Abstract:
Integrable hierarchies of partial differential equations arose as a
tool describing behavior of waves of
special form. However, their solutions appeared to include formal ones
that are generating functions
for natural enumerating problems. According to Sato's construction,
such solutions can be expressed
in terms of Young diagrams and Schur polynomials.
A spectacular example of such a solution is the
Witten–Kontsevich potential, which generates certain geometric
characteristics of moduli spaces
of complex curves. For solutions of this kind, the equations of the
hierarchy can be treated as
recurrence relations allowing for efficient computations of the
coefficients of the formal power
series expansions.
It will be explained how to construct solutions to the
Kadomtsev–Petviashvili
hierarchy by means of Schur polynomials, and examples will be given,
including those found during
the last years, of important enumeration problems leading to these
solutions.
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