Past Junior Number Theory Seminar

22 February 2016
16:30
Alexander Betts
Abstract
Classically, one puts an algebraic structure on certain "congruence" quotients of the upper half plane by interpreting them as spaces parametrising elliptic curves with certain level structures on their torsion subgroups. However, the non-congruence quotients don't admit such a straightforward description.
 
We will sketch the classical theory of congruence modular curves and level structures, and then discuss a preprint by W. Chen which extends the above notions to non-congruence modular curves by considering so-called Teichmueller level structures on the fundamental groups of punctured elliptic curves.
  • Junior Number Theory Seminar
15 February 2016
16:30
Sofia Lindqvist
Abstract

Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used.  I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.

  • Junior Number Theory Seminar
8 February 2016
16:30
Danny Scarponi
Abstract

Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.

 

  • Junior Number Theory Seminar
1 February 2016
16:30
Aled Walker
Abstract

Many theorems and conjectures in prime number theory are equivalent to finding solutions to certain linear equations in primes -- witness Goldbach's conjecture, the twin prime conjecture, Vinogradov's theorem, finding k-term arithmetic progressions, etcetera. Classically these problems were attacked using Fourier analysis -- the 'circle' method -- which yielded some success, provided that the number of variables was sufficiently large. More recently, a long research programme of Ben Green and Terence Tao introduced two deep and wide-ranging techniques -- so-called 'higher order Fourier analysis' and the 'transference principle' -- which reduces the number of required variables dramatically. In particular, these methods give an asymptotic formula for the number of k-term arithmetic progressions of primes up to X. In this talk we will give a brief survey of these techniques, and describe new work of the speaker, partially ongoing, which applies the Green-Tao machinery to count prime solutions to certain linear inequalities in primes -- a 'higher order Davenport-Heilbronn method'. 

  • Junior Number Theory Seminar
23 November 2015
16:00
Alexander Betts
Abstract

Much of the arithmetic behaviour of an elliptic curve can be understood by examining its mod p reduction at some prime p. In this talk, we will aim to explain some of the ways we can define the mod p reduction, and the classifications of which reduction types occur.

Topics to be covered include the classical reduction types (good/multiplicative/additive), the Kodaira-Neron reduction types that refine them, and the Raynaud parametrisation of a semistable abelian variety. Time permitting, we may also discuss joint work with Vladimir Dokchitser classifying the semistable reduction types of 2-dimensional abelian varieties.

  • Junior Number Theory Seminar
16 November 2015
16:00
Jakub Konieczny
Abstract

I will discuss the many appearances of the class of IP sets in classical theorems of combinatorial number theory and ergodic theory. Our point of departure will be the celebrated theorem of Hindman on partition regularity of IP sets, which is crucial for the introduction of IP-limits. We then discuss how existence of certain IP-limits translates into recurrence statements, which in turn give rise to results in number theory via the Furstenberg correspondence principle. Throughout the talk, the methods of ergodic theory will play an important role - however, no prior familiarity with them is required.

  • Junior Number Theory Seminar
9 November 2015
16:00
Philip Dittmann
Abstract

Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.

  • Junior Number Theory Seminar
2 November 2015
16:00
Christopher Nicholls
Abstract

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

  • Junior Number Theory Seminar

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