Mathematical Physics Laboratory

Centre de recherches mathématiques

Working Seminar on Integrable Systems, Random Matrices, and Random Processes

Spring / printemps 2008

Centre de Recherches Mathématiques
Université de Montréal
Pavillon André Aisenstadt, 2920, ch. de la Tour
Salle/Room 4336


Time/Date/Heure: Thursday, May 8; jeudi le 8 mai 2008, at / à 2:00-3:00 p.m. / 14h00-15h00
Title/titre: "Counting coloured 3-D Young diagrams with vertex operators"
Speaker/Conférencier: Benjamin Young (UBC and CRM)
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

This will be an enumerative combinatorics talk, in which we will use the vertex operator methods of Okounkov-Reshetikhin to compute certain generating functions for plane partitions (or 3D Young Diagrams). These generating functions turn out to be Donaldson-Thomas partition functions for the orbifold C^3/G, where G is the group Z/n or Z/2 x Z/2. In the latter case, the calculation is rather delicate, and we need to make a connection with "pyramid partitions" (height functions of the dimer model on the square lattice).



Lecture series / Preparatory course on Tau functions and their applications

Hiver / Winter 2008

Friday from 1:30 to 3:30 p.m. / Vendredi, de 13h30 à 15h30

Centre de Recherches Mathématiques
Université de Montréal
Pavillon André Aisenstadt, 2920, ch. de la Tour
Salle/Room 4336



Session 1

Time/Date/Heure: Thursday, January 17; jeudi le 17 janvier 2008, at / à 4.00pm / 16h00
Title/titre: "The Sato-Segal-Wilson KP Tau function"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : Concordia University, SGW, Library Building, 9th floor, room LB 912 (Math Help Centre)

Content:
- Examples: Schur functions, more general polynomial, tau functions, theta functions, matrix integrals.
- Applications: integrable systems; random matrices, random processes, combinatorics
- The KP hierarchy; Baker functions; Sato's formula
- Bilinear relations
References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985).
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).

Tuesday, January 22 2008 from 3:30 p.m.to 5:00 p.m.
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Exceptional session: "Introduction to Tau functions and their applications"

The KP hierarchy: examples of KP Tau functions

John Harnad, Concordia and CRM

Continuation of Session 1.

Session 2

Time/Date/Heure: Friday, January 25; vendredi le 25 janvier 2008, from/de 13:30 at / à 15:30
Title/titre: "Geometric definition of KP tau functions : the Sato-Segal-Wilson Grassmanian"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Review of finite dimensional Grassmannians, Plücker coordinates, Plücker relations, Schur functions
- The Segal-Wilson Grassmannian, partitions
- Fermi Fock space (Bose-Fermi correspondence)
- The abelian flow group : the (simplified) Sato definition of the KP tau function
- Interpretation as fermionic expectation values
- The Hirota bilinear equations for KP as Plücker relations
- Plücker coordinates and Schur function expansions

References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985).
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).

Session 3

Time/Date/Heure: Friday, February 1; Vendredi le 1er février 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 2: (Part A) Geometric definition of KP tau functions: the Sato-Segal-Wilson Grassmannian:"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Review of finite dimensional Grassmannians, Plücker coordinates, Plücker relations, Schur functions
- The Segal-Wilson Grassmannian, partitions
- The abelian flow group : the (simplified) Sato definition of the KP tau function

References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985).
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).
- E. Date, M. Jimbo, M. Kashiwara and T. Miwa, "Transformation groups for soliton equations", in Nonlinear Integrable Systems, Proc. RIMS Symposium, pgs. 39 -120, eds. T. Miwa and M. Jimbo, (World Scientific, Singapore 1983)
- M. Sato and Y. Sato, "Soliton Equations as Dynamical Systems on Infinite Grassmann Manifold" in: Nonlinear PDE in Applied Science, U.S.-Japan Seminar. Tokyo. Lecture Notes in Appl. Anal. 5, 259-271 (1982)

Session 4

Time/Date/Heure: Friday, February 8; Vendredi le 8 février 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 2: (Part A, continued) The tau function on finite dimensional Grassmannians"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Finite dimensional Grassmannians, partitions
- The abelian flow group in finite dimensions : the Sato definition of the KP tau function
- Finite Schur function expansions
- The fermionic formulation

References:
- M. Sato and Y. Sato, "Soliton Equations as Dynamical Systems on Infinite Grassmann Manifold" in: Nonlinear PDE in Applied Science, U.S.-Japan Seminar. Tokyo. Lecture Notes in Appl. Anal. 5, 259-271 (1982)
- T. Miwa, M. Jimbo and E. Date, "Solitons, differential equations, symmetries and infinite dimensional algebras", Cambridge Tracts in Mathematics 135 (Cambridge, NY, 2000).

Session 5

Time/Date/Heure: Friday, February 15; Vendredi le 15 février 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 2: (Part B) The Sato-Segal-Wilson Grassmannian:"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- The Segal-Wilson Grassmannian, partitions
- The abelian flow group : the Sato definition of the KP tau function
- The fermionic formulation
- Proof of equivalence between the bilinear equations and the Plücker relations

References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985).
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).
- M. Sato and Y. Sato, "Soliton Equations as Dynamical Systems on Infinite Grassmann Manifold" in: Nonlinear PDE in Applied Science, U.S.-Japan Seminar. Tokyo. Lecture Notes in Appl. Anal. 5, 259-271 (1982)
- T. Miwa, M. Jimbo and E. Date, "Solitons, differential equations, symmetries and infinite dimensional algebras", Cambridge Tracts in Mathematics 135 (Cambridge, NY, 2000).

Session 6

Time/Date/Heure: Friday, February 22; Vendredi le 22 février 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 3: (Part A) Examples of tau functions I : KP tau functions of rational, trigonometric and algebro-geometric type"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Schur functions, general polynomial tau functions
- The Calogero-Moser system: rational solutions of KP hierarchy
- KP solitons
- Solutions associated to algebraic curves

References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985).
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).
- M. Sato and Y. Sato, "Soliton Equations as Dynamical Systems on Infinite Grassmann Manifold" in: Nonlinear PDE in Applied Science, U.S.-Japan Seminar. Tokyo. Lecture Notes in Appl. Anal. 5, 259-271 (1982)
- T. Miwa, M. Jimbo and E. Date, "Solitons, differential equations, symmetries and infinite dimensional algebras", Cambridge Tracts in Mathematics 135 (Cambridge, NY, 2000).

Session 7

Time/Date/Heure: Friday, February 29; Vendredi le 29 février 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 3: (Part B) Examples of tau functions I : Integrable systems"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- KP solitons
- Finite gap solutions : algebraic curves
- Multi-KP generalizations

References:
- G.Segal, G. and G. Wilson, G. "Loop Groups and Equations of KdV Type", Publications Math., I.H.E.S., 61, 5-65,(1985)
- L.A. Dickey "Soliton Equations and Hamiltonian Systems", 2nd ed., Advanced Series in Mathematical Physics, Vol. 26 (World Scientific).
- T. Miwa, M. Jimbo and E. Date, "Solitons, differential equations, symmetries and infinite dimensional algebras", Cambridge Tracts in Mathematics 135 (Cambridge, NY, 2000).

Session 8

Time/Date/Heure: Friday, March 7; Vendredi le 7 mars 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 4: (Part A) Examples of tau functions II : Matrix integrals"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Tau functions associated to algebraic curves (continued) - Hermitian Matrix Models
- Schur function expansions of matrix integrals

References:
- A. Gerasimov A. Marshakov, A. Mironov, A. Morozov and A. Orlov A "Matrix models of two-dimensional gravity and Toda theory" Nucl. Phys. B 357 565-618 (1991)
- M. Bertola, B. Eynard and J. Harnad "Partition functions for matrix models and isomonodromic tau functions", J. Phys. A 35, 3067-3084(2003). Apendix B.
- J. Harnad and A. Yu. Orlov, "Fermionic conctruction of partition functions for two-matrix models and perturbative Schur function expansions", J. Phys. A 39, 8783-8809 (2006)

Session 9

Time/Date/Heure: Friday, March 14; Vendredi le 14 mars 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 4: (Part B) Examples of tau functions II : Matrix integrals"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Kontsevich-Witten integral
- Multi-KP tau functions and multimatrix models

References:
- A. Gerasimov A. Marshakov, A. Mironov, A. Morozov and A. Orlov A "Matrix models of two-dimensional gravity and Toda theory" Nucl. Phys. B 357 565-618 (1991)
- M. Kontsevich, "Intersection Theory on the moduli space of curves and the Matrix Airy Function", Commun. Math. Phys. 147, 1-23 (1992)
- C. Itzykson and J.-B. Zuber, "Combinatorics of the modular group II. The Kontsevich Integral", Int. J. Mod. Phys. A7, 5661-5705 (1992)
- J. Harnad and A. Yu. Orlov, "Fermionic conctruction of partition functions for two-matrix models and perturbative Schur function expansions", J. Phys. A 39, 8783-8809 (2006)

Session 10

Time/Date/Heure: Friday, March 21; Vendredi le 21 mars 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 4: (Part B) Examples of tau functions II : Matrix integrals"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Schur function expansions of matrix integrals
- Kontsevich-Witten integral
- Fermionic constructions

References:
-J. Harnad and Orlov A. Yu. "Scalar products of symmetric functions and matrix integrals" Theor. Math. Phys. 137 1676–90 (2003)
- M. Kontsevich, "Intersection Theory on the moduli space of curves and the Matrix Airy Function", Commun. Math. Phys. 147, 1-23 (1992)
- C. Itzykson and J.-B. Zuber, "Combinatorics of the modular group II. The Kontsevich Integral", Int. J. Mod. Phys. A7, 5661-5705 (1992)
- J. Harnad and A. Yu. Orlov, "Fermionic conctruction of partition functions for two-matrix models and perturbative Schur function expansions", J. Phys. A 39, 8783-8809 (2006)

Session 11

Time/Date/Heure: Friday, March 28; Vendredi le 28 mars 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 4: (Part B) Examples of tau functions III : Fermionic construction of 2-matrix partition functions"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Fermionic formulation of 2-Toda / 2-KP Tau function
- Fermionic construction of 2-matrix model partition function
- Double Schur function expansions

References:
-J. Harnad and Orlov A. Yu. "Scalar products of symmetric functions and matrix integrals" Theor. Math. Phys. 137 1676–90 (2003)
- J. Harnad and A. Yu. Orlov, "Fermionic conctruction of partition functions for two-matrix models and perturbative Schur function expansions", J. Phys. A 39, 8783-8809 (2006)

Session 12

Time/Date/Heure: Friday, April 4; Vendredi le 4 avril 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 5: (Part A) Fermi systems in equilibrium and random processes"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Equilibrium statistical models of fermions
- Generating functions for exclusion processes

References:
- J. Harnad and Orlov A. Yu. "Fermionic construction of tau functions and random processes" arXiv:0704.1157 (2007)
- J. Harnad and A. Yu. Orlov "Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy", arXiv:0704.1145 (2007)

Session 13

Time/Date/Heure: Friday, April 11; Vendredi le 11 avril 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 5: (Part A & B continued) Fermi systems in equilibrium and random processes"
Speaker/Conférencier: John Harnad, CRM and Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Generating functions for exclusion processes (asymptotic distributions)
- Equilibrium statistical models of fermions

References:
- J. Harnad and Orlov A. Yu. "Fermionic construction of tau functions and random processes" arXiv:0704.1157 (2007)
- J. Harnad and A. Yu. Orlov "Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy", arXiv:0704.1145 (2007)

Session 14

Time/Date/Heure: Friday, April 18; Vendredi le 18 avril 2008, from/de 13h30 at / à 15h30
Title/titre: "Dispersionless 2-Toda hierarchy, Normal matrix models, tau function of an analytic curve"
Speaker/Conférencier: Ferenc Balogh, Concordia
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
- Cauchy transform of a planar domain, Schwarz function of an analytic curve, harmonic moments
- Moving boundary problem in two dimensions (Laplacian growth), Richardson's theorem
- Connection to normal matrix models
- Dispersionless two-dimensional Toda hierarchy
- Faber polynomials and Grunsky coefficients, Lax-Sato equations for conformal maps
- Tau function of an analytic curve

References:
- P. Wiegmann, A. Zabrodin, Conformal Maps and Integrable Hierar- chies, Comm. Math. Phys. 213, 523-538 (2000)
- I. Kostov, I. Krichever, M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, The τ - function for analytic curves, Random matrix mod- els and their applications, 285-299, Math. Sci. Res. Inst. Publ., 40, Cam- bridge Univ. Press, Cambridge, 2001.
- K. Takasaki, T. Takebe, Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), no. 5, 743-808.
- J. Harnad, I. Loutsenko, O. Yermolayeva, Constrained reductions of two-dimensional dispersionless Toda hierarchy, Hamiltonian structure, and interface dynamics, J. Math. Phys. 46 (2005), no. 11, 112701, 19 pp.
- L-P. Teo, Analytic functions and integrable hierarchies-characterization of tau functions, Lett. Math. Phys. 64 (2003), no. 1, 75–92.
- A. N. Varchenko, P. I. Etingof, Why the boundary of a Round Drop Becomes a Curve of Order Four, AMS, 1992

Session 15

Time/Date/Heure: Friday, April 25; Vendredi le 25 avril 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 7 : (Part A) Isomonodromic tau functions and various relations to matrix models I."
Speaker/Conférencier: Marco Bertola, Concordia and CRM
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
We will define the notion of isomonodromic tau function according to the Japanese school and show their connections to the partition function of various types of matrix models and/or Toeplitz/Haenkel determinants.
- Isomonodromic deformations
- Definition of isomonodromic tau functions; residue formulae
- Spectral curve; isomonodromic invariants as residues

References:
- M. Jimbo, , T. Miwa, and K. Ueno, , "Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coeefficients I." Physica 2D, 306-352 (1981).
- M. Jimbo and T. Miwa, ``Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coeefficients II, III.", Physica 2D, 407-448 (1981); ibid., 4D, 26-46 (1981).

Session 16

Time/Date/Heure: Friday, May 2; Vendredi le 2 mai 2008, from/de 13h30 at / à 15h30
Title/titre: "Chapter 7 : (Part B) Isomonodromic tau functions and various relations to matrix models II."
Speaker/Conférencier: Marco Bertola, Concordia and CRM
Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Content:
We will define the notion of isomonodromic tau function according to the Japanese school and show their connections to the partition function of various types of matrix models and/or Toeplitz/Haenkel determinants.
- Isomonodromic deformations
- Definition of isomonodromic tau functions; residue formulae
- Spectral curve; isomonodromic invariants as residues

References:
- M. Jimbo, , T. Miwa, and K. Ueno, , "Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coeefficients I." Physica 2D, 306-352 (1981).
- M. Jimbo and T. Miwa, ``Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coeefficients II, III.", Physica 2D, 407-448 (1981); ibid., 4D, 26-46 (1981).





Working seminar on Integrable Systems, Random Matrices, Random Processes: Fall 2007

For further information: bertola@CRM.UMontreal.CA or anguelov@CRM.UMontreal.CA