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
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
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
equations with integer coefficients. These kinds of problems will also
be discussed for specific kinds of algebraic numbers such as Pisot and
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.
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