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\begin{document}
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{\bf \LARGE  Some analysis at UCLA}

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Terence Tao (UCLA)\\

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\noindent

UCLA has a diverse analysis and PDE group, including

\bi

\item Greg Eskin (Inverse problems, scattering theory)

\item John Garnett (Complex analysis, harmonic analysis)

\item Michael Hitrik (Semiclassical analysis, Schr\"'odinger equations)

\item Rowan Killip (Spectral theory, random matrices, integrable systems)

\item Jim Ralston (Wave equations, scattering theory)

\item Terry Tao (Harmonic analysis, combinatorics, nonlinear PDE, analytic number theory)

\item Christoph Thiele (Fourier analysis, nonlinear analysis, multilinear integrals)

\item Several postdocs and visitors each year.

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In addition, there is a large number of graduate students actively participating in the analysis
group, for instance through our participating seminar.  

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\noindent

\bi

\item Analysis is connected to several other fields of mathematics.  I'll give just one example, connecting
Fourier analysis and ergodic theory to number theory.

\item {\bf Theorem} (Green, T., 2004) For any $k$, the prime numbers contain infinitely many arithmetic progressions of length $k$.

\item This was previously only established for $k=3$:

\item {\bf Theorem} (van der Corput, 1939) The prime numbers contain infinitely many arithmetic progressions of length 3.

\item The proof of van der Corput's theorem involves harmonic analysis.  Extremely rapid sketch:

\item Let $\Lambda: \Z^+ \to \R$ be the function such that $\Lambda(n) = \log p$ whenever $n$ is a power of a prime $p$, and $\Lambda(n) = 0$
otherwise.  This is the \emph{von Mangoldt function}.  It usefully captures the \emph{unique factorization property} of the natural numbers
via the identity
$$ \log n = \sum_{d \in \Z^+: d \hbox{ divides } n} \Lambda(d).$$
Also, it is more or less supported on the primes (and also the powers of primes, but those are a much sparser set).

\item To prove van der Corput's theorem, it suffices to show that the quantity
$$ \sum_{a=1}^N \sum_{r=1}^N \Lambda(a) \Lambda(a+r) \Lambda(a+2r)$$
grows very quickly with $N$ (in fact it grows like $N^2 / \log^3 N$).

\item We fix a large integer $N$, and identify $\{1,\ldots,N\}$ with the cyclic group $\Z/N\Z$.  Then the above sum is essentially
$$ \sum_{a \in \Z/N\Z} \sum_{r \in \Z/N\Z} \Lambda_N(a) \Lambda_N(a+r) \Lambda_N(a+2r)$$
where $\Lambda_N: \Z/N\Z \to \R$ is some truncated version of $\Lambda$ (the exact details are omitted here).

\item If one then utilizes the Fourier transform
$$ \Lambda_N(a) = \sum_{\xi \in \Z/N\Z} \hat \Lambda_N(\xi) e^{2\pi i a\xi/N};$$
$$ \hat \Lambda_N(\xi) := \frac{1}{N} \sum_{a \in \Z/N\Z} \Lambda_N(a) e^{-2\pi i a \xi/N}$$
then the previous expression can be written (after some computation) as
$$ \sum_{\xi \in \Z/N\Z} \hat \Lambda_N(\xi)^2 \hat \Lambda_N(-2\xi).$$
The proof then proceeds by computing $\hat \Lambda_N(\xi)$ very precisely, taking advantage of identities such as the unique factorization identity.

\ei

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Longer progressions

\bi

\item van der Corput's argument does not extend easily to longer progressions, even to those of length 4.  In that case one wants to compute
something like
$$ \sum_{a \in \Z/N\Z} \sum_{r \in \Z/N\Z} \Lambda_N(a) \Lambda_N(a+r) \Lambda_N(a+2r) \Lambda_N(a + 3r)$$
but if one now tries to use the Fourier transform, one gets a mess.  

\item These types of multilinear expansions have been well studied both in harmonic analysis and in ergodic theory (the study of averages).
A very similar expression in ergodic theory is the \emph{Furstenberg average}
$$ \lim_{N \to \infty} \sum_{r=1}^N  \int_X f(a) T^r f(a) T^{2r} f(a) T^{3r} f(a)\ d\mu(a)$$
where $(X,d\mu)$ is some compact space and $T: X \to X$ is a measure-preserving transformation on $X$.  This limit was successfully understood using
methods from measure theory, and standard inequalities in analysis (e.g. the Cauchy-Schwarz inequality), as long as $f$ is bounded.  As one conclusion one obtains

\item {\bf Szemer\'edi's theorem} Every subset of the integers with positive density contains arbitrarily long arithmetic progressions.

\item This theorem does not apply to the primes, which have density zero; equivalently, the problem is that the von Mangoldt function $\Lambda$
is unbounded.

\item To resolve this problem, one used a stopping time argument (inspired by similar arguments in ergodic theory, harmonic analysis, and combinatorics, and using some number theory) to split $\Lambda$ into a bounded piece $f$ (which can be handled by Szemer\'edi's theorem)
plus an error $g$ which was ``Gowers uniform'', which means roughly speaking that its contribution to the above type of averages is negligible.   
We are skipping lots of details here - the proof is 50 pages long! - but the point is that ideas from many fields in analysis have played a big role in this number-theoretic result.

\ei

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\noindent

See also

\centerline{\tt http://www.math.ucla.edu/~analysis}

for more information on the analysis group.

\end{document}
