Elenco seminari del ciclo di seminari
“WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY”
Lezioni e seminari dei professori invitati alla Winter School che si terrà presso il Dipartimento di Matematica dal 13 al 17 gennaio 2020
The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.
In this talk I shall focus on systems of nonlinear Ordinary Di§erential Equations, and introduce the notion of their solvability by algebraic operations: implying that their general solution, considered as a function of complex time,
feature at most a Önite number of rational branch points, or equivalently deÖne
a Riemann surface with a Önite number of sheets. Some properties of these
systems shall be reviewed, including the subclasses of them featuring such remarkable properties as isochrony or asymptotic isochrony (as functions of real
time). Techniques to identify such systems shall be reviewed, and several examples reported, including new classes of such systems.
References: F. Calogero, Isochronous Systems, Oxford University Press,
2008 (264 pages, paperback 2012); Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge University Press, 2018 (168 pages). F.
Calogero and F. Payandeh, ìPolynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactionsî, J.
Math. Phys. 60, 082701 (2019). F. Calogero, R. Conte and F. Leyvraz, "New
solvable systems of two autonomous Örst-order ordinary di§erential equations
with purely quadratic right-hand sides" (in preparation).
The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.
The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.
The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of
Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects.
A preliminary layout includes
- Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation)
Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements)
The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions)
Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)
The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of
Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects.
A preliminary layout includes
- Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation)
Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements)
The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions)
Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)
The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of
Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects.
A preliminary layout includes
- Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation)
Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements)
The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions)
Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)