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.

Extra Seminar on Monday June 1st, 2015 at 2pm in ABLT2.

June 1st, 2015: Lê Thái Hoàng (Centre de mathématiques Laurent Schwartz, Ecole Polytechnique)

Title: Sums of reciprocals of fractional parts

Abstract: pdf

Spring Term 2015:

January 20th: Eugen Keil (University of Oxford)

Title: Roth's theorem for quadratic forms

Abstract: In this talk I want to outline some of the main steps to understand quadratic forms on dense sets of integers and explain how similar ideas
can help us to solve more general diophantine equations in primes.

January 27th: Chris Smyth (University of Edinburgh)

Title: Conjugate algebraic numbers: connections and conjectures

Abstract. By definition, an algebraic number is the root of an irreducible polynomial with integer coefficients. But what about its
conjugates -- the other roots of the polynomial? How are they related? I discuss some relations that can be satisfied by conjugate algebraic
numbers, and some that cannot be. It turns out that some such questions can be reduced to problems concerning the integer solutions of  linear
equations with integer coefficients. These kinds of problems will also be discussed for specific kinds of algebraic numbers such as Pisot and
Salem numbers.

February 3rd: Efthymios Sofos (University of Bristol)

Title: Refined properties of rational points on cubic hypersurfaces

Abstract: There has been a great progress in the last years regarding the quantitative distribution of the counting function of rational points of bounded height for low dimensional Fano varieties. However little is known regarding more refined quantitative arithmetic properties of the rational points. We will report on new work regarding the Zariski density of almost prime rational points on smooth cubic hypersurfaces and finish by stating some related open problems. This is joint work with Y.Wang.

February 10th: Rainer Dietmann (Royal Holloway)

Title: Small solutions of congruences

February 17th: Francesco Amoroso (Université de Caen)

Title: Lacunary Polynomials and Schinzel-Zilber conjecture

Abstract: We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factor of a sparse polynomial, are also sparse. To obtain this, we give a version of a theorem of Bombieri and Zannier on the intersection of a subvariety of codimension 2 of the multiplicative group with torsion curves, with an explicit dependence on the height of the subvariety. We also apply this to obtain a result in the direction of a conjecture of Bolognesi and Pirola.

February 24th: Jan-Christoph Schlage-Puchta (Universität Rostock)

Title: Abstract Analytic Number Theory

March 3rd: Cecilia Busuioc (Royal Holloway) CANCELLED!

March 10th:  Sanju Velani (University of York) at !!2PM in McCrea 219!!

Title: Metric Diophantine approximation:  Lebesgue versus Hausdorff

Abstract: There are two fundamental results in the classical theory of metric Diophantine approximation:   Khintchine's theorem  and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We discuss these theorems and  show  that Lebesgue statement implies the general Hausdorff statement. The key  is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure theoretic statements. In view of this, the Lebesgue  theory of limsup  sets  is shown to underpin the general Hausdorff theory. This is rather surprising since   the latter theory is viewed to be a subtle refinement of the former.

March 17th: Robert Royals (University of East Anglia)

Title: Khintchine's theorem in positive characteristic

Abstract: A look at various extensions of Khintchine's famous theorem from probability theory, in particular an extension to quadratic extensions of function fields over a finite field.

March 24th: Eira Scourfield (Royal Holloway)

Title: Exact divisors of polynomials with prime argument

Abstract: pdf