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{\bf MATH 663 }
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{\bf Topics in Applied Mathematics }
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{{\bf Instructor:} \ Michael Gekhtman}
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A fascinating aspect of the theory of integrable (soliton)
ODEs and PDEs is that
they describe a wide variety of important models in physics,
biology and engineering, while their conceptual study requires
application of the most advanced tools of pure mathematics.
In turn, integrable models originally used in applied mathematics,
prove to be relevant and useful in differential and algebraic geometry,
representation theory, etc.
This course will be centered on one of the most important examples of
these interrelations,
the celebrated Kadomtsev-Petviashvili (KP) equation, which can be reduced
to many other integrable equations. We'll discuss
alternative approaches
to this equation, based on the algebra of pseudo-differential operators
on the one hand and loop groups and infinite-dimensional Grassmanian
on the other. The ultimate goal of the course is to present
one of the most striking applications of soliton equations, Mulase's
proof of Novikov's conjecture: The KP equation, discovered in 1970
in plasma physics, can be used to give an answer to the one hundred years
old Schottky problem of singling out Jacobians of compact complex
curves among all complex tori.
All background material that falls outside the scope of basic
courses in analysis, ODEs and linear algebra, will be covered
in class.
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