1. Arithmetic progressions of length three. (March 27, 4PM)
A famous theorem of Endre Szemer\'edi asserts that for every $\delta>0$ and every positive integer $k$ there exists $N$ such that every subset of $\{1,2,\dots,N\}$ of size at least $\delta N$ contains an arithmetic progression of length $k$. The special case when $k=3$ was first obtained by Roth. I shall explain his method and show why it breaks down when $k$ is greater than 3.

2. Arithmetic progressions of length four. (March 29, 2:40PM)
The aim of this talk will be to show that Roth's method can, after all, be generalized to deal with progressions of length four and above. I shall give an outline of the entire proof for progressions of length four, discuss some of the steps in detail, and indicate the extra difficulties that must be overcome for progressions of length greater than four. The advantage of this approach over previous arguments of Szemer\'edi and Furstenberg is that it gives far better information about bounds.

3. What can be said about the existence of operators on an arbitrary Banach space? (March 29, 4PM)
Let $X$ be a (separable, infinite-dimensional Banach space). Many of the most interesting open problems in Banach space theory had one of the following two forms. (a) Does $X$ necessarily contain a `nice' subspace (meaning one with more structure than a general Banach space has already)? (b) Can anything non-trivial be said about the space $L(X)$ of bounded linear operators on $X$? Between 1990 and 1995 it was shown that the answers to almost all questions of this kind are no. Surprisingly, however, this phenomenon is intimately related to a {\it positive} solution of a problem of Banach.

All lectures to be held in Tuttleman 103

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