The program, including

• schedule
• titles and abstracts
• details about the virtual poster session

can be found below. For access information please register.

Schedule

Times are Central European Summer Time (UTC+2)

Titles and abstracts

Karim Adiprasito (Hebrew University of Jerusalem, Israel)
Combinatorial approaches to Hodge and Lefschetz theory (video)

Abstract. I will survery recent developments in combinatorial Hodge theory, in particular the combinatorial proofs of the Hodge Riemann relations for matroids, and beyond that the Lefschetz theorems beyond positivity. I will present some new applications to a conjecture of Hibi and Stanley.

Jørgen Ellegaard Andersen (University of Southern Denmark, Odense, Denmark)
Geometric quantization of curved phase spaces via the theory of Resurgence (video)

Abstract. Geometric Quantization of general symplectic manifolds has so far failed to provide a scheme for how to quantize observables and to address the dependence of the choice of the needed polarization in a way which is compatible with quantization of observables. We will provide answers to these two fundamental questions for general symplectic manifolds which admit compatible Kähler structures. Our construction needs the theory of resurgence to make sense of the many divergent series which naturally arise in the construction of the quantization of the observables and also in the construction of the needed generalised infinite order Hitchin connections, which relates the quantization for different Kähler polarizations in a way compatible with the quantization of the observables.

Francesca Arici (Universiteit Leiden, The Netherlands)
A non-commutative approach to the geometry of circle and sphere bundles (video)

Abstract. The theory of C*–algebras offers an elegant setting for many problems in mathematics and quantum physics. In view of Gel’fand duality, their study is often referred to as non-commutative topology: general noncommutative C*–algebras are interpreted as non-commutative spaces. Many classical geometric and topological concepts can be translated into operator algebraic terms, leading to the so-called noncommutative geometry (NCG) dictionary.
In this talk, I will describe how circle and sphere bundles, central objects in the development of algebraic topology, can be realised in terms of modules over operator algebras.

Dorothea Bahns (University of Göttingen, Germany)
Construction of the Sine-Gordon's Haag–Kastler net of observables

Abstract. Starting from the framework of perturbative Algebraic Quantum Field Theory, which allows for the construction of an interacting quantum field theory in the sense of formal power series, I will show how to construct von Neumann algebras of observables satisfying the Haag–Kastler axioms, a set of axioms required of a sensible quantum field theory in an algebraic setting.
This is joint work with Klaus Fredenhagen and Kasia Rejzner.

Pierre Bieliavsky (University Louvain, Belgium)
On a construction by Dorothea Bahns and René Schultz on algebras of tempered distributions (video)

Abstract. D. Bahns and R. Schultz defined spaces of tempered distributions on \( \mathbb R^{2n} \) on which the non-formal Weyl product is well defined. We extend this result for (other) spaces of tempered distributions which are stable under translation-invariant non-formal star products. In particular, I'll describe the case of the normal ordered star product (canonical Kohn–Nirenberg quantization). The latter formally corresponds to the Jordanian twist used in the context of the \( \kappa \) Minkowski space.
This work is joined with Louis De Man and Heiner Olbermann (UCLouvain, Belgium).

Chiara Esposito (University of Salerno, Italy)
\( [ \)Formality, Reduction\( ]=0 \) ? First step (video)

Abstract. In this talk we propose a reduction scheme for multivector fields phrased in terms of L-infinity-morphisms. First, using geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to build up a suitable deformation retract of DGLA’s. As a second step, we construct a Poisson analogue of the Cartan model for equivariant de Rham cohomology. As a consequence we prove the existence of a curved L-infinity morphism between equivariant multivector fields and multivector fields on the reduced manifolds that coincides with the standard Marsden–Weinstein reduction.

Petr Grinevich (Steklov Mathematical Institute, Moscow, Russia)
Total positivity, M-curves and real regular Kadomtsev–Petviashvili II solutions (video)

Abstract. This lecture is based on joint results with Simonetta Abenda, Bologna University.
It is well-known that real regular multiline soliton solutions of the Kadomtsev–Petviashvili II equation can be constructed using two different approaches:
1. By applying the Darboux transformation associated to a point of totally non-negative Grassmannian
2. By degenerating real regular finite-gap solutions, associated to the so-called M-curves (Riemann surfaces equipped with antiholomoprhic involution with maximal possible number of real ovals).
Both objects — totally non-negative Grassmannians and M-curves — arise in many areas of mathematics, but, at first sight, they are not connected. In other words, how to associate a degenerate M-curve with a divisor to a point of Grassmannian with non-negative Plücker coordinates? We show that a bridge between these two objects can be constructed using the parametrization of totally non-negative Grassmannians in terms of Le-networks suggested by Alexander Postnikov. This parametrization was, in particular, used in the study of on-shell Yang–Mills scattering amplitude.

Simone Gutt (Université Libre de Bruxelles, Belgium)
Minimally and maximally non integrable almost complex structures (video)

Abstract. Newlander Nirenberg's theorem tells us that an almost complex structure \( J \) is integrable if and only if its Nijenhuis tensor (also called Nijenhuis torsion), \( N^J \), vanishes, and this amounts to saying that the complex distribution \( T_J^{1,0} \), whose sections are \( \{ X-iJX \mid X \in \chi(M) \} \), is involutive.
On a symplectic manifold \( (M,\omega) \), there always exist almost complex structures which are compatible (i.e. such that \( g(X,Y) := \omega (X,JY) \) is symmetric), and positive (i.e. \( g \) is positive definite). If such a structure is integrable, it induces a Kähler structure on the manifold. Thurston was the first to give examples of symplectic manifolds which do not carry any Kähler structure. Our aim is to characterize special (non integrable) almost complex structures by properties of their Nijenhuis tensor.
In the symplectic setting, for a compatible positive \( J \), the space of tensors at a point \( p \) with the symmetries of \( N^J_p \) is irreducible under the action of linear transformations of the tangent space \( T_pM \) which preserve \( \omega_p \) and commute with \( J_p \) (i.e. the unitary group). Hence no geometrical linear condition can be imposed on the Nijenhuis tensor other than its vanishing.
We study the \( \operatorname{Image} N^J \) distribution on the manifold, given at each point \( p \) by the span of the image of \( N^J_p \). Observing that the derived distribution of \( T_J^{1,0} \) is given by \( T_J^{1,0}+\operatorname{Image} N^J \), we define a notion of maximally or minimally non integrable almost complex strucure \( J \).
This talk is based on some joint works with Michel Cahen, Maxime Gérard, Jean Gutt, Manar Hayyani, and John Rawnsley.

Ryoichi Kobayashi (Nagoya University, Japan)
Probabilistic Riemann–Hurwitz formula and measure concentration phenomenon (video)

Abstract. Let \( (X, D) \) be a pair of an n-dimensional smooth projective variety \(X\) and a very ample divisor \(D\) with at worst simple normal crossings. For each \( l \in \mathbb G (n, \lvert mD \rvert) \) (\(m\) being a positive integer) we associate a holomorphic map \( \mu \colon X \to l^* \cong \mathbb P^n \) by sending \( x \in X \) to a hyperplane in \( l \) consisting of elements of \( l \) passing through \( x \) (this is interpreted as something like a stereo-graphic projection). We consider the totality of \( \mu \)’s parameterized by the Grassmannian \( \mathbb G (n, \lvert mD \rvert) \). Let \( \mu^* K_{\mathbb P^n}^{-1} = K_X^{-1} + R_\mu \) be the Riemann–Hurwitz Formula associated to \( \mu \) (assuming \( \mu \) being finite). We then encounter the following objects to be studied by methods of asymptotic analysis.
• The totality of the ramification divisors \( R_\mu \) on \( X \) parameterized by the Grassmannian \( \mathbb G(n, |mD|) \).
• The totality of integration currents \( R_\mu \) approximately uniform distributed on \( X \) when \( m \) is large.
Suppose moreover that a holomorphic map \( f \colon \mathbb C^k \to X \) is given. Then
• The family of the Nevanlinna/Cartan/Ahlfors theory for \( f_\mu := \mu \circ f \colon \mathbb C^k \to l^* \cong \mathbb P^n \), parameterized by \( \mathbb G (n, \lvert mD \rvert) \).
• The space parameterizing the totality of the relative positions between the jets of \( f \) and the ramification divisor \( R_\mu \) as well as the functions which measures the relationship between these.
If \( m \) is large, the Grassmannian \( \mathbb G (n, \lvert mD \rvert) \) and other related spaces have large dimension. Therefore the measure concentration phenomenon takes place. We examine the Ahlfors’s philosophy that “the Nevanlinna Theory is a quantitative version of the Riemann–Hurwitz Formula” in the above stated situation.

Catherine Meusburger (Erlangen University, Germany)
Turaev–Viro state sums with defects

Abstract. Turaev–Viro invariants are invariants of 3-manifolds constructed from spherical fusion categories. They define topological quantum field theories and are related to the quantisation of moduli spaces of flat connections.
I explain how to construct a Turaev–Viro style state sum model with defects, where lower dimensional submanifolds are decorated with data from higher categories. More specifically, defect planes are labelled by certain bimodule categories over spherical fusion categories, defect lines by certain bimodule functors and defect vertices by bimodule natural transformations.
This is work in progress with John Barrett.

William Mistegaard (Institute for Science and Technology, Austria)
Quantization and quantum topology (video)

Abstract. In this talk we present results in quantum topology obtained by quantization of moduli spaces of flat connections on two-manifolds. We also present a joint project with Andersen and Hausel, in which we define and study the automorphism equivariant Hitchin index. This arise in the context of quantization of the moduli space of Higgs bundles on a Riemann surface.

Pavel Mnev (University of Notre Dame, USA)
Batalin–Vilkovisky effective actions and cutting-gluing (video)

Abstract. I will explain the “BV–BFV formalism” — a refinement of Batalin–Vilkovisky homological approach to perturbative quantization of gauge theories to a version compatible with cutting-gluing on the spacetime manifold. As an example, I will discuss the BV–BFV data of a subclass of AKSZ sigma models (the perturbed BF theories), constructed in terms of configuration space integrals. I will also discuss applications to 2d Yang–Mills theory (recovering the nonperturbative representation-theoretic answer via cutting with corners) and to the holographic correspondence, in the example of Chern–Simons – WZW relation.
The talk is based on joint works with A.S. Cattaneo, R. Iraso, N. Reshetikhin and K. Wernli.

Erik Panzer (Oxford University, UK)
Polylogarithms in Deformation Quantization (video)

Abstract. Several constructions in deformation quantization use integrals over configuration spaces of marked holomorphic discs, to define graphically indexed families of numbers or differential forms. From the geometry of the configuration spaces, these inherit operadic relations that can be useful to solve a deformation problem. The prime example is the universal quantization formula for Poisson structures on \( \mathbb R^n \) due to Kontsevich.
In this talk I will explain joint work with Brent Pym and Peter Banks, arXiv:1812.11649, that explains the numbers (multiple zeta values) and functions (multiple polylogarithms) that arise from such integrals. The talk will require no familiarity with these special functions and should be broadly accessible. I will illustrate the main steps of an algorithm to compute the integrals.

Short talks

Camilo Angulo (University Federal Fluminense in Rio de Janeiro, Brazil)
Gray stability for contact groupoids (video)

Abstract. A Jacobi structure is a Lie bracket on the sections of a line bundle. These brackets encode time-dependent mechanics in the same way Poisson brackets encode mechanics. Contact groupoids are finite-dimensional models for the "integrations" of these infinite-dimensional Lie algebras. In this talk, we explain how, under a certain compactness hypothesis, one can adapt the argument of Gray–Moser to these multiplicative contact structures and point out some applications.

Francis Bischoff (University of Oxford, UK)
Brane quantization of toric Poisson varieties (video)

Abstract. The geometric quantization of a Kähler manifold produces a vector space by taking the holomorphic sections of a prequantum line bundle. In this procedure, the complex structure serves as a polarization, and hence the vector space is insensitive to this choice. However, by quantizing multiples of the symplectic form, we arrive at the homogeneous coordinate ring, and this does encode the complex structure. In this talk, I will explain how a holomorphic Poisson structure allows us to deform the polarization into a generalized complex structure, in such a way that the Kähler structure is deformed to a generalized Kähler structure. Under this deformation the Kähler quantization remains unchanged, but the homogeneous coordinate ring is deformed to a non-commutative algebra quantizing the Poisson structure. In order to implement this deformation, we need to make use of a conjectural construction of a category of generalized complex branes, which relates it to the A-model of a symplectic groupoid. In this talk, I will explain this construction in the special case of toric Poisson varieties. This is joint work with M. Gualtieri.

Nicola Capacci (University of Zurich, Switzerland)
An algebraic approach to scalar theory on manifolds with boundary (video)

Abstract. In this talk I will discuss the problem of quantization on manifolds with boundary, in the accessible example of scalar field theory. With classical constraints placed upon the boundary, this procedure is understood and yields a BD₀ quantization of the bulk observables, while quantizing the boundary in a compatible way requires the use of the BV–BFV formalism, which performs geometric quantization. I will present some ongoing work with Ödül Tetik in understanding this procedure at the level of factorization algebras and factorization homology.

Miquel Cueca (Georg-August-Universität Göttingen, Germany)
Sigma models and Lie bialgebroids (video)

Abstract. It is well known that Poisson manifolds and Courant algebroids give rise to 2d and 3d sigma models. A way of having both structures is by considering a Lie bialgebroid: Integration gives a Poisson groupoid and the Drinfel'd double a Courant algebroid. We will show that their corresponding sigma models are related by dimensional reduction. This is joint work with Alejandro Cabrera.

Andrea Galasso (National Center of Theoretical Sciences in Taipei)
Toeplitz operators on CR manifolds and group actions (video)

Abstract. Let \( X \) be a compact connected orientable CR manifold with non-degenerate Levi curvature. We define Toeplitz operators acting on spaces of \( (0,q) \)-forms with \( L^2 \) coefficients and we study their algebra. Under some conditions we show that they define a deformation of the Poisson algebra of smooth functions on \( X \) lying in the kernel of the Reeb vector field. Given a locally free and transversal action of a compact connected Lie group \( G \) we study the associated algebra of \( G \)-invariant Toeplitz operators for \( (0,q) \)-forms. In the presence of a locally free transversal CR circle action, we study Fourier components of Toeplitz operators. We recovered well-known theorems for \( G \)-invariant Toeplitz operators on circle bundles of quantizable Kähler manifolds. This is based on joint works with Chin-Yu Hsiao.

Pengfei Huang (University of Heidelberg, Germany)
Geometry of base manifold which parametrizes a family of Higgs bundles (video)

Abstract. Studying the moduli space of certain geometric objects from the viewpoint of differential geometry is an important approach to understand the geometry of the moduli space. A basic starting point is to endow the moduli space with a suitable Riemannian metric. If the parametrized geometric object is equipped with a metric, in general, the moduli space could inherit a natural metric called the Weil–Petersson metric as a functional of the metric on the geometric object, some celebrated geometric properties like hyperbolicity of these moduli spaces can be achieved by calculating the curvature of Weil–Petersson metrics, or certain suitable Finsler metrics (also called augmented Weil–Petersson metrics). For the moduli spaces of algebraic varieties, this was done mainly by Ahlfors, Royden, Wolpert for curve case, and mainly by Schumacher, To–Yeung, and Deng for higher dimensional cases.
In this talk, I will present a recent joint work with Dr. Zhi Hu on the investigation of the geometry of the base manifold of a family of stable Higgs bundles over a fixed compact Kähler manifold by calculating the curvature with respect to suitable Weil–Petersson metric (based on Biswas–Schumacher's formula) or Finsler metric.

Kevin Morand (Sogang University, Seoul, Republic of Korea)
Graph complexes and deformation quantization of Lie bialgebroids (video)

Abstract.In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a graph version of the deformation theory for Poisson structures. In this talk, we will review Kontsevich’s graph construction and the rôle it plays in the quantization of Poisson manifolds before introducing a generalisation suitable to address the deformation quantization problem for Lie bialgebroids.

Gabriele Rembado (Hausdorff Centre for Mathematics Bonn, Germany)
Geometry and quantisation of moduli spaces of connections on Riemann surfaces (video)

Abstract. The geometry and quantisation of moduli spaces of unitary flat connections on Riemann surfaces have been widely studied in the past: as the complex structure on the surface is deformed the moduli spaces assemble into a local system of symplectic manifolds, and Kähler quantisation turns it into a projectively flat vector bundle. The complexified version brings about holomorphic connections and hyperkähler manifolds, requiring new ideas in Kähler quantisation; deformation quantisation on the other hand has been carried out in greater generality, namely for moduli spaces of meromorphic connections with irregular singularities. In this talk we will briefly review this story and phrase the singular case in the same geometric language of the nonsingular ones, involving flat symplectic fibre bundles: their bases provide an intrinsic approach to isomonodromic deformations, and their quantisation provides a mathematical approach to irregular singularities in the Wess–Zumino–Novikov–Witten model.

Philipp Schmitt (Leibniz-Universität Hannover, Germany)
Non-formal deformation quantization and Wick rotations (video)

Abstract. Formal deformation quantization of Poisson manifolds has been well-understood since Kontsevich's work of 1997. It can be viewed as a first step in solving the quantization problem in physics, which asks to associate a non-commutative quantum algebra to a classical Poisson algebra. In this sense, I will show how a formal deformation quantization of certain homogeneous manifolds, like the complex projective spaces or the hyperbolic discs, can be made convergent on a space of analytic functions, yielding non-commutative algebras for almost all values of Planck's constant \( \hbar \). I will relate these quantizations of different manifolds by an analytic continuation and rotation in the complex plane, similar to the Wick rotation in physics.

Poster session

The virtual poster session will take place on Tuesday, 10 August, in a virtual poster garden on Gather Town (details are sent to registered participants by email). Titles and abstracts for the poster session are found below — the posters can be viewed ahead of the poster session by clicking on the title of the poster.

Maram Alossaimi (Sheffield University, UK)
Poisson algebras I

Abstract. The concept of Poisson algebra is one of the most important concepts in mathematics that make a link between commutative and noncommutative algebra. Poisson algebras can be defined as Lie algebras which satisfy the Leibniz rule. In our research, we classified a large Poisson algebras class \( A = \Bbbk [t][x,y] \), that is, \( A \) is a polynomial Poisson algebra in two variables \( x \) and \( y \) over the polynomial Poisson algebra \( \Bbbk [t] \), where \( \Bbbk \) is an algebraic closure field with zero characteristic. By using Lemma (Oh, 2006) we concluded that there are three main cases and each case has several subcases. Then we find the Poisson spectrum of \( A \) in each case. In this poster, I identify only the first case.

Miquel Cueca (Georg-August-Universität Göttingen, Germany)
\( BG \) as a symplectic stack

Abstract. If the Lie algebra of the Lie group \( G \) has an \( \operatorname{ad} \)-invariant pairing then \( BG \), the classifying stack, inherits a (2-shifted) symplectic structure. We will show that \( BG \), its symplectic structure and the prequantum bundle can be realized using different models. \( \operatorname{String} (G) \) and the Chern–Simons sigma model appear naturally. This is joint work with Chenchang Zhu.

Arne Hofmann (Georg-August-Universität Göttingen, Germany)
Microlocal analysis and renormalization in QFT

Abstract. Relativistic QFT can be described in terms of families of distributions (generalized functions) such as Wick products or time-ordered products. The "symbol calculus" of Lagrangian or Fourier Integral distributions is a microlocal generalization of the Fourier transform. It provides (with some caveats) a duality between distributions and smooth functions on the wavefront set of the distribution, which is a smooth Lagrangian submanifold of the cotangent bundle.
While the Wightman propagator on Minkowski space is a Lagrangian distribution, in order to describe the singularities of, for instance, time-ordered products, it is necessary to extend the framework of Lagrangian distributions to Lagrangian manifolds with singularities, e.g. unions of cleanly intersecting manifolds. A symbol calculus in this setting should provide the starting point for an investigation of renormalization freedom from a microlocal perspective.

David Kern (Georg-August-Universität Göttingen, Germany)
Reduction of linear Poisson structures and Lie bialgebroids

Abstract. In this poster we give a Poisson approach to infinitesimal ideal systems by reducing the linear Poisson structure on the dual of a Lie algebroid in a linear way. The corresponding reduction of the Lie algebroid can be cast in terms of coisotropic triples. This leads to a slightly more general kind of infinitesimal ideal system and gives a way to reduce Lie bialgebroids. This is joint work with Marvin Dippell and Madeleine Jotz-Lean.

Felix Menke (University of Würzburg, Germany)
Coisotropic algebras, modules and geometry

Abstract. Looking at examples from Poisson geometry and from deformation quantization, we motivate the notion of coisotropic algebras. Essentially, those are triples of algebras which give a description of various reduction procedures in algebraic terms.
We then consider the question which coisotropic algebras arise as the smooth functions on the geometric objects of interest, coisotropic manifolds, and how to obtain coisotropic manifolds from coisotropic algebras.
Furthermore, when studying such coisotropic algebras, it makes sense to investigate also their associated category of coisotropic modules. Here we focus on projective coisotropic modules and aim for a coisotropic Serre–Swan Theorem: For examples of coisotropic algebras from differential geometry we want to relate their projective coisotropic modules to sections of vector bundle-like structures.

Lukas Miaskiwskyi (TU Delft, The Netherlands)
Continuous Lie algebra homology of gauge algebras

Abstract. Quantizations of infinitesimal gauge symmetries are classified in terms of the continuous Lie algebra cohomology group of gauge algebras in degree 2. For gauge bundles with semisimple fibers, this space was calculated by Janssens–Wockel (2013), their method relying heavily on the low degree of the cohomology group. In this poster, we extend these results to homology in higher degree. 
To this end, we review some homological algebra for topological chain complexes and use it to lift the well-known Loday–Quillen–Tsygan-Theorem (1983, 1984) from a statement in algebraic Lie algebra homology to one that takes topological data into account. For globally trivial gauge algebras whose fibres are classical Lie algebras, this calculates a certain stable part of continuous homology. A similar description was given by Feigin (1988), but lacking a detailed proof.
The results for trivial bundles are used in a Gelfand–Fuks-like local-to-global spectral sequence which describes nontrivial gauge algebras. Finally, we discuss how to extract information from this spectral sequence and the obstructions to a closed description of the full homology. This is joint work with Bas Janssens.

Stefano Ronchi (Georg-August-Universität Göttingen, Germany)
Hamiltonian Lie algebroids

Abstract. Hamiltonian Lie algebroids were introduced by Blohmann and Weinsten in [1], in the context of presymplectic manifolds, as a generalization of the notion of hamiltonian action of a Lie algebra. We recall this notion and its counterpart in the context of Poisson geometry, introduced in [2] and set to appear in a forthcoming paper [3], and show how they generalize hamiltonian actions of Lie algebras. In particular, it was shown in [1] that this notion allows coisotropic reduction on a presymplectic manifold. Finally, we provide some examples in the tangent and cotangent Lie algebroids, and give an outlook of the current research. 
References.
[1] Blohmann, C., & Weinstein, A. (2018). Hamiltonian Lie algebroids. arXiv:1811.11109.
[2] Ronchi, S. (2020). Hamiltonian Lie algebroids over Poisson manifolds. Master thesis, University of Bonn.
[3] Blohmann, C., Ronchi, S., & Weinstein, A. (in preparation). Hamiltonian Lie algebroids over Poisson manifolds. 

Matthias Schötz (Université Libre de Bruxelles, Belgium)
Symmetry reduction of states

Abstract. (Based on arXiv:2107.04900) We describe a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra \( \mathfrak g \). The key idea advocated for is that the "correct" notion of positivity on a *-algebra \( A \) is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares \( a^*a \) with \( a\in A \), but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on \( A \) thus depends on this choice of positivity on \( A \), and the notion of positivity on the reduced algebra \( A_{\mathrm{red}} \) should be such that states on \( A_{\mathrm{red}} \) are obtained as reductions of certain states on \( A \). In the special case of the *-algebra of smooth functions on a Poisson manifold \( M \), this reduction scheme reproduces the coisotropic reduction of \( M \), where the reduced manifold \( M_{\mathrm{red}} \) is just a geometric manifestation of the reduction of the evaluation functionals associated to certain points of \( M \). However, other commutative and non-commutative examples like the Weyl algebra are also of interest.