The Working Group in Algebra, Geometry and Quantization is a weekly meeting of the research team of Prof. Martin Schlichenmaier. Its aim is to present both research works and surveys of mathematical areas of common interest.

### Meetings

The working group currently meets on Tuesday from 14h00 to 15h00.
The room is specified next to the date (Maison du Nombre, 6th floor - Belval).

Occassionally we will have talks by invited guests at other times to be announced here.

### Organizers

• Olivier Elchinger
• Francois Petit
• Martin Schlichenmaier

### Upcoming sessions

• 06.06.2017 -- Room S.6A
Anne Pichereau (Université Blaise Pascal, Clermont-Ferrand), "$\mathbb{Z}_2$-graded Poisson structures and their cohomology""
Abstract: This is a joint work with Michael Penkava (Univ. Wisconsin-Eau Claire). We study $\mathbb{Z}_2$-graded Poisson structures, that is Poisson structures defined on graded commutative polynomial algebras, with even generators and odd generators. We will recall the definitions of a Poisson structure and its cohomology, in the classical case, as well as in the $\mathbb{Z}_2$-graded case. Finally, we will compare some explicit bases obtained for these cohomology, in order to highlight differences and analogies between the $\mathbb{Z}_2$-graded context and the classical context.

### Past sessions (2016-2017)

• 23.05.2017 -- Room S.6A
Gaëtan Borot (Max Planck Institute for Mathematics), "The ABCD of topological recursion"
Abstract: Following a recent proposal of Kontsevich and Soibelman, we study the quantization of quadratic Lagrangians in $T^*V$ by a procedure of topological recursion. I will describe 3 geometric sources of such (quantized) Lagrangians: Frobenius algebras, non-commutative Frobenius algebras, and spectral curves. In the latter case, Kontsevich-Soibelman topological recursion retrieves Eynard-Orantin topological recursion.

Based on a joint work with Andersen, Chekhov and Orantin.
• 16.05.2017 -- Room S.6A
Yaël Frégier (Université d'Artois), "Harmonic forms and deformations of embedding of Lie algebras"
Abstract: Bertram Kostant has introduced in Lie theory an analog of the Laplacian leading to harmonic forms in Chevalley Eilenberg cohomology. We want, in a joint work with Olivier Elchinger, to use these to determine the universal deformations of the embeddings of maximal nilpotent algebras in semi-simple complex Lie algebras. The aim of this talk is explain the main ingredients involved in this construction.
• 24.04.2017 -- Room S.6A ~14h-15h30~
25.04.2017 -- Room S.6A ~14h00-15h30~
27.04.2017 -- Room S.6A ~14h00-15h30~
Ian Marshall (HSE Moscow), "Integrable systems, with attention to symplectic and Poisson structures and Hamiltonian reduction"
Joint lecture course with the seminars "Algebraic Topology, Geometry and Physics" and "Geometry and Topology"
Abstract: I want to explain the method of Hamiltonian Reduction with examples from the field of Integrable Systems.
1. Integrable Systems in Classical Mechanics : Liouville’s Theorem, and examples
2. Lax systems
3. Symplectic and Poisson structure and symmetric actions of Lie groups
4. Something about the history of the subject
5. Hamiltonian reduction and applications.
• 04.04.2017 -- Room S.6A
Mauro Porta (University of Pennsylvania), "The derived Riemann-Hilbert correspondence"
Abstract: In this talk, I will discuss a generalization of the classical Riemann-Hilbert correspondence between the category of representations of the fundamental group of an algebraic variety $X$ and the category of vector bundles on $X$ equipped with a flat connection. It has been conjectured by C. Simpson that the Riemann-Hilbert correspondence can be extended to an equivalence between the $\infty$-category of representations of the full homotopy type of $X$ with values in the $\infty$-category of perfect complexes and the $\infty$-category of perfect complexes on $X$ equipped with flat connection. The result I will discuss further generalizes this picture, by considering families of representations, where the base of the family is allowed to be a derived analytic space. The talk is based on the preprint arXiv 1703.03907.
• 06.12.2016 -- Room A17
Elba Garcia-Failde (Max Planck Institute for Mathematics), "Nesting statistics in the O(n) loop model on random maps of any topology"
Abstract: In this talk, I will call maps a certain class of graphs embedded on surfaces. We will consider random maps equipped with a statistical physics model: the O(n) loop model. We investigate the nesting properties of loops by associating to every map with a loop configuration a so-called nesting graph, which encodes all interesting nesting information. We characterize the generating series of maps of genus g with k boundaries and k' marked points realizing a fixed nesting graph. These generating series are amenable to explicit computations in the loop model with bending energy on triangulations, and we describe their critical behavior in the dense and dilute phases. This was first studied for maps with the topology of disks and cylinders and we generalize it to arbitrary topologies making use of a procedure called topological recursion, which will be introduced and applied to our analysis.
• 29.11.2016 -- Room B27
Adrien Brochier (University of Hamburg), "Quantum Character varieties"
Abstract: Among the many applications of quantum groups are the constructions by Reshetikhin-Turaev of topological invariants (links invariants and representations of mapping class groups), and the quantization of the Atiyah--Bott Poisson structure on character varieties of surfaces by several authors. In this talk I will present an explicit description of a canonical quantization of the category of sheaves on character varieties, the quantization of the underlying varieties being recovered by taking global section. Those categories form the two dimensional part of a certain partially defined 4 dimensional topological field theory which in turn recover Reshetikhin--Turaev's construction as some sort of boundary condition. This is based on joint works with D. Ben-Zvi, D. Jordan and Noah Snyder.
• 22.11.2016 -- Room Paul Feidert
Nikolai Tyurin (Max Planck Institute for Mathematics), "Special Bohr - Sommerfeld geometry"
Abstract: We combine two methods in lagrangian geometry: Special Lagrangian geometry of N. Hitchin and J. McLean and ALAG programme of A. Tyurin and A. Gorodentsev. As a result for any 1- connected compact symplectic manifold with integer symplectic form we get certain spaces of Special Bohr - Sommerfeld lagrangian submanifolds, fibered over infinte dimensional projective spaces with discrete fibers. Therefore in general these spaces can be endowed with Kahler structure. Moreover, if our given symplectic manifold admits compatible integrable complex structure then we can derive from these spaces certain finite dimensional objects called moduli spaces of Special Bohr - Sommerfeld cycles. As a byproduct we get a construction of "lagrangian shadows" for ample algebraic divisors in compact algebraic varieties.
• 08.11.2016 -- Room B27 ~14h-15h30~
10.11.2016 -- Room B27 ~14h00-15h30~
10.11.2016 -- Room B27 ~15h30-17h00~
Emil Akhmedov (HSE/ITEP Moscow), "On geometry and dynamics of fields in de Sitter and anti de Sitter space-times"
Joint lecture course with the seminar "Algebraic Topology, Geometry and Physics"
Abstract: The lectures will be mostly addressed to graduate students. I am with physics background, hence, the presentation will be without any pretends for the real mathematical rigor. However, I will do my best to keep my presentation as rigorous as my education allows. I will start with the describtion of the geometry of de Sitter and anti de Sitter space-times. My goal will be to present the way we can quickly (with minimal efforts) see most of the properties of these space-times. Then I will continue with the derivation of the free massive scalar modes and their properties in these space-times. I will end up my lectures with the derivation of the Green functions in these space-times in two different ways.
• 18.10.2016 -- Room B27
Marko Zivkovich (University of Luxembourg), "Introduction to Graph Complexes"
Abstract: Graph complexes are some of the most intriguing objects in homological algebra, connected to many other fields of algebra and topology. They are very simple to define, but their cohomology is very hard to compute and largely unknown at present. I introduce various kinds of graph complexes and give some motivations. I also explain some techniques to study them, namely introducing quasi-isomorphisms between them and extra differentials.
• 11.10.2016 -- B27
Oleg K. Sheinman (Steklov Mathematical Institute, Moscow), "Moduli of matrix divisors on Riemann surfaces."
Abstract: Matrix divisors are introduced by A.Weil (1938) and are considered as a chronologically first approach to the theory of holomorphic vector bundles on Riemann surfaces. Their interrelation is roughly the same as the interrelation between conventional divisors and linear bundles. The list of those who investigated holomorphic vector bundles on Riemann surfaces is impressive. Besides already mentioned A.Weil it includes (in the chronological order) the names of Atiyah (1957), Grotendick (1957), Seshadri (1977), Hitchin (1987), Faltings (1993). Nevertheless something new has been discovered recently with relation to Lax operator algebras (in turn, discovered recently with relation to integrable systems). In my talk, first, I am going to give a brief overview of existing directions in the theory of holomorphic vector bundles on Riemann surfaces. Next I shall present the approach by A.Tyurin (1965-66), which is less known but has certain advantages, and turned out to be quite effective in the theory of integrable systems (Tyurin parametrization of integrable systems). Then I'll pass to my main purpose: to develop the Tyurin ideas for the case of G-bundles where G is a complex semi-simple Lie group. I shall describe the moduli of matrix divisors as a coset of a certain group (thus a homogeneous space), and the tangent space at the unit of this homogeneous space in terms of Lax operator algebras.

### Archive

For former programs see here.