NTS Spring 2013/Abstracts

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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 ap(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.

February 28

David Perry (NSA)
Title: The Cracking of Enigma

Abstract: Having learned in the previous talk (Wed., Feb. 27, 5pm–6pm, Van Vleck B239) how the Enigma cryptodevice worked and was used by the Germans at the beginning of World War II, we will now learn precisely how the Polish mathematicians were able to crack the Enigma, setting off a series of events that changed the course of world history. The history of cryptology was also irrevocably changed, with a growing realization that the future of secrecy would rely on mathematicians and the brand new discipline of computer science. This talk is geared towards those with some undergraduate mathematics experience, but less is required than you might suspect.

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.

March 14

Hang Xue (Columbia)
Title: On the Gan–Gross–Prasad conjecture for U(n) × U(n)

Abstract: In this talk, we shall introduce the Gan–Gross–Prasad conjecture for U(n) × U(n) and sketch a proof under certain local conditions using a relative trace formula. We shall also talk about its refinement and applications to the Gan–Gross–Prasad conjecture for U(n +1) × U(n).

April 4

John Jones (Arizona State)
Title: The tame-wild principle

Abstract: We consider discriminant relations for number fields, i.e., when the discriminant of one field must divide the discriminant of another. If we embed the fields in a Galois extension L/F with Galois group G, this can be phrased in terms of subgroups H and K of the Galois group: does D_{L^H} | D_{L^K}. It is easy to prove results of this type under the assumption that all ramification is tame. We investigate whether consideration of tame ramification is sufficient – whether relations which would always hold for tamely ramified extensions must also hold for wildly ramified extensions. We present successes, failures, and applications (of the successes) to computational questions.

April 11

Andrew Snowden (MIT)
Title: Arithmetic families of torsors

Abstract: Let G be a group scheme over the rational projective line (with some points discarded). Suppose X is a G-torsor such that Xt is trivial for almost all rational numbers t. Can we conclude that X itself is trivial? I will discuss several results, some positive and some negative. This is joint work with Jacob Tsimerman.

April 18

Jerry Wang (Harvard)
Title: Pencils of quadrics and 2-Selmer groups of Jacobians of hyperelliptic curves

Abstract: Since Bhargava and Shankar's new method of counting orbits, average orders of the 2,3,4,5-Selmer groups of elliptic curves over Q have been obtained. In this talk we will look at a construction of torsors of Jacobians of hyperelliptic curves using pencils of quadrics and see how they are used to compute the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves over Q with a rational Weierstrass point.

May 2

Wei Ho (Columbia/Princeton)
Title: Families of lattice-polarized K3 surfaces

Abstract: There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron–Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics.

This is joint work with Manjul Bhargava and Abhinav Kumar.

May 9

Stephen Gelbart (Weizmann Institute)
Title: A p-adic integral for the reciprocal of the p-adic L-function L(s, χ)

Abstract: We introduce an analog of part of the Langlands–Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL(2), and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions.

Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami

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