Scroll down for abstracts for the most recent seminars
Thurs. Sept. 7----Takashi Goto (Tokyo University)
Title: Non-directed graph related to Selmer group of certain elliptic curve
Abstract: Many mathematicians studied the elliptic curve E_n : y^2=x^3-n^2x because it is related to the congruent number problem. Feng and Xiong (2004, J. Number Theory) gave some condition that Selmer rank of E_n is zero. They used terms of graph theory. In this talk, we recall their results and consider some variants of the congruent number problem.
Thurs. Sept. 14----Christian Eholtz (University of London)
Title: Combinatorial prime number theory
Abstract: In this talk we study combinatorial questions about primes. In particular, Ostmann asked whether there exist two sets A and B (with at least two elements each) so that their sumset A+B equals the set of primes, for sufficiently large primes. Using a new version of the large sieve method we show, that such sets A and B would need to have counting functions of size N^(1/2 +o(1)), whereas previously only a lower bound of N^(o(1)) and an upper bound of N^(1+o(1)) was known. This implies, for example, that the set of primes cannot be decomposed into three such sets.
We also look at very thin sets of primes such as primes of the form x^2+y^4 and show that underlying additive structures exist which are larger than one might have expected.
Thurs. Sept. 21----Yannan Qiu (UW-Madison)
Title: Tate Classes on Siegel 3-folds
Abstract: The cycle groups of a variety are always of great interest. For Siegel 3-folds, it turns out that all degree-2 Tate classes defined over abelian number fields are generated by Hilbert modular surfaces. In the talk, we will give a proof for the degree-2 Tate classes contained in the cohomology of compact support.
Thurs. Sept. 28----Ben Kane (UW-Madison)
Title: Quadratic Forms and CM Lifts of Supersingular Elliptic Curves
Abstract: In this talk, we investigate the representability of discriminants by quadratic forms with corresponding theta series in Kohnen's plus space of level 4p, with p a prime. Due to a connection of Gross to supersingular elliptic curves, this gives arithmetic information about CM lifts of a fixed supersingular elliptic curve. Our methods, developed first by Ono and Soundarajan, lead to explicitly computable bounds, conditional upon certain GRH assumptions.
Thurs. Oct. 5----Dihua Jiang (Minnesota)
Title: On Cuspidal Automorphic Forms of SO(2n+1)
Abstract: The speaker will discuss the basic structures of cuspidal automorphic forms of SO(2n+1) in terms of Langlands functoriality. Although the results will be carried over to at least all classical groups, the speaker will concentrate what have been proved by the speaker of his co-workers.
Thurs. Oct. 12----Jianya Liu (Shandong University)
Title: A Weyl-like bound for automorphic L-functions
Asbtract: A central problem in the theory of L-functions is to investigate their
sizes on the critical line. The convexity bound, which follows from the
Phragmen-Lindelof principle, is useless in applications. Therefore much effort has
been made to obtain subconvexity bounds for various L-functions, which have also
been applied to give various equi-distribution results. In the classical case of
the Riemann zeta-function, this convexity bound is |t|^{1/4+\epsilon}, and the
classical subconvexity theorem of Weyl states that the 1/4 can be reduced to 1/6.
This 1/6 has resisted big improvement in the past 80 years, although the famous
Lindelof hypothesis states that it should be reduced to 0. In this talk, I will
report a joint work with Yuk-Kam Lau and Yangbo Ye, in which we obtain a Weyl-like
bound for Rankin-Selberg automorphic L-functions in the spectral aspect.
Thurs. Oct. 19----Kevin Ford (University of Illinois)
Title: Estimating the density of the union of arithmetic progressions
Asbtract: Consider a collection of arithmetic progressions a_1 mod m_1,..., a_k mod m_k and let D be the density of integers which lie in their union. We will discuss recent progress on the problem of estimating, for a given list m_1,...,m_k of moduli, the maximum and minimum of D over all possible choices of residue classes a_1,...,a_k. Under some conditions on the moduli, we can show that the maximum D is less than 1, answering some questions of Erdos, Graham and Selfridge concerning "covering systems" of congruences. The problem of estimating the minimum D is connected with certain divisor problems, one of which was solved recently using new results about Kolmogorov-Smirnov statistics.
Thurs. Oct. 26----Scott Ahlgren (University of Illinois)
Title: Congruences between modular forms of weights two and four
Asbtract: We discuss some conjectures and some results which describe the possible congruences between modular forms of prime level in weights two and four.
Thurs. Nov. 2----Michael Larsen (University of Indiana)
Title: Functoriality and the inverse Galois problem
Asbtract: I will discuss joint work with Chandrashekhar Khare, extending recent work of GaborWiese. The main theorem is that if E is a finite field and n is a positive even integer, there exists a Galois extension of Q whose Galois group is the finite simple group PSp_n(F) for some finite extension F of E. The proof uses group theory, local Langlands, and local and global lifting results from orthogonal groups to GL_n.
Thurs. Nov. 9----Aaron Levin (Brown University)
Title: Ideal Class Groups, Hilbert's Irreducibility Theorem, and Integral Points of Bounded Degree on Curves
Asbtract: We study the problem of constructing and enumerating, for any integers m, n > 1, number fields of degree n whose ideal class groups have "large" m-rank. Our technique, which appears to be new, relies fundamentally on Hilbert's Irreducibility Theorem and results on integral points of bounded degree on curves.
Thurs. Nov. 16----Sharon Garthwaite (UW-Madison)
Title: Arithmetic and analytic properties of exceptional $q$-series
Abstract:
**This is practice for my job talk. This talk is meant for a general audience, and all are invited to attend and give me feedback.**
Despite his early death, Ramanujan had a prolific mathematical career studying exceptional $q$-series. Of particular interest were
partitions and mock theta functions. Many interesting arithmetic and analytic properties of partitions arise from the connection between
the partition generating function and modular forms. On the other hand, the mock theta functions satisfy complex transformation
properties, making them difficult to study. In this talk we draw parallels between the study of partitions and mock theta functions.
We build upon recent work of Zwegers and of Bringmann and Ono placing mock theta functions in the realm of vector-valued modular
forms and harmonic weak Maass forms. In particular, we will study a large class of vector-valued Maass-Poincar\'e series arising from
mock theta functions.
Thurs. Nov. 30----Allison Pacelli (Williams)
Title: High n-Rank in Class Groups of Function Fields
Abstract: For any natural numbers m and n, infinitely many function fields K of degree m over the rational function field have class numbers divisible by n. This is a consequence of stronger results about the structure of the class group. We will look at constructions of function fields with high n-rank, and discuss many open questions in the area.
Thurs. Dec. 7----Frank Thorne (UW-Madison)
Title: Bounded Gaps Between Products of Primes and Applications
Abstract:
Let q_n denote the n-th E_2 number; in recent work, Goldston, Graham, Pintz, and Yildirim proved that the lim inf of q_{n + 1} - q_n is at most 6. We will discuss this work as well as earlier approximations to the twin prime
conjecture, and then present a generalization of the GGPY result to E_r numbers (r \geq 2) where restrictions are allowed on the prime factors. Using results of Ono, Soundararajan, and others, we use this generalization to obtain "bounded gaps" type results relating to class numbers, nonvanishing of L-functions, and ranks of elliptic curves.