Royal Holloway - Number Theory Seminar

Autumn Term 2014:

September 30th: Martin Widmer (Royal Holloway)

Title: Lower bounds for the height of a generator

Abstract: Let d>1 be an integer. In 1998 W. Ruppert asked whether any number field L of degree d has a primitive
element whose absolute multiplicative Weil height is, up to a factor depending only on d, at most the 2(d-1)-th root
of the root discriminant of L. Ruppert showed that the answer is ``yes" for d=2, and we will show that the answer is
negative for composite d. This is joint work with Jeffrey D. Vaaler. 

October 7th: Manfred Madritsch (Université de Lorraine, Nancy)

Title: Construction of normal numbers by pseudo-polynomial sequences over the primes

Abstract: pdf

October 14th: Rainer Dietmann (Royal Holloway)

Title: Random Thue and Fermat equations

Abstract: We consider families of Thue and Fermat equations of the form ax^k-by^k=1 and ax^k+by^k+cz^k=0,
respectively, and obtain upper and lower bounds on the number of such equations of bounded height
that are locally soluble everywhere. Assuming the abc-conjecture, we can then show that almost all
locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion
holds true for Fermat equations of degree at least six. (joint work with Oscar Marmon)

October 21th: James McKee (Royal Holloway)

Title: All totally real algebraic integers are eigenvalues of graphs - part I

Abstract: In 1992, Estes showed that every totally real algebraic integer is the eigenvalue of an integer symmetric
matrix.  Combined with earlier work of Hoffman, this showed that every totally real algebraic integer is the eigenvalue
of a graph. (Recently Salez gave an alternative, more direct, proof of this latter result.)

The short series of lectures will go through the proof of the Estes theorem, following a slightly later paper of Estes and
Guralnick (also 1992), and will also cover Hoffman’s 1975 work.  The rough plan is for the first lecture to be an overview,
for the second lecture to focus on lattices over semilocal rings, and for the third lecture to give the meat of the Estes-Guralnick

October 28th: James McKee (Royal Holloway)

Title: All totally real algebraic integers are eigenvalues of graphs - part II

November 4th: James McKee (Royal Holloway)

Title: All totally real algebraic integers are eigenvalues of graphs - part III

November 11th: Victor Beresnevich (University of York)

Title: Sums of reciprocals of fractional parts of arithmetical progressions

Abstract: Finding estimates for the sums mentioned in the title is a classical topic in analytic number theory with many applications,
e.g. in the theory of uniform distribution. I will first recall some well (and possibly less well) known estimates due to Hardy, Litlewood,
Schmidt, Vaaler and others and then give an account of recent results due to myself, Haynes and Velani. Time permitting I will describe
the techniques for establishing the results.

November 18th: Mark Wildon (Royal Holloway)

Title: Open problems on enumerating partitions and permutations

Abstract: pdf

November 25th: Rachel Player (Royal Holloway)

Title: Ring Learning With Errors

Abstract: The talk introduces the Learning With Errors problem and its variant Ring Learning With Errors. The aim is to show how algebraic
number theory is applied in a cryptographic way and to provide the necessary prerequisites for Martin’s talk the following week.

December 2nd: Martin Albrecht (Royal Holloway)

Title: Implementing Candidate Multilinear Maps from Ideal Lattices

Abstract: Cryptographic multi-linear maps - if they exist - are versatile tools in cryptography. Recently, "graded encoding schemes" were
introduced as candidate approximations of cryptographic multi-linear maps. In this talk, we will first introduce graded encoding schemes
and give a simple application. We then describe how we might construct such a graded encoding scheme from ideal lattices and discuss
algorithmic number-theoretic challenges we faced when implementing this construction. In particular, we discuss algorithms for computing
with elements in the ring of integers of a cyclotomic number field of order 2^k.

December 9th: Jens Bolte (Royal Holloway)

Title: Arithmetic hyperbolic surfaces and their Laplace spectra

Abstract: The aim of this talk is to survey various spectral properties of Laplace-Beltrami operators on surfaces of constant negative curvature.
Some time ago it was noticed that eigenvalue correlations are contrary to what was expected, when the fundamental group of the surface is
arithmetic. I will describe the setting and explain why arithmetic groups play a special role. I will also discuss an explicit form of a Jacquet-Langlands
correspondence and its role in the context of Laplace spectra.