Royal Holloway - Number Theory Seminar
The NT seminar takes place
on Wednesday at 4pm in Horton HLT1.
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
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
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
Abstract: Let z be a totally positive algebraic integer that has
degree d and trace t. The absolute trace of z is t/d.
trace problem aks: what is the smallest limit point of the set of
absolute traces of totally positive algebraic integers? The
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
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.