[Fall 2020]
Prerequisites
* Elementary combinatorics (graph theory);
* Group theory (group actions and quotients, Lie groups and algebras);
* Differential geometry (tensor calculus, supermanifolds);
* Poisson geometry (Hamiltonian mechanics, symplectic geometry);
* Basic cohomology theory (differentials and complexes).
Having the respective MasterMath courses already taken (Lie Groups, Lie Algebras, Poisson Geometry, Symplectic Geometry, Differential Geometry), by itself ideal, is not required.
Sufficient basics can be obtained from:
- J.Lee (2013) "Introduction to smooth manifolds" (2nd ed.), Graduate Texts in Mathematics 218, Springer, NY. (Chapters 1-5, 7-12, 14-17, 19, 21)
- F.W.Warner (1983) "Foundations of differentiable manifolds and Lie groups", Graduate Texts in Mathematics 94, Springer-Verlag, NY-Berlin. (Chapter 3, or any other introductory book on smooth manifolds)
Aim of the course
The course is designed equally for mathematicians and theoretical or mathematical physicists.
The course is aimed to introduce the use of combinatorial structures, primarily graphs, in Poisson geometry and in deformation theory of associative structures (as part of noncommutative geometry), and to acquaint students with the Feynman path integral technique as it appears in the deformation quantization context.
Part 1 of the course is focused on the structures of differential graded Lie algebra (dgLa) on spaces of unoriented graphs, as well as on the graph complexes, L-infinity structures, and graph orientation morphism that takes graph cocycles to symmetries of Poisson brackets on arbitrary affine manifolds.
Part 2 describes Kontsevich's construction of the associativity-preserving deformation quantisation for the commutative product of functions on affine Poisson manifolds. Using the hyperbolic plane geometry, the construction of star-product is expressed in terms of oriented graphs whose weights are intrinsically related to the (multiple-) zeta values. The task is to study the components of the L-infinity morphism which takes Poisson bivectors to associative noncommutative star-products.
Part 3 is aimed to put the construction of star-product in the context of Feynman path integral: its heuristic idea is described and practical calculations then show how, in the Ikeda-Izawa Poisson sigma-model, writing the star-product formula in terms of oriented graphs amounts to a perturbative expansion of correlation functions which are defined over a model of hyperbolic plane in the disk.
Mid-way (in early November 2020), the calculus of (un)oriented graphs and the algebra of star-products encoded by weighted oriented graphs will be illustrated by using the available software packages.
At the end of the course, at least one class will be devoted to open problems in the theory of graph complexes (e.g., Willwacher's relation of graph cocycles to the generators of Grothendieck--Teichmueller Lie algebra), or about symmetries of Poisson brackets and Poisson cohomology, and number-theoretic properties of the graph weights in the star-products.
Lecturers
A.V. Kiselev (RuG),
prof. W. Zudilin (RU Nijmegen).