Here’s a small sample of what we’ve had so far.
Before the conference even began, Vitaly Bergelson gave two introductory talks on the ergodic approach to additive number theory problems: the first one about the Poincaré recurrence theorem, the second about equidistribution, the “van der Corput trick” and Weyl differencing. While the lectures were accessible to students, they also included a good number of things that the senior mathematicians in the audience might not have heard before, from the history of the subject to a somewhat unexpected “quadratic” point of view on the theorems of Roth on arithmetic progressions and Sárközy on square differences in dense sets.
Vitaly’s conference talk started with an introduction to Szemerédi’s theorem. (There were reportedly people in the audience who didn’t know Szemerédi’s theorem – which is great! We’re always happy to see new people around and to introduce them to the subject.) He then went on to discuss his work with Alexander Leibman and Randall McCutcheon on ergodic theorems and an extension of the polynomial Szemerédi theorem to “generalized polynomials” – functions that can be built by iterating polynomials and the floor function.
To continue with the ergodic theory theme – Tamar Ziegler talked about her work with Terry Tao on the polynomial Szemerédi theorem in the primes, and Bernard Host gave a lecture on his joint work with Bryna Kra on nilsequences and their applications in ergodic theory and additive combinatorics. We have two more “ergodic” talks scheduled tomorrow, by Nikos Frantzikinakis and Maté Wierdl.
Ben Green and Terry Tao both gave talks about the progress on their program to prove the Dickson (or Hardy-Littlewood) conjecture on the asymptotic number of solutions to linear equations in the primes. In an earlier paper, they reduced the “non-degenerate” case of the conjecture to proving two statements, which they dubbed the inverse Gowers norm conjecture and the Möbius-nilsequences conjecture. (“Non-degenerate” means that the system does not either include or encode implicitly any equation in two variables. For instance, the twin primes conjecture involves the equation , which is degenerate.)
According to Terry Tao, they have now resolved the Möbius-nilsequences conjecture – this was the subject of his two lectures. The proof consists of a “number-theoretic” part where specific information about the Möbius function is exploited via the circle method, and a “dynamical” part involving a new Ratner-type theorem for nilmanifolds. I’m sure that we will be hearing more about this soon!
Ben Green gave an update on the inverse Gowers conjecture. For the norm, this conjecture was proved by Green and Tao in 2005 in and in finite fields of characteristics , and by Samorodnitsky in characteristics 2. Last year, Lovett-Meshulam-Samorodnitsky and (independently and about the same time) Green-Tao found finite fields counterexamples for norms with . However, these examples only work in finite fields of low characteristics. In a recent paper on distribution of polynomials over finite fields, Green and Tao proved that this particular type of counterexamples can’t occur if the characteristic of the fields is large enough; in particular there is still a good chance that the conjecture will be true in . (See Terry’s blog post on the subject.)
Akshay Venkatesh gave a “speculative” (his words) lecture about his joint work with Jordan Ellenberg on modelling number-theoretic phenomena by the statistics of seemingly unrelated random objects. For instance, the distribution of zeroes of L-functions can be modelled by the distribution of eigenvalues of random matrices chosen from a fixed subgroup of SL(N) for some large N; this goes back to Montgomery in the case of the Riemann zeta function, and to Katz-Sarnak and Cohen-Lenstra for more general L-functions. Another example: there are parallels between the statistics of arithmetic functions (e.g. partition or divisor functions) and certain phenomena in algebraic geometry. Most of this is heuristic rather than rigorous, but the numerical evidence is reported to be quite compelling and the heuristic considerations do suggest intriguing questions. In a follow-up Math Department colloquium talk, Akshay described some specific results of this type (joint with Jordan Ellenberg and Craig Westerland) at the interface of number theory, algebraic geometry and topology.
To be continued…