The NT seminar takes place on Wednesday at 4pm in Horton HLT1.

Here is a campus plan.

September 27:

Abstract: In this talk, I will explain how certain results from model theory (o-minimality) may be used to solve

diophantine problems of a certain type - namely the Manin-Mumford-Andre-Oort type problems.

October 5:

Abstract: This talk is about a joint work with Thomas Lachmann on a uniform distribution property

of "second order" which is called the Poissonian pair correlation property. Particular attention will paid

towards recent progress on a putative Khinchin-like zero-one law for a metric problem in this area. By

improving on earlier results due to J. Bourgain and A. Walker, we shall see that the conjectured threshold

can, in general, not be increased.

October 11: No NT-Seminar

October 18:

Abstract: In 2005 Doug Lind generalized the concept of Mahler measure to an arbitrary compact abelian group.

For a given group one can ask for the minimal non-trivial measure; the counterpart of the classical Lehmer Problem

for the usual Mahler measure. For a finite abelian group this corresponds to the smallest non-trivial integral group

determinant. After a quick survey of existing results I will present some new congruences satisfied by the

Lind Mahler measure for p-groups. These enable us to determine the minimal measure when the p-group has one

particularly large component and to compute the minimal measures for many new families of small p-groups.

This is joint work with Mike Mossinghoff of Davison College. If there is time I will also mention some 3-group

results from a summer undergraduate research project with Stian Clem which may hint at what is going on in general.

October 25:

"Algebraic Number Theory for Coding: From Fermat to Shannon to 5G"

Abstract: Algebraic number theory has emerged as a new foundation of modern coding theory, due to its connection

with Euclidean lattices. In wireless communications, it is the key mathematic tool to construct powerful error-correction

codes over mobile fading channels. This talk presents an overview of the constructions of codes from number fields for

fading and MIMO (multi-input multi-output) channels, and introduces a novel framework to achieve the Shannon

capacity of these channels. If time permits, a glimpse at the applications to multi-user communications in next-generation

networks will be given.

November 1: No NT-Seminar

November 8:

Abstract: What can we say about rational points of bounded height on the graph of a transcendental analytic function?

Extending his work with Bombieri, Pila proved a bound that cannot be improved in general. But it can be improved for

various restricted classes of functions. I'll discuss recent work in this direction.

November 15:

Abstract. Modular symbols have been a useful tool to study the space of holomorphic cusp forms of weight 2, and the

homology of modular curves. They have been the object of extensive investigations by many mathematicians including

Birch, Manin, and Cremona. Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical

properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these

conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. In joint work with

Morten S. Risager we prove these on average using analytic properties of Eisenstein series twisted with modular symbols.

We also prove another conjecture predicting the Gaussian distribution of normalised modular symbols ordered according

to the size of the denominator of the cusps.

November 22: No NT-Seminar

November 29: James McKee (Royal Holloway)

Abstract: Let z be a totally positive algebraic integer that has degree d and trace t. The absolute trace of z is t/d. The Schur-Siegel-Smyth

trace problem aks: what is the smallest limit point of the set of absolute traces of totally positive algebraic integers? The smallest

known limit point is 2: is this the smallest? This problem is still very much open, but in joint work with Pavlo Yatsyna we settle

the analogous problem for totally positive integer symmetric matrices.

December 6:

Abstract: As a way to encode asymptotic aspects of a group, one can define several "zeta functions" associated to counting problems on

groups. I will give an overview of the theory of these zeta functions and, through basic yet intriguing examples, I will present a way of

computing two of these zeta functions.