Mathematics 224, Fall 2012/Spring 2013

TuTh 9:30am-11:00am , Room 3 Evans

Professor  Alberto Grunbaum
Telephone: (510) 642-5348
Office: 903 Evans Hall
Office hours: Wed 11:30-13:00, Thursday 11:00-12.30

Description of the class

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

Riesz- SzNagy Functional Analysis

Lax Functional Analysis

Teschel Mathematical methods in quantum mchanics

Reed-Simon Functional Analysis

Karlin-Taylor Stochastic processes

Newell Soliton Mathematics