Schedule and abstracts

• Barbara Fantechi, Lagrangians on moduli spaces of vector bundles on K3 surfaces

Abstract TBA

• Alice Fialowski, On the cohomology of restricted Heisenberg Lie algebras - slides

In this talk I will show that the Heisenberg Lie algebras over an algebraically closed field of positive characteristic admit a family of restricted Lie algebras. First, I will introduce the notion of restricted Lie algebra with examples, and point out the importance of restricted structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg algebras by giving explicit cocycles. Also describe the restricted 1-dimensional central extensions, including exact formulas for the Lie brackets and [p]-operators. The work is joint with T. Evans and Y. Yang.

• Karl-Hermann Neeb, Algebraic Quantum Field Theory on causal homogeneous spaces - slides

Lorentzian symmetric spaces and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are nets of operator algebras. In our talk we report on an ongoing project concerned with the construction of such nets on causal homogeneous spaces. A key result asserts that the modular groups of local algebras that arise in this context are generated by Euler elements (they define 3-gradings of the Lie algebra). Conversely, Euler elements in semisimple Lie algebras specify causal symmetric spaces and so-called wedge domains, that can in turn be used to construct nets of operator algebras. We thus obtain a rich geometric environment for nets of operator algebras. This is based on joint work with V. Morinelli, J. Frahm and G. Olafsson.

• Silvia Sabatini, Topological properties of (tall) monotone complexity one spaces - slides

In symplectic geometry it is often the case that compact symplectic manifolds with large group symmetries admit indeed a Kähler structure. For instance, if the manifold is of dimension 2n and it is acted on effectively by a compact torus of dimension n in a Hamiltonian way (namely, there exists a moment map which describes the action), then it is well-known that there exists an invariant Kähler structure. These spaces are called symplectic toric manifolds or also complexity-zero spaces, where the complexity is given by n minus the dimension of the torus.

In this talk I will explain how there is some evidence that a similar statement holds true when the complexity is one and the manifold is monotone (the latter being the symplectic analog of the Fano condition in algebraic geometry), namely, that every monotone complexity-one space is simply connected and has Todd genus one, properties which are also enjoyed by Fano varieties. These results are largely inspired by the Fine-Panov conjecture and are in collaboration with Daniele Sepe [2].

Moreover, with Isabelle Charton and Daniele Sepe [1], we completely classify monotone complexity one space that are "tall" (no reduced space is a point), and prove that the torus action extends to a full toric action, that each of these spaces admits a Kähler structure and that there are finitely many such spaces, up to a notion of equivalence that will be introduced in the talk.

References:

[1] I. Charton, S. Sabatini, D. Sepe, "Compact monotone tall complexity one T-spaces", arXiv:2307.04198 [math.SG].

[2] S. Sabatini, D. Sepe, "On topological properties of positive complexity one spaces", Transformation Groups 9  (2020).

• Ivan Struchiner, Families of G-Structures with Symmetries - slides

Families of G-structures with (good) symmetries give rise to G-structure groupoids. Their infinitesimal counterparts, G-structure algebroids, are the data of a moduli problem known as Cartan's Realization Problem. In this talk I will discuss the geometry of families of G-structures and also the role of Lie theory in finding a versal family of complete 1-connected solutions to a Cartan's Realization Problem. The talk is based on joint work with Rui Loja Fernandes and on an ongoing project with Wilmer Smilde.

• Barbara Tumpach, Banach Poisson-Lie groups and integrable systems - slides

This talk is a journey towards the theory of Banach Poisson-Lie groups and their relation to integrable systems. The talk will be organised around take-away examples that allows to understand the problems arising in the infinite-dimensional setting. Examples related to the restricted Grassmannian, an infinite-dimensional Hermitian symmetric space appearing in Fermionic second quantization theory, will be given. In this example, the choice of an infinite-dimensional commutative subgroup of the restricted general linear group will produce an integrable system of equations like in the case of the Korteweg-De Vries hierarchy. This difficult and fascinating field is paved with open problems, some of which we will mention during the presentation. 

• Stefan Waldmann, Entire Functions on Lie Groups and Convergent Star Products

In this talk I will report on recent progress in understanding the convergence properties of star products on cotangent bundles of a Lie groups. The Lie algebraic part, the convergence properties of the Gutt star product on the dual of the Lie algebra, is now well-understood by a previous work of Chiara Esposito, Paul Stapor and myself. Using this, the standard ordered star product on the cotangent bundle can be extended from polynomial functions to certain completions, resulting in a Fréchet algebra with nice properties and analytic dependence on the deformation parameter $\hbar$. A crucial role is played by the entire functions on G, a class of analytic functions determined by their Lie-Taylor coefficients at the identity. I will provide some new ideas about this class of functions, which should be interesting also from general considerations. The results are joint work with Michael Heins and Oliver Roth.

• Luca Accornero, Integrability obstructions in differential geometry - slides

A geometric structure on a smooth manifold is called integrable when it admits a local model, typically described in terms of a "standard" version of the same geometric structure on the Euclidean space. In G-structure theory, the local models are provided by linear structures. Integrability is controlled by certain systems of PDEs; obstructions to the existence of solutions for such PDEs offer integrability obstructions for the underlying geometric structure. For G-structures, these obstructions can be understood in terms of the torsion of a compatible connection. 

In this talk, we explain how to encode geometric structures in terms of principal groupoid bundles, present a construction of integrability obstructions that goes beyond G-structures, and discuss the Morita invariance of such obstructions.

Based on joint work with Francesco Cattafi, Marius Crainic, and María Amelia Salazar.

• Lory Aintablian, Differentiation of elastic diffeological groupoids - slides

I will discuss a differentiation procedure from elastic diffeological groupoids to generalized Lie algebroids in the category of diffeological spaces. First, I will explain the differentiation procedure of groupoid objects in any category with an abstract tangent structure in the sense of Rosick{\'y}. Since the left Kan extension of the tangent structure of smooth manifolds defines a tangent structure on elastic diffeological spaces, this construction applies to elastic diffeological groupoids. This is joint work with Christian Blohmann.

• Ilias Ermeidis, Deformations of ideals in Lie algebras - slides

The main aim of this talk is to introduce the differential graded Lie algebra which controls the deformation problem of an ideal inside a Lie algebra. Making use of this algebraic structure, natural geometric questions will be answered such as: under which assumptions a Lie ideal is rigid/stable and how much differs the deformation theory of Lie ideals from their underlying deformation theory as Lie subalgebras. Last but not least, we will enlighten the reason why this classical deformation problem had not been solved so far, by mentioning how the deformation theory of Lie ideals goes beyond the category of Lie algebras and fits naturally into the wider context of Lie 2-algebras. 

This is a joint work (in progress) with Madeleine Jotz.

• Ioannis Gkeneralis, "Standard" isotropy subgroups of $SU(2)^n$-representations - slides

We start with Euclidean spaces $\mathbb{R}^n$ on which we can define a multiplication such that, if $x, y \not= 0$, then $xy \not= 0$. If we insist that the multiplication is associative, then $n = 1, 2, 4$ from Frobenious Theorem. In the first case we have the multiplication of the reals, in the second case we have the complex multiplication and in the third case, we have the multiplication on the quaternions which is not commutative. The corresponding spheres, in each case, are the elements of length one and for $n = 1$ is the group of two elements $\mathbb{Z}_2$, for $n = 2$ is the unit circle $S^1$, for $n = 4$ is the unit sphere in $\mathbb{R}^4$, $S^3$, and they are all groups with the induced multiplication. In each of these dimensions we have the corresponding torus, $Z^n = \mathbb{Z}_2{\times} \dots \times \mathbb{Z}_2$, $T^n = S^1 {\times} \dots {\times} S^1$ and $Q^n = S^3{\times} \dots {\times}S^3$. In each case, starting with a polyhedron, we can construct a space on which the tori act and the quotient space is the original polyhedron \cite{dj}. These manifolds are called standard models. The local action is given in fact by the corresponding multiplication.

In the quaternionic case, difficulties arise when we deal with the non-commutativity. More precisely, the representation theory of $Q^n$ is more complicated than that of $T^n$. The basic result that is pivotal in the characterisation of the local action is that the irreducible real representations of $S^3 \cong SU(2)$ are only the representations of dimension divisible by $4$ that appear, then we can show that the action is by quaternionic multiplication.

The talk is based on a joint paper with Professor Prassidis Efstratios on "Toplogical Rigidity of Quoric Manifolds".

• Lennart Obster, Rigidity (conjecture) for a class of Lie algebroids

This talk is about the ongoing work of proving a rigidity result for a class of Lie algebroids. Using the Nash-Moser fast convergence method, the rigidity problem relies on proving “tameness inequalities” of certain homotopy operators. Important elements are therefore:

- Which class of Lie algebroids are we considering?

- What homotopy operators are we looking for?

- How do we find explicit formulas for such homotopy operators?

In the talk we will address these questions and focus on the progress made.

• Tobias Simon, Polynomial growth of holomorphic extensions of principal series representations at the boundary of the crown - slides

In view of the Krötz-Stanton Extension Theorem, which allows one to holomorphically extend orbit maps of irreducible unitary representations of semisimple Lie groups to a holomorphic K_\C-principal fibre bundle over the complex crown domain, we report on our results of distributional limits as one approaches the boundary of this bundle. The main ingredient are polynomial growth rates of the holomorphically extended Iwasawa components which allows one to prove polynomial growth estimates in principal series representations.

• Wilmer Smilde, Geometric structures and Lie theory - slides

Already implicit in the work of Elie Cartan, there is a deep connection between classification problems for geometric structures (coframes) and modern Lie theory. Many classification problems arise as a 'PDE in the structure functions of a coframe'. In rare (but not unnatural) cases, this PDE is of so-called finite type, and defines a Lie algebroid that classifies the problem. The problem of classifying all complete (global) solutions is essentially the same as determining whether the Lie algebroid is integrable. This deep connection between Lie theory and geometric classification problems has been worked out by Fernandes and Struchiner.

In this talk, we will propose a generalization of Lie algebroids, a Bryant algebroid, that is designed to capture the essential structure of the 'infinite type PDEs in the structure functions of a coframe'. The theory of Bryant algebroids makes the connection between generic classification problems for geometric structures and Lie theory clear and precise. This is ongoing work with Rui Loja Fernandes.

• Maximilian Stegemeyer, String topology of Lie groups and symmetric spaces - slides

String topology studies algebraic structures on the homology of the free loop space of a closed manifold. The most well-known operations are the Chas-Sullivan product and the string topology coproduct. One can understand the homology of the free loop space by studying the Morse-Bott theory of the energy functional of a Riemannian metric on the manifold. With Lie-theoretic methods one can do so for the symmetric metric on the free loop space of a compact symmetric space. We use this idea to study the string topology of compact symmetric spaces and in particular of compact Lie groups. We show that the Chas-Sullivan product on compact symmetric spaces is related to the iteration of closed geodesics and that the string topology coproduct vanishes for compact Lie groups of higher rank.

• Janina Bernardy, Homotopy reduction of multisymplectic structures in lagrangian field theory

• Alfonso Garmendia, Frames of VB-groupoids

• Michael Heins, Entire functions on Lie groups

• João Nuno Mestre, Deformations of complex Lie groupoids

• Maarten Mol, Kähler metrics and toric Lagrangian fibrations

• Thu Hien (Hisha) Nguyen, Analytic closure of sets of hyperbolic polynomials 

• Leonid Ryvkin, Not afraid to replace O(t^2) with t^2=0 in differential geometry

• Lucas Seco, Counting Geodesics on Compact Symmetric Spaces

• Aldo Witte, Classifying Lie algebroids