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Tau Functions, Correlation Functions and Applications: August 30 – September 3, 2021

Participant List schedule

Organized by: Marco Bertola, John Harnad, Jacques Hurtubise, Alexander Its and Dmitry Korotkin  

 Tau functions are key ingredients in the modern theory of integrable systems. In the classical  dynamical systems setting, they appear as generating functions for solutions of integrable hierarchies, such as the KP or Toda hierarchy, which are realized as isospectral deformation families of linear differential or pseudodifferential operators. They also appear as generators of solutions to finite deformation systems of linear differential operators that preserve their monodromy around singular points. In many cases, they arise as partition functions or correlators for various quantum and statistical systems, and have representations as fermionic or bosonic vacuum expectation values. They may also serve as generating functions, in the combinatorial sense, for various geometrical or topological enumerative invariants. A common feature is that they are interpretable as regularized determinants of a deformation class of Fredholm operators.

Applications of tau functions range through a variety of domains of physics and mathematics, including: the spectral theory of random matrices, the generation of enumerative invariants relating to Riemann surfaces and their discretization, quantum spin and lattice models, integrable random point processes, lattice recursions, tilings and many others.  

The principal tool in the analysis of such systems and their asymptotics is the inverse spectral method, based on the matrix Riemann-Hilbert problem. Recent developments have linked their determination to other currently active areas of mathematical physics, such as: 2D conformal field theory and supersymmetric Yang-Mills systems, leading to dramatic breakthroughs in our understanding.

This workshop will bring together many of the world’s leading experts in the subject, both to report on their latest results and compare the various approaches used. It is expected that this will help advance our understanding of some of the outstanding open questions remaining in the field, such as:

1) Determining fully the relation between tau functions of isospectral and isomonodromic type.

2) Relating the Grassmannian interpretation of continuous integrable hierarchies to discrete integrable systems defined by lattice recursions.

3) Finding a satisfactory explanation of the mysterious double role played by tau functions, linking classical with quantum integrable systems.

4) Clarifying the role of tau functions in the “cluster algebra” structure underlying many families of discrete integrable systems.

Talk Schedule

Time Title Speaker
8:45am Introduction Remarks Marco Bertola, Sasha Abanov
9:00am KP integrability of triple Hodge integrals Alexander Alexandrov
10:00am Generalized Cluster Structures And Periodic Difference Operators Michael Gekhtman
11:00am On a class of KdV tau functions related to integrable probability Mattia Cafasso
12:00pm Lunch Break N/A
1:00pm Triangular Ice: Combinatorics and Limit Shapes P. Di Francesco
2:00pm Limit shapes in large tensor products Nicolai Reshetikhin
Time Title Speaker
9:00am Bilinear expansions of lattices of KP Tau-function in BKP Tau-functions: a fermionic approach John Harnad
10:00am Spectral curves for KP tau functions of hypergeometric type Sergey Shadrin
11:00am Reductions of KP type hierarchies, related to conjugacy classes of the Weyl group of classical Lie algebras

Johan van de Leur

12:00pm Lunch Break N/A
1:00pm Planar directed networks and quantum loop equations Leonid Chekhov
2:00pm Duality in Macdonald theory and the q-Whittaker limit of spherical DAHAs Rinat Kedem
Time Title Speaker
9:00am Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk

Simonetta Abenda

10:00am Correlation functions for unitary invariant ensembles and Hurwitz numbers.

Tamara Grava

11:00am Isomonodromic deformations: Confluence, Reduction and Quan- tization Marta Mazzocco
12:00pm Lunch Break N/A
1:00pm Multinomial models Richard Kenyon
2:00pm Diagrams, nonabelian Hodge spaces and global Lie theory Philip Boalch
Time Title Speaker
9:00am On the fermonic basis and its classical meaning for integrable models Smirnov Fedor
10:00am Noncommutative cluster integrability (joint w/ N.Ovenhouse and S.Arthamonov) Michael Shapiro
11:00am Beyond tau-functions, or how to tame anyons out of free fermions, with a little bit of blowups and confinement Nikita Nekrasov
12:00pm Lunch Break N/A
1:00pm A soliton interacting with a regular gas of solitons Ken McLaughlin
2:00pm Logarithmic Painlevé functions and Mathieu stability chart Oleg Lisovyi
Time Title Speaker
9:00am On tau-functions for the KdV hierarchy Di Yang
10:00am Quantization of Monodromy data for confluent Painlevé systems, Calabi-Yau 3-algebras and degenerated non-commutative del Pezzo surfaces Vladimir Roubtsov
11:00am Topological recursion/intersection numbers correspondences Gaetan Borot