Herein the interested web surfer will find a list of some basic background references regarding some of the topics of the conference. This is not a complete bibliographic listing, but if we have missed something, please feel free to let us know by email.
A. Integrable Systems
(1) Survey Articles
1. J. Moser, Various aspects of integrable Hamiltonian systems.
Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978),
pp. 233289, Progr. Math. 8, Birkhauser, Boston, Mass.,1980.
[This is a survey on various aspects underlying the integrability
of a Hamiltonian system such as group representation, symplectic
reduction and isospectral deformation. Several finite dimensional
systems are discussed illustrating the various aspects. The
survey ends with a section on the inverse spectral theory of
the Hill's equation.]
2. H. McKean, Integrable systems and algebraic curves. Global
analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ.
Calgary, Calgary, Alberta, 1978), pp.83200, Lecture Notes
in Math., 755, Springer, 1979. [This is a detailed survey
on the link between some special manybody problems and shallow
water waves to algebraic geometry. The paper begins with the
FermiPastaUlam lattice and explains how it has led Kruskal
and Zabusky to their numerical experiment. A good part of the
paper is devoted to the KdV equation, in particular with regard
to its solution in the periodic case and its connection to the
CalogeroMoser systems. Other topics surveyed include:
(a) a parallel theory for the periodic Toda lattice, (b) Boussinesq
equation on a circle, (c) the Neumann system, (d) grouptheoretic
ideas.
3. P. Deift, Integrable Hamiltonian systems. Dynamical systems
and probabilistic methods in partial differential equations
(Berkeley, CA, 1994), 103138, Lectures in Appl. Math. 31,
Amer. Math. Soc., 1996. [This is a series of 4 lectures which
aims to give a rapid introduction to the subject. The content
includes a survey of the following topics: (a) notion of complete
integrability and the LiouvilleArnold theorem, (b) the Toda lattice
and its generalization, (c) classical rmatrices, (d) inverse
scattering of the defocusing nonlinear Schrodinger (NLS) equation
as a RiemannHilbert problem, (e) long time asymptotics of the
defocusing NLS equation via the nonlinear steepest descent
method.
4. A. Reyman and M. SemenovTianShansky, Grouptheoretic methods
in the theory of finitedimensional integrable systems. Dynamical
systems VIIm Encyclopaedia of Mathematical Sciences, (V.I. Arnold
and S.P. Novikov, eds.), vol. 16, SpringerVerlag, 1994, pp.116
225. [This is a survey of a general grouptheoretic scheme (a.k.a.
classical rmatrix theory) to construct integrable Hamiltonian
systems and their solutions in a systematic way. Many finite
dimensional examples are given to illustrate the general theory.
At the end of each section, careful historical notes are given
to guide the reader to the original literature.]
5. New developments in the theory and application of solitons (a
discussion organized and edited by Sir Michael Atiyah, F.R.S.,
J.D. Gibbon, and G. Wilson), Phil. Trans. R. Soc. Lond. A,
volume 315, pp. 333469 (19850. [This includes contributions
from J.D. Gibbon, J.B. Keller, P. van Moerbeke, I.B. Frenkel,
G. Wilson, N. M. Ercolani and H. Flaschka, A.C. Scott, L.F.
Mollenauer, R.S. Ward, Sir Michael Atiyah, F.R.S. and N.J.
Hitchin, as well as a transcript of an interactive discussion
moderated by J.T. Stuart, F.R.S. and M. Tabor]
6. Important developments in soliton theory, edited by A. Fokas
and V. Zakharov, Springer, 1993. [This book consists of a
collection of survey papers which covers the following aspects
of the subject: (a) method of solution, (b) asymptotic methods,
(c) algebraic aspects, (d) quantum and statistical mechanical
methods, (e) near integrable methods and computational aspects]
7. A. Its, The RiemannHilbert problem and integrable systems, Notices of AMS, 50(11), 13891400, 2003. [This article presents some new developments in the RiemannHilbert formalism related to integrable systems using Painleve equations as a main example.]
(2) Texts
1. P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge
University Press, Cambridge, 1989. [This is a short introductory
text which covers the essence of soliton theory, including:
the discovery of solitary waves, the inverse scattering method for
the KdV equation, the AKNSZS scheme, Hirota's method, Painleve property,
and numerical methods.]
2. A.C. Newell, Solitons in Mathematics and Physics, SIAM, 1985.
[This is an introductory
text on soliton theory that begins with a historical chapter followed by a
chapter concerning the derivation of various soliton equations from physical
models and a chapter about the inversescattering transform and its use
(including some perturbation theory). This book then departs from this
standard material, culminating with two chapters addressing the importance
of symmetries in soliton theory, finally linking together Hirota's theory of
taufunctions with the WahlquistEstabrook method and the theory of loop
groups and vertex algebras.]
3. L. Faddeev and L. Takhtajan, Hamiltonian methods in the
theory of solitons, SpringerVerlag, Berlin, 1987.[This book presents
the Hamiltonian approach to soliton theory. The main point of view
is that the soliton equations are infinite dimensional completely
integrable Hamiltonian systems. The first part of the book covers
the nonlinear Schrodinger equation in detail, including the
formulation of the inverse problem as a RiemannHilbert problem
and the construction of actionangle variables. In the second
part, the authors apply the same ideas to other equations including
: the SineGordon equation, the Toda lattice, the Heisenberg
ferromagnet and the LandauLifschitz equation. The book ends with a
sketchy chapter on the Lietheoretic approach to integrable models.]
(3) Papers
1. P. Lax, Integrals of nonlinear equations of evolution and solitary
waves. Comm. Pure Appl. Math. 21 (1968), 467490 [This is a
very significant paper in which the author introduced the
wellknown "Lax pair" for the KdV equation. Nowadays,
the method of Lax pairs underlies much of the modern theory
of integrable systems]
2. V. Zakharov, L. Faddeev, The Kortewegde Vries equation is a
fully integrable Hamiltonian system, Funct. Anal. Appl. 5,
280287 (1971), translated from Funk. Anal. Priloz 5 (4),
1627 (1971) [In this important paper, the authors showed
that the variables which linearize the KdV equation can be
interpreted as actionangle variables. Thus this establishes
the KdV equation as an infinite dimensional integrable Hamiltonian
system]
3. V. Zakharov, A. Shabat, Exact theory of twodimensional self
focusing and onedimensional selfmodulation of waves in
nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 6269;
translated from \v Z.Eksper. Teoret. Fiz. 61 (1971), no. 1,
118134(Russian)81.35 [The authors showed that the nonlinear
Schrodinger equation also has a Lax pair formulation. Prior
to this work, it was not clear if there exist other equations
besides KdV for which the Lax pair formulation is possible]
4. C. Gardner, J. Greene, M. Kruskal, R. Miura, Kortewegde Vries
equation and generalization VI. Methods for exact solution.
Comm. Pure Appl. Math. 27 (1974), 97133 [In a short note
which appeared in 1967, the authors announced a nonlinear
change of variables under which the dynamics of the KdV
equation becomes linear. In this seminal contribution, the complete
detail of the inverse scattering method was worked out in the
context of the KdV equation. Subsequently, the method was found
to be applicable to other important integrable PDEs and
lattice models]
5. H. Flaschka, The Toda lattice I. Existence of integrals. Phys. Rev. B
(3) 9 (1974), 19241925. [In this paper, the author made the important
change of variable which allowed him to rewrite the equations of motion of
the Toda lattice as an isospectral deformation of a Jacobi matrix. This has
led to many subsequent developments in the area]
6. H. Flaschka, The Toda lattice II. Inverse scattering solution.
Progr. Theoret. Phys. 51 (1974), 703716. [In this work, an inverse
scattering method is developed for the solution of the doublyinfinite
Toda lattice. This is the first time the inverse scattering method
was shown to be applicable to a lattice model]
B. Random Matrices
(1) Survey Articles
1. Random matrix models and their applications, edited by P. Bleher and A. Its, MSRI Publications, 40, Cambridge University Press, 2001. [This proceeding contains several survey papers by both mathematicians and physicists about various aspects of random matrix theory and its applications such as combinatorics, universality, solvable physical models and enumeration of Fyenman graphs.]
2. Journal of Physics A, Volume 36, Number 12, 2003. Special issue: Random Matrix Theory, edited by P. J. Forrester, N. C. Snaith and J. J. M. Verbaarschot. [This special issue contains survey and original papers for many different aspects of random matrix theory. Namely applications to number theory, applications to statistical mechanics, integrable systems, mesoscopic physics and disordered systems, quantum chaos and nonHermtian matrices are discussed. Also contained is an article by the editors reviewing the main historical developments in random matrix theory.]
3. W. Kšnig, Orthogonal polynomial ensembles in probability theory, Probability Surveys, Volume 2, 385447, 2005. [This paper surveys a number of models from physics, statistical mechanics, probability theory and combinatorics, which turned out to be described in terms of random matrix theory. This paper is aimed at nonexperts with background in probability who want a quick survey of the field.]
4. P. Deift, Integrable systems and combinatorial theory, Notices of AMS, 47(6), 631640, 2000. [This paper surveys the connections between increasing subsequences in combinatorics, random matrix theory and integrable systems.]
5. N. Katz and P. Sarnak, Zeroes of zeta functions and symmetry,
Bull. Amer. Math. Soc. (N.S.) 36(1), 126, 1999. [This is an expository paper concerning
the connections between the zeros of zeta functions and random matrix theory.]
(2) Texts
1. M. L. Mehta, "Random Matrices", Academic Press. [This is a classic book on random matrices. The book contains many models and results from a physicist's view point.]
2. P. A. Deift, "Orthogonal Polynomials and Random Matrices", AMS. [This lecture note focuses on the orthogonal polynomial approach for universality question in Hermitian random matrices. A detailed account of RiemannHilbert analysis is presented.]
3. P. Forrester, Loggases and Random matrices, available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html [This book in progress covers more recent topics that are not in the Mehta's book. Also one can find more details of proofs.]
4 N. Katz and P. Sarnak, "Random Matrices, Frobenius Eigenvalues, and Monodromy", AMS Colloquium Publications.
(3) Papers
1. C. A. Tracy and H. Widom, "Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices", J. Stat. Phys., volume 92, pp 809835 (1998). [This paper presents an elementary proof of some of most fundamental algebraic formulas for correlations functions and gap distributions in random matrix theory.]
