# 2005 Chern Lectures

*The 2005 Chern Lectures will be delivered by Terence Tao on November 7, 9, 10, and 17, 2005.*

Department of Mathematics, University of California, Berkeley, presents

The 2005 Chern Lectures

Terence Tao

Professor of Mathematics

University of California, Los Angeles

**Talk 1: Global regularity of wave maps**

*November 7, 60 Evans, 4:00-5:00pm*

Wave maps are a natural time-dependent analogue of harmonic maps, which
have been studied extensively by many analysts, topologists, and
geometers (including Chern!). They also appear in general relativity.
We shall describe some recent progress in the global regularity theory
for such wave maps, which combines state-of-the-art harmonic analysis
methods with geometric renormalization techniques to make the non-linear
component of the wave map equation analytically tractable. One
interesting feature in the large energy, negatively curved case is that
the heat flow for the harmonic map equation must be used to renormalize
the wave map equation.

**Talk 2: Global solutions of the Maxwell-Klein-Gordon and Yang-Mills equations**

*November 9, 10 Evans, 4:00-5:00pm*

The Maxwell-Klein-Gordon and Yang-Mills equations are the basic
geometric equations for evolutions of connections in spacetime. Our
understanding of their global behaviour (and in particular in their
possible asymptotics or singularity formulation) lags behind that of the
wave maps equation and simpler models such as the non-linear wave
equation. Nevertheless, there has been some recent progress by Igor
Rodnianksi and myself (for Maxwell-Klein-Gordon) and Jacob Sterbenz and
Joachim Krieger (for Yang-Mills) in obtaining small data smooth
solutions in six and higher dimensions; work is now in progress
concerning the critical case of obtaining small energy smooth solutions
in four dimensions. A key new tool is the introduction of covariant,
scale-invariant Strichartz estimates which take into account the
structure of the connection; this is combined with existing techniques
such as frequency decomposition and the Coulomb gauge.

**Talk 3: Higher-dimensional singularity removal for the Yang-Mills equation**

*November 10, 60 Evans, 4:00-5:00pm*

Yang-Mills connections play an important role in the geometry of
bundles, and also in integrable systems. In four and higher dimensions,
singularities can form in these connections, even if the energy is
finite and even after using gauge transformations to remove artificial
co-ordinate singularities. In the critical four-dimensional case, a
fundamental theorem of Uhlenbeck shows that only a finite number of
isolated singularities (instantons) can occur, with each singularity
absorbing a certain minimum amount of energy. In particular if the
energy is sufficiently small then no singularities form. We shall
discuss this result and the extension (by Gang Tian and myself) to
higher dimensions, in which a monotonicity formula of Price and an
iterative gauge averaging and gluing argument is used to establish
regularity when the energy is small, and controls the curvature density
on the singular set when the energy is large. In contrast with the
time-dependent theory, the theory here is substantially more difficult
in high dimensions than in the critical four-dimensional case.

**Talk 4: Long arithmetic progressions in the primes**

*November 17, 60 Evans, 4:00-5:00pm*

A famous and difficult theorem of Szemeredi asserts that every subset of
the integers of positive density will contain arbitrarily long
arithmetic progressions; this theorem has had four different proofs
(graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic),
each of which has been enormously influential, important, and deep. It
had been conjectured for some time that the same result held for the
primes (which of course have zero density). I shall discuss recent work
with Ben Green obtaining this conjecture, by viewing the primes as a
subset of the almost primes (numbers with few prime factors) of positive
*relative* density. The point is that the almost primes are much
easier to control than the primes themselves, thanks to sieve theory
techniques such as the recent work of Goldston and Yildirim. To
"transfer" Szemeredi's theorem to this relative setting requires that
one borrow techniques from all four known proofs of Szemeredi's theorem,
and especially from the ergodic theory proof.

**Reception in 1015 Evans Hall following the November 7 lecture.**