Additional to the Learning Seminar, during the year, we also meet in a Research Seminar where the members of the group explore their own projects and where Guests also present their latest/on-going research.
Upcoming Seminars:
17.06. Christoph Balcerzak (Universität zu Köln)
Title: T.B.A.
Where/When: Seminarraum 1 (005) from 14h to 15:30h
Abstract
T.B.A.
24.06. Annika Tarnowsky (Max-Planck-Institut für Mathematik, Bonn)
Title: T.B.A.
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
T.B.A.
08.07. Pedro Frejlich (Universidade Federal do Rio Grande do Sul, Brazil)
Title: T.B.A.
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
T.B.A.
15.07. Pedro Frejlich (Universidade Federal do Rio Grande do Sul, Brazil)
Title: T.B.A.
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
T.B.A.
Previous Talks:
03.06. Ruben Louis (Jilin University (China) & Georg-August-Universität Göttingen)
Title: On Nash blow-up of Lie algebroids and singular foliations
Where/When: Seminarraum 1 (005) from 14h to 15:30h
Abstract
We prove that any Lie algebroid \(A\) admits a Nash-type blow-up \(\operatorname{Nash}(A)\) that sits in a nice short exact sequence of Lie algebroids \(0\to K\to\operatorname{Nash}(A)\to D\to 0\) with \(K\) a Lie algebra bundle and \(D\) a Lie algebroid whose anchor map is injective on an open dense subset. The base variety of \(\operatorname{Nash}(A)\) is a blowup determined by the singular foliation of \(A\). This construction is inspired by the work of O. Mohsen, applied in non-commutative geometry, and by a classical method developed by the mathematician J. Nash, primarily used in algebraic geometry for desingularization. We provide concrete examples.
27.05. Marvin Dippell (Università degli Studi di Salerno, Italy)
Title: A Novel Approach to the HKR Theorem in Differential Geometry
Where/When: Stefan Cohn-Vossen Raum (313) from 14h to 15:30h
Abstract
The classical Hochschild–Kostant–Rosenberg (HKR) Theorem in differential geometry gives a quasi-isomorphism between the differentiable Hochschild complex of the algebra of smooth functions on a manifold and the multivector fields on it. The HKR morphism plays an important role in deformation theory, in particular it appears as the lowest degree in Kontsevich’s formality. Even though the HKR Theorem has been known for a long time, available proofs are often of a local nature and are hard to generalize to more structured situations. I will present a novel proof for the HKR Theorem using a symbol calculus and a van Est-double complex. This strategy will allow for an explicit global homotopy and can easily be adapted to various situations. As an example, I will present how the HKR deformation retract can be applied to understand homogeneous star products on the total space of a vector bundle.
20.05. Dan Wang (Max-Planck-Institut für Mathematik, Bonn)
Title: Degeneration of quantum spaces along the geodesic rays
Where/When: Seminarraum 1 (005) from 14h to 15:30h
Abstract
Geometric quantization on symplectic manifolds plays an important role in representation theory and mathematical physics, and is deeply related to symplectic and differential geometry. A crucial problem is to understand the relationships among geometric quantizations associated with different polarizations. In this talk, we will discuss the existence of mixed polarizations and degeneration of quantum spaces along the geodesic rays on toric manifolds.
13.05. Lory Aintablian (Max-Planck-Institut für Mathematik, Bonn)
Title: Differentiation of groupoid objects in tangent categories
Where/When: Seminarraum 1 (005) from 14h to 15:30h
Abstract
The infinitesimal counterpart of a Lie group(oid) is its Lie algebra(oid). I will show that the differentiation procedure works in any category with an abstract tangent structure in the sense of Rosicky, which was later rediscovered by Cockett and Cruttwell. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category \(C\) with a scalar \(R\)-multiplication, where \(R\) is a ring object of \(C\). Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc. This is joint work with Christian Blohmann.
06.05. Wilmer Smilde (University Illinois Urbana-Champaign, United States of America)
Title: Relative algebroids and classification problems for geometric structures
Where/When: Seminarraum 1 (005) from 14h to 15:30h
Abstract
Many geometric structures—like Riemannian metrics satisfying a diffeomorphism-invariant curvature condition—can be described by differential equations with symmetries. In many of these problems, algebroid-like structures show up almost magically, as observed by Bryant. This turns out to be very useful, especially when looking for complete solutions rather than just local ones. In this talk, I will introduce relative algebroids, a framework that systematically explains why this happens and unifies Lie algebroids with the formal theory of PDEs.
22.04. Mykola Matviichuk (Max-Planck-Institut für Mathematik, Bonn)
Title: New examples of quantum projective spaces
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the q-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich’s conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.
08.04. Alfonso Garmendia (Max-Planck-Institut für Mathematik, Bonn)
Title: E-symplectic and almost regular Poisson manifolds
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
This talk investigates interactions among three notions; almost regular foliations, and two generalizations of b-symplectic manifolds: E-symplectic manifolds and almost regular Poisson structures. E-manifolds are almost regular foliations with a symplectic structure. They give rise to a Poisson manifold. Almost regular Poisson manifolds are Poisson manifolds whose symplectic foliation is almost regular. These two notions differ but they are deeply connected. The holonomy groupoid of the almost regular foliation has a natural Poisson structure that one can write explicitly. In some cases this Poisson groupoid gives enough information to get the Symplectic groupoid integrating the underlying Poisson manifold.
15.04. Christoph Balcerzak (Universität zu Köln)
Title: Compact Real Lie Algebras, Cartan Involutions and the Iwasawa Decomposition
Where/When: Übungsraum 2, Gyrhofstraße from 10h to 11:30h
Abstract
In this talk I want to classify compact real Lie algebras, where Cartan involutions play an important role. I want to show the existence and uniqueness (up to conjugacy) of those, from that the conjugacy of compact real forms follows. Furthermore, Cartan Involutions lead to the so called Iwasawa decomposition of a semisimple real Lie algebra, that I want to define as well.
06.03. Christoph Balcerzak (Universität zu Köln)
Title: Classification of real and complex Lie algebras
Where/When: Seminarraum 3 (314) from 10:30h to 12h
Abstract
This is the second talk in a series on Lie algebras and their classifications. This time around, we shall discuss the classification of real and complex Lie algebras using the root systems discussed in the previous talk.
18.02. Christoph Balcerzak (Universität zu Köln)
Title: Root systems, Cartan matrices and Dynkin diagrams
Where/When: Seminarraum 3 (314) from 10:30h to 12h
Abstract
In this talk I want to discuss root systems and how you can classify them with Dynkin diagrams, as a preparation for following talks about semisimple Lie algebras.
05.02. Žan Grad (Instituto Superior Técnico, Lisbon)
Title: Double complexes of representation-valued forms
Where/When: Seminarraum 3 (314) from 11h to 12h
Abstract
In this talk, we discuss representation-valued forms on the nerve of a Lie groupoid comprising the Bott-Shulman-Stasheff complex, and its infinitesimal analogue, the Weil complex. More precisely, we discuss double complex structures compatible with the simplicial differentials. We focus on the special case when the representation is a subrepresentation of the adjoint representation on the isotropy bundle, in which case a multiplicative Ehresmann connection may be used instead of just an invariant linear connection. The columns of the double complexes are then formed by the horizontal exterior covariant derivative, whose interesting properties and applications will be presented.
05.02. Simon Fischer (Institute of Mathematics, Georg-August Universität Göttingen)
Title: Classifying (formal) singular foliations in a neighbourhood around a leaf
Where/When: Seminarraum 3 (314) from 10h to 11h
Abstract
We will discuss how to classify singular foliations in a formal setting, given a fixed leaf with a given transverse structure at a point. One way to classify those is via multiplicative Yang-Mills connections, a connection appearing in curved gauge theory; that is, with tools from a generalised version of gauge theory we will argue that there are not as many foliations as one might think. This is a joint work with Camille Laurent-Gengoux from Université de Lorraine.
04.02. Jorn van Voorthuizen (Universität zu Köln)
Title: Multipole moments in stationary spacetimes
Where/When: -119 from 10:30h to 12h
Abstract
We review the construction of the Geroch–Hansen multipole moments for stationary asymptotically flat vacuum spacetimes, and discuss their well-definedness. This hinges on the uniqueness of the one-point conformal completion in Geroch’s asymptotic flatness definition. Based on Geroch’s approach, we formulate and prove a revised uniqueness result, thereby filling in some gaps in the original approach.
21.01. Lennart Obster (Universidade de Coimbra & Universidade do Porto)
Title: A cohomology theory for multiplicative tensors
Where/When: Besprechungsraum 1 from 10:30h to 12h
Abstract
Multiplicative tensors (possibly symmetric or exterior) are sections of certain vector bundles compatible with a multiplication of a Lie group(oid). Examples include multiplicative differential forms (e.g. symplectic forms), multiplicative multivector fields (e.g. multiplicative Poisson structures), multiplicative complex structures and multiplicative Ehresmann connections. The infinitesimal counterparts of multiplicative tensors are called infinitesimally multiplicative tensors. Infinitesimally multiplicative tensors are sections of certain vector bundles compatible with a Lie bracket of a Lie algebra/oid. In the talk we will describe a cohomology theory for tensor powers of vector bundles over Lie groupoids. In this (global) cohomology theory, multiplicative tensors appear as cocycles. Similarly, we describe the infinitesimal counterpart of this cohomology theory. The plan is to discuss some properties of these cohomology theories, including the relation between the global and infinitesimal theories, and to talk about a few applications.
11.12. Dan Agüero (Scuola Internazionale Superiore di Studi Avanzati, Trieste)
Title: On complex Lie algebroid with constant real rank
Where/When: Besprechungsraum 1 from 10:30h to 12h
Abstract
In this talk, we shall present the basic notions of complex Lie algebroids. We associate a real Lie algebroid to them and briefly describe their local structure. If time allows, we shall introduce the operation of complex sum of real Lie algebroids. This talk is based in arXiv:
2401.05274.
03.12. Karandeep Singh (Julius-Maximilians-Universität Würzburg)
Title: Stability of leaves and differential graded Lie algebras
Where/When: Besprechungsraum 2 Gyrofstr. from 10:30h to 12h
Abstract
Given a geometric structure inducing a (singular) foliation on a manifold, and a leaf of the foliation, it is natural to ask when the leaf is stable under deformations of the geometric structure.
In many cases (e.g. Poisson structures, Lie algebroids, Dirac structures…), this is a special case of the following question about differential graded Lie algebras (dgLa): when does the inclusion of a dg-Lie subalgebra of a dgLa induce a locally surjective map on Maurer-Cartan elements up to equivalence?
Under a finite-dimensionality assumption, which is only satisfied for zero-dimensional leaves, I have given a sufficient cohomological condition for a positive answer to the general question.
In this talk, I will discuss the result in the above-mentioned finite-dimensional setting, and report on some progress on dropping the finite-dimensionality assumption, the latter of which should provide a unified approach to the stability of leaves of more general structures, generalizing results of Marius Crainic and Rui Fernandes for Lie algebroids and Poisson manifolds.
21.11. Andreas Schüßler (KU Leuven)
Title: Lifting geometric properties to real projective blowups
Where/When: Besprechungsraum 1 from 10:30h to 12h
Abstract
Real projective blowups replace a closed, embedded submanifold of a manifold \(M\) by a submanifold of codimension 1. Given a (singular) geometric structure (e.g. a Poisson structure) on \(M\), one can ask if it lifts to a geometric structure on the blowup, possibly making it more regular. I will discuss known results on the lift of foliations and Poisson structures, and the work-in-progress with Marco Zambon on Dirac structures.