Some analysis at UCLA

Terence Tao (UCLA)

UCLA has a diverse analysis and PDE group, including • Greg Eskin (Inverse problems, scattering theory)

• John Garnett (Complex analysis, harmonic analysis)

• Michael Hitrik (Semiclassical analysis, Schr¨’odinger equations)

• Rowan Killip (Spectral theory, random matrices, integrable systems)

• Jim Ralston (Wave equations, scattering theory)

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

• Christoph Thiele (Fourier analysis, nonlinear analysis, multilinear integrals)

• Several postdocs and visitors each year. In addition, there is a large number of graduate stu-

dents actively participating in the analysis group, for instance through our participating seminar.

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

• Theorem (Green, T., 2004) For any k, the prime numbers contain infinitely many arithmetic progressions of length k.

• This was previously only established for k = 3:

• Theorem (van der Corput, 1939) The prime numbers contain infinitely many arithmetic progressions of length 3.

• The proof of van der Corput’s theorem involves harmonic analysis. Extremely rapid sketch:

• Let Λ : Z^{+} → R be the function such that Λ(n) = log p whenever n is a power of a prime p, and Λ(n) = 0 otherwise. This is the von Mangoldt function. It usefully captures the unique factorization property of the natural numbers via the identity

log n = ∑_{d∈Z^{+}:d divides n} Λ(d).

Also, it is more or less supported on the primes (and also the powers of primes, but those are a much sparser set).

• To prove van der Corput’s theorem, it suffices to show that the quantity

∑_{a=1}^{N} ∑_{r=1}^{N} Λ(a)Λ(a + r)Λ(a + 2r)

grows very quickly with N (in fact it grows like N^{2}/ log^{3} N).

• We fix a large integer N, and identify {1, …, N} with the cyclic group Z/NZ. Then the above sum is essentially

∑_{a∈Z/NZ} ∑_{r∈Z/NZ} Λ_{N}(a)Λ_{N}(a + r)Λ_{N}(a + 2r)

where Λ_{N} : Z/NZ → R is some truncated version of Λ (the exact details are omitted here).

• If one then utilizes the Fourier transform

Λ_{N}(a) = ∑_{ξ∈Z/NZ} Λ̂_{N}(ξ)e^{2πiaξ/N};

Λ̂_{N}(ξ) := \frac{1}{N} ∑_{a∈Z/NZ} Λ_{N}(a)e^{−2πiaξ/N}

then the previous expression can be written (after some computation) as

∑_{ξ∈Z/NZ} Λ̂_{N}(ξ)^{2}Λ̂_{N}(−2ξ).

The proof then proceeds by computing Λ̂_{N}(ξ) very precisely, taking advantage of identities such as the unique factorization identity.

Longer progressions

• 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

∑_{a∈Z/NZ} ∑_{r∈Z/NZ} Λ_{N}(a)Λ_{N}(a+r)Λ_{N}(a+2r)Λ_{N}(a+3r)

but if one now tries to use the Fourier transform, one gets a mess.

• 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 Furstenberg average

lim_{N→∞} ∑_{r=1}^{N} ∫_{X} f(a)T^{r}f(a)T^{2r}f(a)T^{3r}f(a) dμ(a)

where (X, dμ) is some compact space and T : X → 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 • Szemerédi’s theorem Every subset of the inte-

gers with positive density contains arbitrarily long arithmetic progressions.

• This theorem does not apply to the primes, which have density zero; equivalently, the problem is that the von Mangoldt function Λ is unbounded.

• 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 Λ into a bounded piece f (which can be handled by Szemerédi’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 numbertheoretic result.

See also

   http://www.math.ucla.edu/ analysis

for more information on the analysis group.