Royal Holloway - Number Theory Seminar

The NT seminar takes place on Wednesday at 4pm in Horton HLT1.
Here is a campus plan.

Autumn Term 2017

September 27: Andrei Yafaev (UCL)

"Some interactions of model theory and diophantine geometry"

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: Niclas Technau (Graz) (!!Thursday Oct 5th at 2pm in Moore 0-07!!)

"On Exceptional Sets in the Metric Poissonian Pair  Correlations problem"

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: Christopher Pinner (Kansas State Univ.)

"The Lind-Lehmer Constant for Finite Abelian Groups"

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: Cong Ling (Imperial)

"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: Gareth Jones (Manchester)

"Rational values of analytic functions"

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: Yiannis Petridis (UCL)

"Arithmetic Statistics of modular symbols"

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)

"The Schur-Siegel-Smyth trace problem for integer symmetric matrices"

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: Matteo Vannacci (Düsseldorf)

"Two proofs of a remarkable identity and some zeta functions associated to groups"

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.