This is the second meeting of Oregon Number Theory Days, a triannual number theory seminar rotating between the University of Oregon, Oregon State University, and Portland State University. The main speaker will be Ken Ono of Emory University, who will give two talks. Lunch will be provided and there will also be a talk by postdoc Asif Zaman of Stanford University. If you are interested in attending this meeting, please register by following the instructions at http://people.oregonstate.edu/~petschec/ONTD/
Ken Ono, Lecture I: Polya's Program for the Riemann Hypothesis and Related Problems. Abstract: In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. We prove the hyperbolicity of 100% of the Jensen polynomials of every degree. Moreover, we prove the GUE random matrix prediction for the distribution of the zeros in 'derivative aspect'. These results come from a general theorem which models such polynomials by Hermite polynomials. This general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
Ken Ono, Lecture II: Can you feel the Moonshine? Abstract: Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s.
Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.
Asif Zaman: A new formulation of the Chebotarev density theorem. Abstract: First established in 1926, the Chebotarev density theorem is a broad generalization of the prime number theorem describing the asymptotic distribution of prime splitting behavior in number fields. Lagarias and Odlyzko (1977) proved an effective version that has been applied in many situations, such as for modular forms, elliptic curves, and binary quadratic forms. However, the asymptotic holds for quite a restrictive range. We describe more of its history, a new formulation that closely parallels the classical case of arithmetic progressions, and a uniform improvement that expands its range of validity. The same methods are also applicable for general families of L-functions. This talk represents joint work with Jesse Thorner.