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In a few weeks (and more precisely, starting Friday, September 24), I will begin teaching Math 245A, which is an introductory first year graduate course in real analysis. (A few years ago, I taught the followup courses to this course, 245B and 245C.)  The material will focus primarily on the foundations of measure theory and integration theory, which are used throughout analysis.  In particular, we will cover

1. Abstract theory of $\sigma$-algebras, measure spaces, measures, and integrals;
2. Construction of Lebesgue measure and the Lebesgue integral, and connections with the classical Riemann integral;
3. The fundamental convergence theorems of the Lebesgue integral (which are a large part of the reason why we bother moving from the Riemann integral to the Lebesgue integral in the first place): Fatou’s lemma, monotone convergence theorem, and the dominated convergence theorem;
4. Product measures and the Fubini-Tonelli theorem;
5. The Lebesgue differentiation theoremabsolute continuity, and the fundamental theorem of calculus for the Lebesgue integral.  (The closely related topic of the LebesgueRadon-Nikodym theorem is likely to be deferred to the next quarter.)

See also this preliminary 245B post for a summary of the material to be covered in 245A.

Some of this material will overlap with that seen in an advanced undergraduate real analysis class, and indeed we will be revisiting some of this undergraduate material in this class.  However, the emphasis in this graduate-level class will not only be on the rigorous proofs and on the mathematical intuition, but also on the bigger picture.  For instance, measure theory is not only a suitable foundation for rigorously quantifying concepts such as the area of a two-dimensional body, or the volume of a three-dimensional one, but also for defining the probability of an event, or the portion of a manifold (or even a fractal) that is occupied by a subset, the amount of mass contained inside a domain, and so forth.  Also, there will be more emphasis on the subtleties involved when dealing with such objects as unbounded sets or functions, discontinuities, or sequences of functions that converge in one sense but not another.  Being able to handle these sorts of subtleties correctly is important in many applications of analysis, for instance to partial differential equations in which the functions one is working with are not always a priori guaranteed to be “nice”.

In my previous post, I briefly discussed the work of the four Fields medalists of 2010 (Lindenstrauss, Ngo, Smirnov, and Villani). In this post I will discuss the work of Dan Spielman (winner of the Nevanlinna prize), Yves Meyer (winner of the Gauss prize), and Louis Nirenberg (winner of the Chern medal). Again by chance, the work of all three of the recipients overlaps to some extent with my own areas of expertise, so I will be able to discuss a sample contribution for each of them. Again, my choice of contribution is somewhat idiosyncratic and is not intended to represent the “best” work of each of the awardees.

As is now widely reported, the Fields medals for 2010 have been awarded to Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani. Concurrently, the Nevanlinna prize (for outstanding contributions to mathematical aspects of information science) was awarded to Dan Spielman, the Gauss prize (for outstanding mathematical contributions that have found significant applications outside of mathematics) to Yves Meyer, and the Chern medal (for lifelong achievement in mathematics) to Louis Nirenberg. All of the recipients are of course exceptionally qualified and deserving for these awards; congratulations to all of them. (I should mention that I myself was only very tangentially involved in the awards selection process, and like everyone else, had to wait until the ceremony to find out the winners. I imagine that the work of the prize committees must have been extremely difficult.)

Today, I thought I would mention one result of each of the Fields medalists; by chance, three of the four medalists work in areas reasonably close to my own. (Ngo is rather more distant from my areas of expertise, but I will give it a shot anyway.) This will of course only be a tiny sample of each of their work, and I do not claim to be necessarily describing their “best” achievement, as I only know a portion of the research of each of them, and my selection choice may be somewhat idiosyncratic. (I may discuss the work of Spielman, Meyer, and Nirenberg in a later post.)