Meeting 2023

The 2023 meeting is taking place October 10—12, 2023, at AUB.

Organizers: Rafael Andrist, Floriand Betrand, Giuseppe Della Sala, Georges Habib, Sylvie Paycha

Schedule

Tuesday, October 10

Wednesday, October 11

Thursday, October 12

Abstracts

• Gaofeng Huang (University of Bern)
  Title: Symplectic holomorphic density property for Calogero-Moser spaces
  Abstract: The n-th Calogero-Moser space is the phase space of a system of n indistinguishable particles on a complex line, interacting through inverse square potential. By a work of Wilson, its mathematical model is known to be a smooth complex affine variety with a holomorphic symplectic structure, coming from complex symplectic reduction and this variety is diffeomorphic to the Hilbert scheme of n points on the affine plane. We consider four completely integrable symplectic holomorphic vector fields on this manifold, then compute new symplectic vector fields by taking linear combination and Lie bracket. It turns out that this process does generate all algebraic fields, thus giving the symplectic holomorphic density property of this Stein manifold. Joint work with Rafael B. Andrist.

• Tamer Tlas (American University of Beirut)
  Title: Yang-Mills as a constrained Gaussian.
  Abstract: I will discuss the recent work on rewriting the Yang-Mills path integral in the form of a Gaussian integral with a constraint.
• Pierre Clavier (Université de Haute-Alsace)
  Title: Tree zeta values
  Abstract: Recently, some efforts have been made to generalise multizeta values (MZVs) to more general structures than words and in particular to rooted forests and cones. I will present a new generalisation named tree zeta values (TZVs), which associate numbers to rooted forests. After briefly introducing TZVs, I will detail some of their properties that have been recently studied and present their relations to other generalisations of MZVs. Work in collaboration with Dorian Perrot.
• Isabelle Chalendar (Université Gustave Eiffel)
  Title: Discrete vs Continuous for linear and continuous operators on Banach spaces
  Abstract: Abstract: Given a linear and continuous operator T on a Banach space X, one can consider the discrete semigroup from its (positive) integer powers and a delicate problem consists in finding whether T can be embedded in a continuous semigroup T_t parametrised by a non-negative parameter t. There are obvious sufficient conditions involving spectral properties of T but in general we only have a few examples such as the famous Volterra operator V on various Banach spaces. This operator will be at the center of the talk and compactness properties will be analyzed.
  This is joint work with Ihab Al Alam, Fida El Chami, Emmanuel Fricain, Georges Habib and Pascal Lefèvre. Our collaboration was funded by “Théorale”, Cèdre, Project Hubert Curien.
• Paolo Aschieri (University of Piemonte Orientale)
  Title: Noncommutative principal bundles, gauge transformations and connections
  Abstract: We review some mathematics and physics motivations for the study of noncommutative geometry and the study of the differential geometry of noncommutative principal bundles. In particular we consider infinitesimal gauge transformations for noncommutative principal bundles with mild noncommutativity (i.e. noncommutativity compatible with equivariance under a triangular Hopf algebra). These include the instanton bundle as well as the frame bundle on the Connes-Landi noncommutative sphere S⁴_ϑ. Gauge transformations close a braided Lie algebra and lead to an Atiyah sequence of braided Lie algebras (or braided Lie algebroid). Connections are introduced as splittings of the sequence and their curvature studied. (Based on joint works with G. Landi and C. Pagani).
• Karma Dajani (University of Utrecht)
  Title: Minkowski Normal Numbers
  Abstract: In 1909 Émile Borel proved that under the Lebesgue measure, almost every number is normal in base b. This means that if one looks at the expansion of a uniformly chosen point x in the unit interval of the form x = ∑_i=1^∞ a_i bⁱ, a_i ∈ {0, 1, ···, b−1}, then any arbitrary chosen block d₁ ··· d_n ∈ {0, 1, ···, b−1}n appears with frequency b − n in the expansion of x. We start by rephrasing this concept in the language of Ergodic Theory allowing us to generalize the notion of normality to other types of expansions (like continued fractions) using other measures to count frequencies. Ergodic Theory tells us that normality is a generic property, but with this theory one is unable to exhibit a specific normal number, so that is where the challenge lies. We then proceed to give a historical overview of results on the construction of normal numbers, and we end by explaining a specific construction of an irrational number whose continued fraction expansion is normal with respect to the Minkowski measure, which is singular with respect to Lebesgue measure. (The construction is joint work with M. de Lepper and E.A. Robinson.)
• Felipe Leitner (University of Greifswald)
  Title: Kohn-Dirac operator and twistor spinors in CR geometry
  Abstract: In my talk I plan to discuss CR geometry from the viewpoint of differential geometry. In fact, CR geometry is a parabolic geometry, closely related to conformal geometry via the Fefferman construction. In this setting I will introduce spinor field equations, like Dirac and twistor equations. We study aspects of their solutions, like vanishing theorems, obstructions to positive scalar curvature, related to Kohn-Rossi cohomology; and integrability conditions for the existence of twistors, which are related to eigenvalue estimates for the Dirac operator.

• Jean Ruppenthal (University of Wuppertal)
  Title: The L²-Stokes Theorem on Complex Varieties and Dolbeault Cohomology
  Abstract: Let X be a singular complex space. Roughly speaking, by definition, one says that the L²-Stokes-Theorem holds on X for the exterior derivative d or for the ∂-operator, respectively, if partial integration is possible with respect to these operators on L²-forms, i.e., the singular set should be negligible in a certain sense. It is conjectured, that the L²-Stokes-Theorem holds for d on projective varieties, and it is known to fail in general for ∂. In this talk, we will discuss the state of the art and present an idea of how to overcome the problem for the ∂-operator.