GAP XVI — Timișoara

Schedule

Mini-courses

Anton Alekseev (University of Geneva):
The Kashiwara-Vergne theory

The Duflo Theorem states the isomorphism of the center of the universal enveloping algebra with the ring of invariant polynomials. The Kashiwara-Vergne (KV) problem on the properties of the Baker-Campbell-Hausdorff series is one of the strategies of proving the Duflo Theorem. Surprisingly, it turns out that the KV problem is related to many other fields of mathematics.

In this mini-course, we will start by reviewing the Duflo Theorem and the KV problem, and then we will explain the connection between the KV problem and the Goldman bracket and Turaev cobracket defined by intersections and self-intersections of curves on 2-manifolds. Our toolbox will include the Knizhnik-Zamolodchikov connection and the non-commutative version of the divergence.

The mini-course is based on joint works with B. Enriquez, N. Kawazumi, Y. Kuno, E. Meinrenken, F. Naef and C. Torossian.

Philip Boalch (Université Paris-Sud):
Symplectic manifolds and isomonodromic deformations

Given any connection on an algebraic vector bundle on a smooth complex curve, there are various pieces of topological data one can define, and then use to classify such connections. I'll compare various approaches and show how the notion of “wild monodromy” leads to a simple presentation of the resulting moduli space (the wild character variety). Many varieties well-known in representation theory occur as examples. Then I'll review the TQFT-type approach to their symplectic/Poisson structures, using the framework of quasi-Hamiltonian geometry (complementing the original analytic approach from 1999). Finally I'll discuss the notion of “wild Riemann surface” that involves recognising that some of the boundary conditions behave exactly like the moduli of the curve, and how varying this structure leads to the “wild mapping class group” and its action by algebraic Poisson automorphisms on the wild character variety, generalising the usual (tame) case. The final step in this project (the twisted case) is joint work with D. Yamakawa (arXiv:1512.08091).

Eckhard Meinrenken (University of Toronto):
Deformations spaces and normal forms

We will use the deformation to the normal cone' construction to prove old and new normal form theorems, ranging from the Morse Lemma and the Weinstein splitting theorem to local normal forms for Lie algebroids and related structures. A central ingredient of these proofs is a linearization lemma for Euler-like vector fields.

Plenary talks

Ruggero Bandiera (University of Rome):
Baker-Campbell-Hausdorff product in the free pre-Lie algebra

We describe a simple algorithm to compute the expansion of the BCH product in the Lyndon basis of the free Lie algebra. After works by Murua and Casas-Murua, this problem is reduced to the one of computing the expansion of the BCH product of the generators in a free pre-Lie algebra of rooted trees. Our approach is based on a pre-Lie analog of the usual formula BCH(X,Y)=log(e^Xe^Y) and on a combinatorial study of pre-Lie logarithms. Based on joint works with Florian Schaetz and Luca Simi.

Daniel Beltita (Institute of Mathematics "Simion Stoilow" of the Romanian Academy):
On an inverse problem in representation theory of nilpotent Lie groups

We show that every Heisenberg group is uniquely determined by its unitary dual space, within the class of exponential Lie groups. We also announce further progress on the inverse problem in representation theory of nilpotent Lie groups, that is, the open problem of establishing to what extent nilpotent Lie groups can be recovered from their representation theory. The presentation is based on recent joint work with Ingrid Beltita.

William Riley Casper (Louisiana State University):
Representation Theory and the Matrix Bochner Problem

The Matrix Bochner Problem is to classify all sequences of orthogonal matrix polynomials which are eigenfunctions of a second-order matrix differential operator. For certain special rank 1 symmetric pairs (G,K), each irreducible representation π of K gives rise to such a sequence of polynomials. In this talk, we will recall a representation-theoretic construction of orthogonal matrix polynomials and discuss various properties the associated differential operators and matrix weights are known to possess. Next, we will discuss recent work (joint with Milen Yakimov) where structural results for low dimensional, semiprime PI rings are used to prove under mild hypotheses that every sequence of orthogonal matrix polynomials which are eigenfunctions of a matrix differential operator arises as a noncommutative bispectral Darboux transformation of a direct sum of classical orthogonal polynomials. We will also demonstrate how our classification explains various properties observed in the representation-theoretic construction.

Ion I. Cotaescu (West University of Timisoara):
External symmetry of covariant fields on curved spacetimes

I present the theory of external symmetry that defines the induced representations transforming the covariant fields under isometries on pseudo-Riemmanian manifolds

Benjamin Enriquez (University of Strasbourg):
TBA

TBA

Vladimir Fock (University of Strasbourg):
Riemann geometry without indices. (Joint work with P.Goussard)

Working with Levi-Civita connection, Christoffel symbols, Riemann and Einstein tensors is know to be rather painful. We suggest a technique to simplify many computations using a hidden action of the algebra sl(2)×sl(2) on the space of tensors and (almost) avoiding using indices. As an application we will discuss the algebroid arising in gravity theory replacing the gauge group.

Stefan Haller (University of Vienna):
Rank two distributions of Cartan type on 5-manifolds

A rank two distribution on a 5-manifold is said to be of Cartan type if it is as non-integrable as possible, that is, if it is bracket generating with growth vector (2,3,5). These structures can equivalently be described as regular normal parabolic geometries of type (G2,P) where G2 denotes the split real form of the exceptional Lie group and P is a particular parabolic subgroup. Over a 5-manifold equipped with a rank two distribution of Cartan type every irreducible representation of G2, thus, gives rise to a sequence of natural differential operators called (curved) BGG sequence. In this talk we will discuss recent results concerning the analysis of these operators. BGG sequences are not elliptic, but hypoelliptic. They share many key analytic features with elliptic sequences. On closed 5-manifolds this permits to link local and global geometry with spectral theory.

Motivation for studying these hypoelliptic sequences comes from the fundamental question: Which 5-manifolds admit a rank two distribution of Cartan type? For open manifolds this question admits a simple answer. In the closed case the topological aspect of this problem can be settled. However, it remains unclear to what extent the corresponding h-principle holds true on closed 5-manifolds.

This talk is based on joint work with Shantanu Dave.

Bas Janssens (Delft University of Technology):
Equivariant positive energy representations of gauge groups

We classify projective unitary positive energy representations of the gauge group — the group of vertical automorphisms of a principal fibre bundle. It is natural to assume equivariance with respect to a group G of transformations of the base, to be thought of as the group of space-time transformations. The gauge group representation is said to be of positive energy if G contains a 1-parameter subgroup (the time translations) whose generator (the Hamilton operator) has positive spectrum. We are able to give a full classification if the one-parameter group of time translations acts freely. Otherwise, we obtain localisation theorems that effectively reduce the representation theory to base manifolds of lower dimension.

Madeleine Jotz (University of Göttingen):
On ideals in Lie algebroids

I will describe a notion of ideal in Lie algebroids, named 'infinitesimal ideal system' and motivate from several point of views why this is in my opinion the 'right' notion of symmetry in the Lie algebroid setting: among other features, I will discuss quotients by infinitesimal ideal systems.

Then I will sketch an obstruction to the existence of an infinitesimal ideal system structure on a given Lie pair, in terms of the Atiyah class of the Lie pair. I will finally talk about some more obstructions to the existence of an infinitesimal ideal system in a given Lie algebroid.

The first part of the talk is based on joint work with Ortiz and Drummond.

Jiang-Hua Lu (The University of Hong Kong):
Total positivity and Poisson cluster structures on shifted big cells in flag varieties

We introduce a cluster structure on every shifted big cell in the flag variety of a complex semisimple Lie group, and we discuss the compatibility between this cluster structure and Lusztig's total positivity as well as the standard Poisson structure on the flag variety.

Gwenael Massuyeau (University of Bourgogne):
Bottom tangles in handlebodies and an extension of the Kontsevich integral

(Joint work with Kazuo Habiro.) At first glance, the set of knots does not show in itself a very rich algebraic structure ; however, there exist several highly-structured categories that encapsulate the set of knots. In this talk, we will consider the category B of "bottom tangles in handlebodies" which will be compared to the category T of usual "tangles". The linearization of the category B has a natural filtration and it turns out that the associated graded is universal among symmetric linear categories with a distinguished "Casimir-Hopf algebra" (e.g., categories of representations of a quadratic Lie algebra). After an overview of the necessary background and definitions, we will sketch the proof of this result which is based on an extension of the Kontsevich integral (from T to B) for every Drinfeld associator.

Nikita Nikolaev (University of Geneva):
Abelianisation of Logarithmic Connections

I will describe an approach to studying meromorphic connections on vector bundles called abelianisation. This technique has its origins in the works of Fock-Goncharov (2006) and Gaiotto-Moore-Neitzke (2013), as well as the WKB analysis. Its essence is to put rank-n connections on a complex curve X in correspondence with much simpler objects: connections on line bundles over an n-fold cover Σ→X. The point of view is similar in spirit to abelianisation of Higgs bundles, aka the spectral correspondence: Higgs bundles on X are put in correspondence with rank-one Higgs line bundles on a spectral cover Σ→X. However, unlike Higgs bundles, abelianisation of connections requires the introduction of a new object, which we call the Voros cocycle. The Voros cocycle is a cohomological way to encode objects such as ideal triangulations that appeared in Fock-Goncharov, spectral networks that appeared in Gaiotto-Moore-Neitzke, as well as the matrices appearing in the WKB analysis. By focusing our attention on the simplest case of logarithmic singularities with generic residues in rank 2, we will describe an equivalence of categories, which we call the abelianisation functor, between sl(2)-connections on X satisfying a certain transversality condition and rank-one connections on an appropriate 2-fold spectral cover Σ→X. This presentation is based on the work completed in my thesis (2018) and recent extensions thereof.

Pavol Severa (University of Geneva):
Quantization of Poisson Lie groups and of (easy) Poisson groupoids

There is a rather simple description of (all) Hopf algebras in terms of braids. It is closely related to the idea of Hopf algebra valued holonomies on surfaces. A similar description works also for Hopf algebroids with a commutative base. As an application we get an explicit quantization of Poisson Lie groups (or of Lie bialgebras) to Hopf algebras, and also a quantization of Poisson groupoids with a commutative base to Hopf algebroids.

Reyer Sjamaar (Cornell University):
Transversely symplectic Riemannian foliations

Examples of transversely symplectic Riemannian foliations include K-contact manifolds, cosymplectic manifolds, symplectic mapping tori, and Prato's toric quasifolds. The leaf space of such a foliation is an étale symplectic stack. I will survey my recent work with Benjamin Hoffman and Yi Lin about actions of Lie groups and Lie 2-groups on such spaces, such as a convexity theorem and an Atiyah-Bott-Berline-Vergne type localization theorem.

Alice Barbara Tumpach (University of Lille 1):
Bruhat-Poisson structure of the restricted Grassmannian and the KdV hierarchy

In the first part of this talk, we explore the notion of Poisson structure in the Banach context. We show that the Leibniz rule for a Poisson bracket on a Banach manifold does not imply the existence of a Poisson tensor. The existence of a Hamiltonian vector field on a Banach manifold endowed with a Poisson tensor is also not guaranteed. We propose a definition of a (generalized) Poisson structure on a Banach manifold which includes all weak symplectic Banach manifolds.

The second part of the talk is devoted to the study of particular examples of generalized Poisson manifold, namely Banach Poisson-Lie groups related to the Korteweg-de-Vries hierarchy. We construct a generalized Banach Poisson-Lie group structure on the unitary restricted group, as well as on a Banach Lie group consisting of (a class of) upper triangular bounded operators. We show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the unitary restricted group, and that the action of the triangular Banach Lie group on it by “dressing transformations” is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.