This is a year-long graduate class on a set of topics intended for an
audience of mathematics, physics, chemistry and other physical
sciences students. It combines material that has in the recent past
been covered in Math 203, Math 220 and a one semester 224.
The first semester will give an infinite dimensional view of linear algebra,
also known as Functional Analyisis. We will replace a finite dimensional
space by a Hilbert space, look at the issue of solving Ax=b in this
setup and then go all the way to proving the spectral theorem
for selfadjoint as well as for unitary operators.
The approach will be
inclusive: I will try to accomodate people that care mostly about proofs as
well as people that want to see concrete illustrations by means of physical
problems of what is being proved. I will also keep numerical implementations
of the results in mind trying to illustrate how concrete problems and their
numerical aspects have motivated the development of the theory.
The first semester will start with some general theory aimed at giving
a solid foundation that the rest of the course. We will then move into
the spectral study of second
order differential operators as well as their discrete analogs. A theme that
we will revisit over and over is the importance of boundary conditions
in getting a precise formulation and solution of the physical problem at
hand. Many of the general notions of the first part of this semester become
quite concrete at this point.
Maybe we will look at problems where periodicity is
important, such as Galileo's pendulum or wave
propagation in lattices. If we do this we will learn (or recall) some beatiful pieces
of elliptic functions theory due to Jacobi and Weierstrass.
In the second semester I will illustrate how the theory is used as a
way of looking at stochatics processes, including classical as well as
quantum walks, diffusion processes such as Brownian motion, the Orstein-
Uhlenbeck process, etc. Throught this course I will try to be fairly
concrete and to apply what we learn in specific instances: a good part of
the theory comes historically from some of these examples. We will deal
with things like the Bessel equation, the Gauss hypergeometric equation and
some of its special cases.
With these examples at hand we will
discuss issues such as asymptotic
expansions, Stokes phenomena, monodromy problems, etc. One of the treats
will be a presentation of R. Feyman's approach to solving partial
differential equations by means of integration in path space. Another such
treat will be to look into M. Kac's version of the question " Can you hear
the shape of a drum ?".
If time ( and the interest of the audience) allows it I would like to
touch on the deep connections between integrable nonlinear equations
such as Korteweg-deVries, Toda, nonlinear Schroedinger and others
and the subject of inverse spectral theory. This is a great story that
illustrates once more the fact that good solid mathematical
tools never die, they just re-emerge somewhere else and keep the whole enterpise alive.
In summary, I will try to cover a large body of work which has proved time
and again to be an extremely useful toolkit in the physical sciences.
Books being considered
Michael Vaughan, Wiley, Introduction to Mathematical Physics
Akhiezer-Glazman Theory of operators in Hilbert space