Royal Holloway - Number Theory Seminar

The seminar takes place on Tuesday at 4pm in ABLT2 (Arts Building Lecture Theatre 2).
Here is a campus plan.

Spring Term 2016:

January 12: Matthew Palmer (University of Bristol)

Title: Diagonal approximation and the Duffin-Schaeffer theorem

Abstract: In classical Diophantine approximation, the Duffin-Schaeffer theorem is a
generalisation of Khinchin's theorem from monotonic functions to a wider class of approximating
functions. In recent years, there has been some interest in finding analogues of the theorems of
classical approximation in different settings - one of those settings is that of number fields. A very
general analogue of Khinchin's theorem was proven in number fields by David Cantor in the
1960s - however, so far the only versions of the Duffin-Schaeffer theorem proven have been
for very restrictive choices of number fields. In this talk, I will discuss a version of the Duffin-Schaeffer
theorem for all number fields.

January 19: No Seminar!

January 26: Martin Widmer (Royal Holloway)

Title: Weakly admissible lattices,  primitive lattice points, and Diophantine approximation"

Abstract: After surveying some impressive results of Skriganov on counting
lattice points in aligned boxes for weakly admissible lattices we present
some new counting results in a more general framework. Our error estimates
are inferior to Skriganov's regarding the dependence on the volume of the box but superior
regarding the dependence on the lattice. It is this improvement that allows
for counting results for primitive lattice points. When time permits we will also
discuss applications to Diophantine approximation by primitive points
as studied by Chalk and Erdős and more recently by Dani, Laurent, and Nogueira.

February 2: Philipp Habegger (University of Basel)

Title: On Singular Moduli that are Algebraic Units

Abstract: A singular moduli is the j-invariant of an elliptic curve
with complex multiplication, e.g. 0 or 1728 or -3375. A
classical result states that all singular moduli are algebraic
integers. It seems to be unknown whether there are singular moduli
that are algebraic units. But there can be at most finitely many, as
we will see in this talk. Moreover, I will present a related
finiteness statement reminiscent of Siegel's Theorem on integral
points on curves of positive genus.  If time permits, I'll also
discuss a version for curves in genus two, which is joint work
with Fabien Pazuki.

February 9: Alex Wilkie (University of Manchester)

Title: Rational points on subanalytic sets-some uniform results

Abstract: It has been known for some time that all  rational points of
height at most H lying on an (n-1)-dimensional subanalytic subset S of
[-1,1]^n all satisfy Q(X)=0, where Q is a polynomial  of degree at most
C(logH)^{B}. In this talk I will present a recent result of Pila, Cluckers
and myself which shows that the constants C,B maybe taken to be
independent of the parameters appearing in the definition of S.

February 16: Jens Bolte (Royal Holloway)

Title: A Gutzwiller trace formula for large hermitian matrices

Abstract: A Gutzwiller trace formula expresses spectral functions of certain
self-adjoint operators in terms of suitable geometric or dynamical data. Some
well known examples are the classical Selberg trace formula and the Duistermaat-
Guillemin trace formula. After an extended motivation I will consider the case
of hermitian matrices, where the Hilbert space in which the operators are defined
is finite-dimensional. The matrices will be realised as Weyl quantisations of symbols
on a torus, and the related dynamics are Hamiltonian flows on the torus. The talk is
based on joint work with Sebastian Egger and Stefan Keppeler.

February 23: Roger Heath-Brown (University of Oxford)

Title: Vinogradov's Mean Value and the Riemann Zeta-Function

Abstract: We shall describe the recent results of Wooley, and of Bourgain
Demeter and Guth, on Vinogradov's mean value, and how they lead to
substantial improvements in estimates for the Riemann Zeta-function.

March 1: Gary Greaves (Tohoku University)

Title: Eigenvalues of integer symmetric matrices

Abstract:  We present results about the eigenvalues of symmetric matrices with integer entries. 
We will discuss necessary and sufficient conditions for the characteristic polynomials and minimal
polynomials of such matrices.  A special case of interest is when all entries are either +1 or -1. 
This special case is related to the study of so-called "equiangular line systems" (sets of lines such
that the angle between any pair of lines is constant).

March 8: Seminar cancelled!

March 15: Christopher Frei (Graz University of Technology)

Title: The Hasse norm principle for abelian extensions

Let L/K be a normal extension of number fields. The Hasse norm
principle is a local-global principle for norms. It is satisfied if any
element x of K is a norm from L whenever it is a norm locally at every
place. For any fixed abelian Galois group G, we investigate the density
of G-extensions violating the Hasse norm principle, when G-extensions
are counted in order of their discriminant. This is joint work with Dan
Loughran and Rachel Newton.