List of talks Summer Semester 2025

April 16th 2025 - Camilo Angulo (University of Göttingen), "Examples of Poisson manifolds with compactness properties".

Abstract: Poisson geometry lies in the intersection of symplectic geometry, foliation theory and Lie theory. As in each of these areas compactness hypotheses yield a wealth of results, it would be desirable to have a notion of compactness in Poisson geometry that simultaneously subsumes the theory of compact semisimple compact Lie groups and compact symplectic manifolds. This goal has been recently achieved by Crainic, Fernandes and Martinez-Torres, who defined a Poisson manifold of compact type (PMCTs) to be a Poisson manifold whose integrating symplectic groupoid is proper. The wonderful properties of these PMCTs lie in contrast to their relative scarcity. The geometric and topological constraints that go into building a PMCT make their definition rather demanding, and in so, constructing a PMCT beyond the trivial case of a compact symplectic manifold with finite fundamental group has proven a challenging problem. In this talk, after properly explaining the elements that go into play, we explain how by allowing for other geometric structures to "integrate" Poisson manifolds, one can get more examples while preserving most of the compactness properties.

April 25th 2025 - Coline Emprin (École Normale Supérieure de Paris), "A prop structure on partitions" (online).

Abstract: PROPs were introduced by Mac Lane in 1965 as a special type of category whose objects are natural numbers, endowed with an additional horizontal composition of morphisms beyond the usual categorical composition. In this talk, I will present a specific PROP structure that emerges from the combinatorics of partitions. The construction of this structure is closely related to the Karoubi envelope of a certain category, which I will introduce along the way. This PROP is of particular interest in the context of functor homology, as its composition corresponds to the Yoneda product of extension groups between exterior power functors. I will conclude by discussing how such constructions can be used to compute extension groups between simple functors defined on free groups.
This is joint work with Dana Hunter, Muriel Livernet, Christine Vespa, and Inna Zakharevich.

April 30th 2025 (12:00 am) - Carlos Perez-Sanchez (Universität Heidelberg), "Towards random noncommutative geometry".

Abstract: The well-known question whether one can "hear the shape of a drum", posed by Marek Kac, has, also famously, a negative answer constructed by Gordon, Webb and Wolpert. In noncommutative geometry, classical dynamics depends only on the spectrum, in that case, of an operator of Dirac type. If, additionally, algebraic data are provided and some axioms verified - building what is known as spectral triple - this structure does allow to reconstruct a manifold, thus answering a weaker version of Kac's question positively.

Spectral triples are relevant in Connes' noncommutative geometric setting, whose path integral quantisation that "averages over noncommutative geometries" shall rely on the concept of ensembles of Dirac operators. This is to be contrasted with a path integral over Riemannian metrics in quantum gravity. In this talk I first explore what an ensemble of noncommutative geometries on a fixed graph is (gauge fields are on, while gravity is still off). Using elements of quiver representation theory

• we associate a Dirac operator to a quiver representation (in a category that emerges in noncommutative geometry);
• we derive the constraints that the set of Wilson loops satisfies (generalised Makeenko-Migdal equations);
• and explore the consequences of the positivity of a certain matrix of Wilson loops ("bootstrap").

In the special case that our graph is a rectangular lattice and our physical action quartic, we obtain Wilsonian lattice Yang-Mills theory, hence the terminology. Unsurprisingly, our ensembles (for an arbitrary graph) boil down to integrating noncommutative polynomials against a product Haar measure on unitary groups. The classical aspects of this theory were constructed in [2401.03705], and the loop equations in [2409.03705].

May 9th 2025 - Nicolai Rothe (Technische Universität Berlin), "Cosmological solutions to the semiclassical Einstein equation with Minkowski-like vacua".

Abstract: We will discuss some newly found solutions to the full massless semiclassical Einstein equation (SCE) in a cosmological setting (with Λ=0). After a short introduction to the relevant notions, we present the SCE in a particular shape which allows for the construction of certain vacuum states. These states may be viewed as the least possible generalization of the Minkowski vacuum to generic cosmological space-times. In this setting, solving the SCE breaks down into solving a certain ODE which can be approached numerically and, at least generically, we obtain solutions that well fit physical expectations. Moreover, these solutions indicate dark energy as a quantum effect back-reacting on cosmological metrics and, since in our model m=Λ=0, this may not be traced back to the usual, obvious dark-energy/cosmological constant effect of a quantum field. Also we will shortly address some related results obtained in our setting.

May 16th 2025 (10:00 am) - Harprit Singh (Universität Wien), "Rough Geometric Integration".

Abstract: Combining ideas from Whitney’s geometric integration theory and rough analysis, we introduce spaces of rough differential \(k\)-forms on \(d\)-manifolds which are formally given by \(f=\sum_If_Idx^I\) where \((f_I)^I\) belong to a class of genuine distributions of negative regularity. These rough \(k\)-forms have several properties desirable of a notion of differential forms:

• they can be integrated over suitably regular \(k\)-manifolds,
• they form a module under point-wise multiplication with sufficiently regular functions,
• exterior differentiation as well as the Stokes theorem extend to these spaces,
• they come with natural embeddings into distribution spaces,
• they contain classes of form valued distributional random fields.

Finally, these spaces unify several previous constructions in the literature. In particular, they generalise spaces of \(\alpha\)flat cochains introduced by Whitney and Harrison, they contain the (rough) \(k\)-forms \(f\cdot dg_1\wedge\dots\wedge dg_k\) introduced by Züst using Young integration, and for \(d=2\) and \(k=1\), they are close to the spaces which Chevyrev et al. use to make sense of Yang–Mills connections. Lastly, as a technical tool we introduce a ‘simplicial sewing lemma’, which provides a coordinate invariant formulation of the (known) multi-dimensional sewing lemma.

This is a joint work with A. Chandra.

May 30th 2025 - Alfonso Garmendia (MPI, Bonn), "E-symplectic and almost regular Poisson manifolds".

Abstract: Singular symplectic manifolds have been studied to model dynamical systems and classical mechanics for spaces with singularities. One particular example is for manifolds with boundary, the dynamics will be given by a seemingly not smooth symplectic form, but considering only vector fields tangent to the boundary this form is well defined and smooth, therefore this dynamical systems are given by a symplectic form only defined on the dual of an specific class of vector fields (the ones tangent to the boundary). Similarly one can consider symplectic forms on different classes of vector fields. We will consider in this talk classes coming from a singular foliation and compare with the structure that the symplectic form brings on the groupoid that integrates the foliation.