NTS Spring 2013/Abstracts

January 24

Tamar Ziegler (Technion)

Title: An inverse theorem for the Gowers norms

Abstract: Gowers norms play an important role in solving linear equations in subsets of integers - they capture random behavior with respect to the number of solutions. It was conjectured that the obstruction to this type of random behavior is associated in a natural way to flows on nilmanifolds. In recent work with Green and Tao we settle this conjecture. This was the last piece missing in the Green-Tao program for counting the asymptotic number of solutions to rather general systems of linear equations in primes.

January 31

William Stein (U. of Washington)

Title: How explicit is the explicit formula?

Abstract: Consider an elliptic curve E. The explicit formula for E relates a sum involving the numbers a_p(E) to a sum of three quantities, one involving the analytic rank of the curve, another involving the zeros of the L-series of the curve, and the third, a bounded error term. Barry Mazur and I are attempting to see how numerically explicit – for particular examples – we can make each term in this formula. I'll explain this adventure in a bit more detail, show some plots, and explain what they represent.

(This is joint work with Barry Mazur).

February 7

Nigel Boston (Madison)

Title: A refined conjecture on factoring iterates of polynomials over finite fields

Abstract: In previous work Rafe Jones and I studied the factorization of iterates of a quadratic polynomial over a finite field. Their shape has consequences for the images of Frobenius elements in the corresponding Galois groups (which act on binary rooted trees). We found experimentally that the shape of the factorizations can be described by an associated Markov process, we explored the consequences to arboreal Galois representations, and conjectured that this would be the case for every quadratic polynomial. Last year I gave an undergraduate, Shixiang Xia, the task of accumulating more evidence for this conjecture and was shocked since one of his examples behaved very differently. We have now understood this example and come up with a modified model to explain it.

February 14

Tonghai Yang (Madison)

Title: A high-dimensional analogue of the Gross–Zagier formula

Abstract: In this talk, I will explain roughly how to extend the well-known Gross–Zagier formula to unitary Shimura varieties of type (n − 1, 1). This is a joint work with J. Bruinier and B. Howard.

March 7

Kai-Wen Lan (Minnesota)

Title: Galois representations for regular algebraic cuspidal automorphic representations over CM fields

Abstract: After reviewing what the title means (!) and providing some preliminary explanations, I will report on my joint work with Michael Harris, Richard Taylor, and Jack Thorne on the construction of p-adic Galois representations for regular algebraic cuspidal automorphic representations over CM (or totally real) fields, without hypothesis on self-duality or ramification. The main novelty of this work is the removal of the self-duality hypothesis; without this hypothesis, we cannot realize the desired Galois representation in the p-adic étale cohomology of any of the varieties we know. I will try to explain our main new idea without digressing into details in the various blackboxes we need. I will supply conceptual (rather than technical) motivations for everything we introduce.