Higher Geometric Structures along the Lower Rhine XVIII
23-24 January 2025 - Radboud University Nijmegen

Date

The workshop starts Thursday 23 January 2025 at 14:00 and ends Friday 24 January 2025 at 16:30.

Location

The talks on Thursday are in room HG00.303 and those on Friday are in HG00.307. Both rooms are in the Huygensgebouw, Heyendaalseweg 135, Nijmegen, The Netherlands (Travel Directions)

Program

23 January (rooom HG00.303) 24 January (room HG00.307)

Registration

It is no longer possible to register for this workshop.

Description

This is the eighteenth in a series of short workshops jointly organized by geometers and topologists from Bonn, Cologne, Nijmegen, and Utrecht, all situated along the Lower Rhine. The focus lies on the development and application of new structures in geometry and topology such as Lie groupoids, differentiable stacks, Lie algebroids, generalized complex geometry, topological quantum field theories, higher categories, homotopy algebraic structures, higher operads, derived categories, and related topics.

Previous Meetings

Earlier editions of these workshops were: I (MPIM Bonn), II (Nijmegen), III (Utrecht), IV (MPIM Bonn), V (Nijmegen), VI (Utrecht), VII (Leuven), VIII (MPIM Bonn), IX (Nijmegen), X (Utrecht), XI (MPIM Bonn), XII (Nijmegen), XIII (Utrecht), XIV (MPIM Bonn), XV (Nijmegen), XVI (Utrecht), XVII (MPIM Bonn)

Organizers

The workshop series is organized by Christian Blohmann, Marius Crainic, Ioan Mărcuț, Ieke Moerdijk, and Steffen Sagave, and the local organizer for this workshop is Steffen Sagave.

Titles and abstracts

  • Lory Aintablian (MPIM Bonn): Differentiation of groupoid objects in tangent categories
    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. Mainly, I will construct the abstract Lie algebroid of a differentiable groupoid in a cartesian tangent category with a scalar \(R\)-multiplication, where \(R\) is a ring object in the ambient category. Examples include differentiation of infinite-dimensional Lie groups, elastic diffeological groupoids, etc.

    This is joint work with Christian Blohmann.

  • David Gepner (Johns Hopkins): Elementary \((\infty,\infty\))-category theory
    Abstract:

    While many of the basic notions of higher category theory have been established for some time, it is less clear how to work with these objects in practice. We will recall key constructions such as the lax Gray tensor product, the lax join, and fibrations of \((\infty,\infty)\)-categories, and explain how to carry out various computations involving certain generating objects.

    Much of this talk will report on joint work with Hadrian Heine.

  • Aaron Gootjes-Dreesbach (U Utrecht): Jet configuration spaces and weightings
    Abstract:

    Blowing up the diagonal of \(M^2\) for a manifold \(M\) yields a configuration space that remembers the collision axis of collided configurations. Fulton and MacPherson famously generalized this construction to configurations of more than two points. In joint work with Á. del Pino, we build configuration spaces for jets of maps \(M\to N\) in the same spirit, but adapted to the main structure on jet space: The Cartan distribution and the Lie filtration it generates. Higher jet orders necessitate the use of weighted blow-ups, which we tackle within the recent differential-geometric framework of weightings due to Loizides and Meinrenken. In this talk, I first intuitively illustrate this motivation and our construction. I will then give an introduction to weightings and discuss how we use them as far as time allows.

  • Fabian Hebestreit (U Bielefeld): Idempotent ideals and derived localisations
    Unfortunately, this talk had to be canceled due to illness.
  • Josefien Kuijper (U Utrecht): The Dehn invariant for spherical scissors congruence as spectral Hopf algebra
    Unfortunately, this talk had to be canceled due to illness.
  • Ieke Moerdijk (U Utrecht): A Categorical Koszul Duality
    Abstract:

    Tom Leinster explained long ago how to define infinity-algebras in a symmetric monoidal category with a notion of weak equivalence. I will present an analogous notion of linear infinity-operads, defined by replacing the familiar Segal condition by an additional structure map. These operads are parametrised by a category of trees, which admits a bar-cobar (or "Koszul") duality between its presheaves and the presheaves on the opposite category. Some other categories turn out to have this property as well, although it is not clear (to me, yet) what characterises such categories.

    The talk is based on ongoing work with Eric Hoffbeck, Paris.

  • Ahina Nandy (RU Nijmegen): Some Remarks on the Special Linear and Metalinear Algebraic Cobordism
    Abstract:

    In classical topology, different cobordism theories can be thought of as uni- versal cohomology theories with certain orientations. For example, complex cobordism (\(\textrm{MU}\)) is the universal complex oriented cohomology theory; that is cohomology theories with Thom isomorphism for every complex vector bundle. The analogous idea of different notions of orientations, and corresponding alge- braic cobordism theories are well studied in \(\mathbb A^1\)-homotopy theory. I will mostly talk about special linear, and metalinear orientations, and their corresponding universal algebraic theories in the context of \(\mathbb A^1\)-homotopy theory. I would like to draw a parallel between the classical theories like special unitary, and string cobordism, and the algebraic theories that come up in the talk.

    This is a joint work in progress with Egor Zolotarev.

  • Lennart Obster (U Coimbra): A cohomology theory for multiplicative tensors
    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.

  • Dan Wang (MPIM Bonn): Geometric quantization of mixed polarizations on toric manifolds
    Abstract:

    Geometric quantization on symplectic manifolds plays an important role in representation theory and mathematical physics, deeply relating to symplectic geometry and differential geometry.  A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated with different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. In this talk, I will introduce the quantum spaces associated with mixed polarizations and explore their relationships with those associated with Kähler polarizations on toric manifolds.