This year's Marker Lectures will be given by Timothy Gowers.  (The
abstracts are listed below.)  The lectures will be in 109 Osmond,
Monday-Thursday March 11-14.  The first lecture is Monday at 8:00 PM,
and the following ones are Tuesday, Wednesday and Thursday at 4:30 PM.
Host: Bryna Kra

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Monday (the one for non-mathematicians):

Title: Can Mathematics be Taught?

Summary: It is obvious that, in one sense, mathematics
can be taught - why else would there be mathematics
teachers and lecturers all round the world? However,
there is a widespread perception that mathematical
{\it aptitude} is something one either has or does not
have, and that if you cannot do mathematics' then
no amount of teaching will help. A closely related
view is that there is something fundamentally mysterious
about the process of solving a mathematical problem.

In this talk I shall defend a very different position:
there is nothing mysterious about doing mathematics,
the subject is open to anybody who is reasonably
intelligent and prepared to work hard, and, more
contentiously, most of what mathematicians currently
do could in the future be done by computers without
the need for a major breakthrough in artificial
intelligence.

The other three talks.

Title: Arithmetic progressions of length three.

Summary: A famous theorem of Szemer\'edi asserts that for
every $\delta>0$ and every positive integer $k$ there exists
$N$ such that every subset of $\{1,2,...,N\}$ with at least
$\d N$ elements contains an arithmetic progression of length
$k$. The first non-trivial case of this theorem, when $k=3$,
was proved by Roth. I shall discuss Roth's proof as well as
the background to the problem.

Title: Arithmetic progressions of length four.

Summary: I shall demonstrate, by means of an example, that
Roth's method cannot be directly generalized to deal with
progressions of length four or more. However, there is a
less obvious generalization that works, though it needs
two important extra ingredients: Weyl's inequality and
an interesting theorem of Freiman on the structure of
sets of integers with small sumsets.

Title: What can be said about a general Banach space?

Summary: The Banach spaces in this talk will be separable
and infinite-dimensional. I shall talk about several
discoveries in the 1990s which showed that there are
spaces with very little structure indeed. This statement
can be made precise: it turns out that there are spaces
$X$ such that every continuous linear operator on $X$
is a small perturbation of a multiple of the identity,
in a useful sense of the word small'. This discovery
led to negative solutions of many open problems, and,
indirectly, to a characterization of Hilbert spaces
conjectured by Banach.