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Tau Functions, Correlation Functions and Applications: August 29 – September 3, 2021

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.