Let me first tell you what 620-629 is
not
about.
Integrable Models (in the sense discussed in 620-629) have
very little to do with modeling physical phenomena (in
the sense of courses on continuum mechanics, for example). The
subject may have started historically that way (by describing
concrete physical phenomena, such as solitary waves in shallow
water channels) but since then the emphasis has steadily drifted
away from these historical origins.
Furthermore, the integrable nonlinear PDE's that are studied in
620-629 are
so very special that the lessons that
we learn from solving them
do not extend in any useful
way to more general nonlinear PDE's. The latter definitely require
perturbative methods of different types (regular and singular),
which are the subject of entirely different studies.
Now, let me try to give you a rough idea of what the course
is about.
As mentioned above, it is a fact of life that almost all PDE's
cannot be solved exactly. Definitely, almost all
nonlinear PDE's cannot be solved exactly. Like it or not,
perturbative techniques remain the one and only way to say something
meaningful about general PDE's.
However, there are some very, very special classes of nonlinear PDE's
that, amazingly enough, can be solved exactly. These nonlinear PDE's
are extremely special and not at all representative of general PDE's
that one meets in applied mathematics, mathematical physics, modeling
physical phenomena,
etc. So, why should we study them?
There are two diagonally opposite answers to this question:
1. One can say that these PDE's and the phenomena that they describe
are too special to be worth studying, and that we should turn our
attention to general PDE's, learn all that we can about perturbative
methods,
etc.
2. One can also say that these PDE's and the phenomena that they
describe are so special that we definitely need to learn as much
as we can about them.
While the first answer is a perfectly valid one from the viewpoint
of an honest applied mathematician, in 620-629, we adopt the second.
There are two reasons for that.
The first is that, once we decide to fully understand how to solve
these very special nonlinear PDE's, a
very rich panorama
of mathematics opens up to us: Representation theory, complex
analysis, algebraic combinatorics, algebraic geometry, symplectic
geometry, and more, all come together as tools that are necessary
to fully understand what is going on.
My point is that the mathematical experience that we gain from
a serious study of
Integrable Models makes this study
very well worth it.
The second reason is that, it is a fact that the very basic ideas
of Classical (and Quantum) Integrable Models, such as tau functions,
vertex operators, infinite dimensional hierarchies,
etc,
etc, have recently (and totally unexpectedly) appeared
in modern mathematical physics (Seiberg-Witten theory, low dimensional
string theory, topological string theory,
etc) and in pure
mathematics (the study moduli spaces of Riemann surfaces, Geometric
Langlands,
etc).
To mathematical physicists (of the more mathematical type), these
are very interesting developments with intriguing inter-connections
that beg to be fully understood.
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