Higher moments of arithmetic functions in short intervals: a geometric perspective

Speaker: 

Vlad Matei

Institution: 

University of Wisconsin

Time: 

Tuesday, November 29, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

In joint work with Daniel Hast, we recast the paper of Jon Keating and Zeev Rudnick "The variance of the number of prime polynomials in short intervals and in residue classes" by studying the geometry of these short intervals through an associated highly singular variety. We manage to recover their results for a a general class of arithmetic functions up to a constant and also obtain information about the higher moments. Recently work of Brad Rodgers in "Arithmetic functions in short intervals and the symmetric group" gives new insight into the geometry of our variety.

Torsion subgroups of elliptic curves in elementary abelian 2-extensions

Speaker: 

Ozlem Ejder

Institution: 

USC

Time: 

Tuesday, November 22, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

Let E be an elliptic curve defined over Q. The torsion subgroup of E over the compositum of all quadratic extensions of Q was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of 31 possible groups and Fujita proved that the list of 20 different groups is complete.

In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. Q(i) and Q(\sqrt{-3}).

Dynamically distinguishing polynomials

Speaker: 

Derek Garton

Institution: 

Portland State University

Time: 

Tuesday, February 28, 2017 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

Given two polynomials with integer coefficients, for how many primes p do the polynomials induce nonisomorphic dynamical systems mod p? This question will lead us to the study of the statistics of wreath products, the Galois theory of dynatomic polynomials, and other topics. This work is joint with Andrew Bridy.

Kloosterman sums and Maass cusp forms of half-integral weight

Speaker: 

Nick Andersen

Institution: 

UCLA

Time: 

Tuesday, December 6, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

Kloosterman sums play an important role in modern analytic number theory. I will give a brief survey of what is known about the classical Kloosterman sums and their connection to Maass cusp forms of weight 0 on the full modular group. I will then talk about recent progress toward bounding sums of Kloosterman sums of half-integral weight (joint with Scott Ahlgren) where the estimates are uniform in every parameter. Among other things, this requires us to develop a mean value estimate for coefficients of Maass cusp forms of half-integral weight. As an application, we obtain an improved estimate for the classical problem of bounding the size of the error term in Rademacher’s formula for the partition function.

Counting problems and homological stability [Please Note Special Day and Time]

Speaker: 

Jesse Wolfson

Institution: 

University of Chicago

Time: 

Monday, January 23, 2017 - 3:00pm to 4:00pm

Host: 

Location: 

RH 340N

The framework of the Weil conjectures establishes a correspondence between the arithmetic of varieties over finite fields and the topology of the corresponding complex varieties. Many varieties of interest arise in sequences, and a natural extension of the Weil conjectures asks for a relationship between the asymptotic point count of the sequence over finite fields and the limiting topology of the sequence over C.  In this talk, I'll recall the Weil conjectures and explain the basic idea of these possible extensions.  I'll then give a survey of ongoing efforts to understand and exploit this relationship, including Ellenberg-Venkatesh-Westerland's proof of the Cohen-Leinstra heuristics for function fields, a ``best possible'' form of this relationship in the example of configuration spaces of varieties (joint with Benson Farb), and a counterexample to this principle coming from classical work of Borel and recent work of Lipnowski-Tsimerman. 

On some algebraic constructions of extremal lattices

Speaker: 

Lenny Fukshansky

Institution: 

Claremont McKenna College

Time: 

Tuesday, October 18, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

A lattice in a Euclidean space is called extremal if it is a local maximum of the packing density function in its dimension. An old theorem of Voronoi gives a beautiful characterization of extremal lattices in terms of their geometric properties. We will review Voronoi's criterion, and then apply it to exhibit families of extremal lattices coming from some algebraic and arithmetic constructions.

Wild symbols in local class field theory

Speaker: 

Michiel Kosters

Institution: 

UC Irvine

Time: 

Monday, May 23, 2016 - 4:00pm to 5:00pm

Location: 

RH 340N

Let K be a local field with residue field of characteristic p>0. Our goal is to understand the cyclic extensions of K of degree a power of p. If K has characteristic 0 and contains a p^m-th primitive root of unity, then one can use class field theory and Kummer theory to construct a symbol which helps us to understand the ramification of cyclic extensions of degree p^m. If K has characteristic p, then one can construct a symbol, using class field theory and Artin-Schreier-Witt theory, which helps us to understand the cyclic extensions of degree p^m for any m. We will discuss both symbols in more detail and discuss methods for computing these symbols.

Almost Divisibility of Selmer Groups

Speaker: 

Ralph Greenberg

Institution: 

University of Washington

Time: 

Tuesday, May 3, 2016 - 2:00pm to 3:00pm

Location: 

RH 340P

There is a classical theorem of Iwasawa which concerns certain modules X for the formal power series ring Λ = Zp[[T]] in one variable.  Here p is a prime and Zp is the ring of p-adic integers.  Iwasawa's theorem asserts that X has no nonzero, finite Λ-submodules. We will begin by describing the modules X which occur in Iwasawa's theorem and explaining  how the theorem is connected with the title of my talk. Then we will describe generalizations of this theorem for  certain Λ-modules (the so-called "Selmer groups")  which arise naturally in Iwasawa theory.  The ring Λ can be a formal power series ring over Zp in any number of variables, or even a non-commutative analogue of such a ring. 

The distribution of consecutive primes

Speaker: 

Robert Lemke Oliver

Institution: 

Stanford University

Time: 

Tuesday, May 31, 2016 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic.  We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well.  We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle.  This is joint work with Kannan Soundararajan.

Pages

Subscribe to RSS - Number Theory