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: tba

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.