Noncommutative Geometry meets Topological Recursion (20w5127)

Organizers

(University of Western Ontario)

(Max Planck Institute for Mathematics)

Hannah Markwig (University of Tuebingen)

(Universitat Munster)

Description

The Institute of Advanced Study in Mathematics will host the "Noncommutative Geometry meets Topological Recursion" workshop in Hangzhou, China from September 27 to October 02, 2020.


This workshop intends to be a first meeting point for specialists and young researchers active in non-commutative geometry, free probability, and topological recursion. In the two first areas, one often wants to compute expectation values of a large class of non-commutative observables in random ensembles of (several) matrices of size N, in the large N limit. The motivations come from the study of various models of 2d quantum gravity via random spectral triples, or from the problem of identifying of interesting factors via approximations by matrix models. Topological recursion and its generalizations provide a priori universal recipes to make and compactly organize such computations, not only for the leading order in N, but also to all orders of expansion in 1/N.in such a way that bridges to other domains where topological recursion has been applied (like enumerative geometry, tropical geometry, mirror symmetry, topological and more generally low-dimensional quantum field theories) become clear.

Concretely, the last 10 years have witnessed the developement of analytic techniques to establish the existence of large N asymptotic expansions, of applications of the topological recursion to a growing class of matrix models which now include some of direct interest in the study of random spectral triples and in non-commutative probability, and of connections between the combinatorics of free probability (i.e. higher order free cumulants) and the topological recursion together with symplectic transformations acting on it. The workshop will explore the consequences of these remarkable algebraic structures axiomatized in topological recursion, which also has a counterpart for non-commutative observables, to address problems in non-commutative geometry and free probability. Knowledge will also flow in the other direction, as the very nature of topological recursion hints at connections to (non-commutative) algebraic geometry, to Hopf algebraic structures and Connes-Kreimer renormalization which deserve exploration.

The mathematical models and phenomena under consideration are common to all these fields, and we wish to unite the strength of probabilistic/asymptotic, algebraic/geometric and combinatorial approaches for the benefit of all the communities involved. This interaction should in fine lead to a better geometric understanding, more powerful computational tools, and new results.


The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).