that is, symplectic up to homotopy) in the same way the cotangent groupoid is canonically a symplectic groupoid, and the tangent bundle of a manifold is canonically symplectic. This makes our cotangent space a good global model for a class of symplectic Q-manifolds that appear in some TQFTs. We will then discuss various applications, including a definition of hamiltonian actions of Lie 2-groupoids. This talk is based on joint work in progress with Miquel Cueca and Chenchang Zhu.
May 17th 2024 - Fabrizio Zanello (University of Potsdam), "A leisurely stroll through the multisymplectic approach to Lagrangian field theories".
Abstract: The mathematical toolbox of multysimplectic geometry was introduced in order to generalize the symplectic formulation of classical mechanics to Lagrangian field theories. After a substantial body of foundational works for first order field theories, though, the program came to a standstill due to technical difficulties. In this talk we give an overview of the motivations, techniques and limitations of the traditional results and present some of the core ideas of a new approach, recently proposed by Blohmann, to overcome those technical difficulties. The talk is based on an ongoing joint project with Antonio Miti.
May 10th 2024 - Luisa Herrmann (University of Potsdam), "On a problem of optimal transport under marginal martingale constraints".
Abstract: Based on an article by Mathias Beigelböck and Nicolas Juillet, I will first describe the martingale version of the optimal transport problem after which I will define the convex order and give some examples. I will then discuss the central question as to whether the set of all martingale transport plans is nonempty and present a constructive proof of the existence under certain assumptions.
May 3rd 2024 - Alejandro Peñuela Diaz (University of Potsdam), "Construction of marginally outer trapped surfaces".
Abstract: In physics, marginally outer trapped surfaces (MOTS) can be understood as quasi-local versions of black hole boundaries. Therefore, these surfaces are crucial to understand black holes and are widely used in black hole simulations. From a mathematical point of view, MOTS are prescribed mean curvature surfaces, that is a generalization of a minimal surface. I will talk about marginally outer trapped surfaces, introduce some of their properties, and explain the challenges when trying to construct them analytically.
April 26th 2024 - Alberto Bonicelli (University of Pavia), "Convergence results in the stochastic sine-Gordon model: an algebraic viewpoint".
Abstract: The importance of the sine-Gordon model in 1+1 spacetime dimensions resides in the integrability of the field theory that it describes. A recent result showed how, within the setting of algebraic quantum field theory, this property translates into a convergence result for both the formal series associated to the S-matrix and to the interacting field of the quantum field theory. After introducing an algebraic approach to the perturbative study of singular stochastic PDEs, I will show how an adaptation of the aforementioned convergence results yields convergence of the momenta of the solution to a stochastic version of the sine-Gordon equation. Interestingly enough, our two-step procedure passes through the quantum theory and recollects the stochastic information via the classical limit.
April 19th 2024 - Leonard Schmitz (MPI Leipzig), "Free generators and Hoffman's isomorphism for the two-parameter shuffle algebra".
Abstract: Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from B∞-algebras and a novel two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions. This is joint work with Nikolas Tapia.
April 12th 2024 - Alexander Schmeding (NTNU, Trondheim), "On manifolds of Lie group valued continuous BV-functions".
Abstract: Functions of bounded variation (BV) with values in a Banach space are a classical topic of analysis with specific applications for example in rough path theory. In the theory of rough paths one considers routinely even BV-functions with values in non-linear spaces such as manifolds and (finite and infinite-dimensional) Lie groups. In this talk we will explain how the well known construction of manifolds of mappings carries over to the world of BV-functions. As a consequence we are able to generalise the construction of current groups to the BV-setting. This also strengthens known regularity properties a la Milnor for Banach Lie groups. Joint work with H. Glöckner and A. Suri (Paderborn).
