The Number Theory Study Group is an informal forum for the Number Theorists at York and their visitors to discuss and collaborate on current research problems.

The study group consists of weekly meetings in which a pre-chosen topic or research paper is discussed during the course of an entire term. The starting level is reasonably elementary so as to clarify the importance of the work under discussion. The aim is to provide an invigorating research climate within the Number Theory group leading to new research and further collaborations. The slot is also used for specialist research seminars in Number Theory and Analysis.

Mail forum:  maths-ntsg-group@york.ac.uk

Upcoming seminar:

Programme for this term:

Date and time	Title	Speaker	Room
April 26, 2013, 14:15 - 16:15	When does the nearest root of a polynomial to a point of some field belong to this field?	Natalia Budarina (Maynooth, Ireland)	G/010
May 16, 2013, 13:45 - 15:15	Applications of metric Diophantine approximation to the distribution of discriminants and resultants of polynomials	Vasili Bernik (Minsk, Belarus)	G/002
May 16, 2013, 15:45 - 17:15	p-adic Littlewood conjecture: how can potential counterexamples look like?	Dmitry Badziahin (Durham)	G/002
May 20, 2013, 11:00 - 12:30	The distribution of algebraic numbers	Vasili Bernik (Minsk, Belarus)	G/002
May 21, 2013, 16:30 - 18:00	Dense and non-dense orbits in compact spaces, with applications	Alan Haynes (Bristol)	G/010
June 4, 2013, 15:15 - 17:15	TBA	Tom Sanders (Oxford)	G/002

Click here for the archive of seminars since 2007

Older records:

Spring term 2011:

Date: 21 Mar 2011
Title: Some problems in the theory of mixed approximations, III
Speaker:  Stephen Harrap (York)
Time and Place:  16:30pm    Rm G/112

Date: 16 Mar 2011
Title: TBA
Speaker:  Maurice Dodson (York)
Time and Place:  16:00pm    Rm P/L/006

Date: 14 Mar 2011
Title: Some problems in the theory of mixed approximations, II
Speaker:  Stephen Harrap (York)
Time and Place:  16:30pm    Rm G/112

 Date: 7 Mar 2011

Title: Some problems in the theory of mixed approximations
Speaker:  Stephen Harrap (York)
Time and Place:  16:30pm    Rm G/112

Date: 28 Jan 2011
Title: On multiplicative Diophantine exponents
Speaker:  Oleg German (Moscow State University)
Time and Place:  16:30pm    Rm G/112

Date: 21 Jan 2011
Title: Additive shifts of multiplicative subgroups
Speaker:  Ilya Shkredov (Moscow State University)
Time and Place:  16:30pm    Rm G/112
Abstract: Let $p$ be a prime number, and $R$ be a multiplicative subgroup in $\mathbb{Z}/p\mathbb{Z}$. Many authors such as Heath-Brown, Bourgain, Tao, Konyagin, Garcia, Voloch, Glibichuk and others studied various additive properties of such subgroups, for example, the cardinality of sumsets $|R+R|$, uniform distribution of $R$, its basis properties and so on. We prove that $$|R \cap R+x_1 \cap .... \cap R+x_k| < |R|^{1/2+a_k},$$ $a_k$ tends to zero as $k$ tends to infinity. Also we obtain a new result on sumsets : $|R+R| > |R|^{5/3} / \log p$, provided that $|R|<p^{1/2}$. Our proof uses Stepanov's method and a recent idea of Katz-Koester from additive combinatorics.

Date: 14 Jan 2011
Title: Moments of the Dedekind Zeta Function, II
Speaker:  Winton Heap (York)
Time and Place:  16:30pm    Rm G/112

Date: 7 Feb 2011
Title: Khintchine's theorem for mixed well-approximable sets
Speaker:  Tatiana Yusupova (York)
Time and Place:  16:30pm    Rm G/112

Date: 31 Jan 2011
Title: Moments of the Dedekind Zeta Function
Speaker:  Winton Heap (York)
Time and Place:  16:30pm    Rm G/112

Spring Term 2010

Reading course on P-adic Number Theory

A study group to read through Cassels' Local Fields will be run through the Autumn Term. Each week we will focus on a different chapter of the book and a member of the Study Group will give an overview of the salient points of that week's reading to the group.

Tentative Chapter Schedule:

1. Chapters 1 & 2

2. Chapter 4

3. Chapter 6

4. Chapter 7

5. Chapter 8

6. Chapter 12

Autumn Term 2009

Date: 10 December 2009
Title: On algebaric points near smooth curves
Speaker:  Olga Kukso (Minsk)
Time and Place:  10:15am    Rm D/130

Date: 10 December 2009
Title: On a problem of Nesterenko.
Speaker:  Vasil Bernik (Minsk)
Time and Place:  11:15pm    Rm G/001

A series of seminars on the Links between Number Theory and Dynamics (inlcuding some open problems).

Alan Haynes (York) will give a term long series of seminars on some very recent results in dynamics and their links with Number theory.

Abstract:

The goal of this sequence of talks is to explain in a simple way the connections between several conjectures in number theory and their counterparts in dynamical systems. Unlike much of the literature which is out there, these lectures are intended to be self contained. We will not assume more than a basic knowledge of analysis.

The starting point will be the recent results of Einsiedler, Katok, and Lindenstrauss on the exceptional set in the Littlewood Conjecture. As will be explained, this is related to the dynamical properties of the action of the subgroup of positive diagonal matrices on the space of unimodular lattices. The Margulis Conjecture, which implies the Littlewood Conjecture, asserts that the only measures which are invariant under this action are the trivial ones.

Next we will move on to discuss the Furstenberg Conjecture, which is a simplified analogue of the Margulis Conjecture. The Furstenberg Conjecture is related to a conjecture of de Mathan and Teulie known as the “mixed” Littlewood Conjecture. This is the assertion that for any real number   and for any distinct primes p1 pk  ,

liminfn np1  npk  n =0        

Furstenberg’s results from the 1960’s are enough to prove the mixed Littlewood Conjecture whenever k 2 Quite recently Bourgain, Lindenstrauss, Michel, and Venkatesh proved some quantitative results along these lines which allow us to show for k =2 that there exists a 0 for which

liminfn (logloglogn) np1  np2  n =0 

This result is on the cutting edge of current knowledge, which raises the question- how can it be improved?

All seminars will take place on the Thursday of the corresponding week and will start at 11:15 - ending at 1:15. The seminars will be held in:

Week 2 – G/001

Week 3 – G/001

Week 4 – PT/003

Week 5 – PT/003

Week 6 – G/001

Week 7 – G/001

Week 8 – G/001

Week 9 – G/001

Week 10 – G/001

Summer Term 2009

Spring Term 2009

Autumn Term 2008

Summer Term 2008

Bob Vaughan FRS will be visiting us from 27th April to 24th May.

His visit is being supported by the York Distinguished Visitors Scheme. Apart from collaborating with various people in the Department, he will give a mini-course and a general lecture in the Pure Mathematics Seminar programme.
 

Mini-course summary
As usual let $p_n$ denote the $n$th prime. The aim is to give a "complete" proof of Goldston-Pintz-Yildirim that $\liminf (p_{n+1}-p_n)/\log p_n =0$, by which I mean an account of Large Sieve, Bombieri-Vinogradov and the application to GPY all distilled down to its simplest form with details and notes.

The paper of Goldston-Pintz-Yildirim is due to appear in the Annals and can be found at:
http://annals.math.princeton.edu/issues/2006/FinalFiles/GoldstonPintzYildirimFinal.pdf



All talks will take place in Room G020. The dates and times are as follows:

Thursday   1  May  15.15 -- 17.15
Tuesday    6  May  11.15 -- 13.15
Thursday   8  May  15.15 -- 17.15
Monday   12  May  15.15 -- 17.15
Thursday 15  May  15.15 -- 17.15

Number Theory Afternoon:  Wednesday 14 May 2008

The following talks will take place in the Pure Mathematics Seminar
series:

Title:  Changes of sign of the error term in the prime number theorem
Speaker: Hugh Montgomery (Michigan/Bristol)
Time and place:  2:45 - 3:45  Rm G/109

Title: A History of Paucity
Speaker: Bob Vaughan (Penn State University)
Time and place:  4:15 - 5:15  Rm G/109

Abstract: The solutions of diophantine equations can often be
classified into two subsets. Trivial solutions, e.g. those lying on sets
of lines on the surface, and non-trivial solutions. The exisitence, or
otherwise, of non-trivial solutions is often of great interest, as in
the Fermat equation. Paucity occurs when the non-trivial solutions are scarce or non-existent.

Other talks:

Title:  Shrinking Target Properties
Speaker: Anish Ghosh (Bristol)
Time and place:  3.30 - 4:45  Rm G/109

Abstract:  Many interesting problems in Diophantine approximation lend themselves to natural dynamical interpretations, and can be studied using dynamical techniques. I will speak about a class of examples which can be described in terms of the frequency with which orbits of certain dynamical systems visit sets which shrink in a prescribed manner. If time permits, I will discuss recent developments.

Spring Term 2008

There is no general theme this term. Instead a series of seminar talks will be given in which current research directions will be discussed. The seminars are absolutely informal with no time boundary.


Date: 17 January 2008
Title: A basic introduction to the Riemann zeta function 
Speaker:  Christopher Hughes
Time and Place:  5.15pm    Rm G/001

Abstract: This talk will develop some of the very basic properties of the Riemann zeta function, such as the Euler product, analytic continuation, the functional equation, and location of its zeros.

Date: 24 January 2008
Title:  Statistics of the Riemann zeros 
Speaker:  Christopher Hughes
Time and Place:  5.15pm    Rm G/001

Abstract: This talk will introduce random matrix theory, and present evidence (numerical,theoretical and heuristic) that RMT can be used to answer statistical questions about the distribution of non-trivial zeros of the Riemann zeta function.

Date: 31 January 2008
Title: Modelling the Riemann zeta function 
Speaker:  Christopher Hughes
Time and Place:  5.15pm    Rm G/001

Abstract: This talk will survey some of the recent results on modelling the Riemann zeta function using characteristic polynomials of random unitary matrices.I will end by discussing how this work can be generalized and expanded, and list some open problems in the field.

Date: 7 February 2008
Title: Schmidt's game, linear forms and fractals
Speaker:  Lior Fishman (Ben Gurion University, Israel)
Time and Place:  4.15pm    Rm G/013

Date: 14 February 2008
Title: Digit frequency and Diophantine approximation 
Speaker:  Mumtaz Hussain 
Time and Place:  4.15pm    Rm G/013

Date: 20 February 2008
Title: Twisted Inhomogeneous Diophantine approximation 
Speaker:  Stephen Harrap
Time and Place:  5.15pm    Rm G/001


Date: 27 February 2008
Title: More Twisted Inhomogeneous Diophantine approximation 
Speaker:  Stephen Harrap
Time and Place:  4.15pm    Rm G/00

A continuation of a series of talks on recent developments in the area of twisted inhomogeneous Diophantine approximation.

Autumn Term 2007

There is no general themethis term. Instead a number of seminar talks will be given by external speakers. The seminars are absolutely informal with no time boundary.

Date: 7 December 2007
Title: On representations of integers by quadratic forms
Speaker: Natalia Budarina (Maynooth)
Time and Place:  4pm    Rm G/002


Date: 10 December 2007
Title: On the size and the arithmetic structure of the discriminants of integral polynomials
Speaker: Vasili Bernik (Minsk)
Time and Place:  10.15am    Rm G/045

Summer Term 2007

General theme: Heterogenous ubiquitous systems

The idea of a ubiquitous system is one of the major concepts used when trying to establish Hausdorff measure/dimension results in metric number theory. During this session we will concentrate on the notion of heterogeneous ubiquitous systems as developed in:

Date: 10 May 2007
Title: Heterogenous ubiquitous systems: an overview of multi-fractals
Speaker: Jason Levesley
Time and Place:  3pm    Rm G/002


Date: 17 May 2007
Title: Heterogenous ubiquitous systems: continued
Speaker: Jason Levesley
Time and Place:  2pm    Rm G/002


Date: 24 May 2007
Title: Heterogenous ubiquitous systems: the results
Speaker: Jason Levesley
Time and Place:  3pm    Rm G/002


Date: 31 May 2007
Title: Heterogenous ubiquitous systems: weakly redundant and upperbounds
Speakers: Mumtaz Hussian and Jason Levesley
Time and Place:  3pm    Rm G/002


Date: 31 May 2007
Title: Heterogenous ubiquitous systems: weakly redundant and upperbounds (II)
Speakers:  Jason Levesley
Time and Place:  3pm    Rm G/002


Date: 14 June 2007
Title: Heterogenous ubiquitous systems: weakly redundant and upperbounds (II)
Speakers:  Jason Levesley
Time and Place:  3pm    Rm G/002


Date: 21 June 2007
Title: Heterogenous ubiquitous systems: the lowerbounds (I)
Speakers:  Mumtaz Hussian
Time and Place:  3pm    Rm G/002




Talks outside of general theme and/or by other speakers:

Date: 22 May 2007
Title: Introduction to some metrical ideas
Speaker: Alan Haynes
Time and Place:  3.30pm    Rm G/002


Date: 24 May 2007
Title: Introduction to some metrical ideas (II)
Speaker: Alan Haynes
Time and Place:  1.30pm    Rm G/002

Date: 30 May 2007
Title: Martingales and Continued Fractions (I)
Speaker: Alan Haynes
Time and Place:  3.30pm    Rm G/002

Date: 31 May 2007
Title: Martingales and Continued Fractions (II)
Speaker: Alan Haynes
Time and Place:  1.30pm    Rm G/002

Date: 6 June 2007
Title: Simultaneous Diophantine Approximation on Surfaces
Speaker: Detta  Dickinson (NUI -- Maynooth)
Time and Place:  3.30pm    Rm G/002

Date: 6 June 2007
Title: Upper bounds for Hausdorff dimension -- an introduction
Speaker: Basil Bernik (Minsk)
Time and Place:  5.45pm    Rm G/002


Date: 7 June 2007
Title: Upper bounds for Hausdorff dimension
Speaker: Basil Bernik (Minsk)
Time and Place:  1.30pm    Rm G/002


Date: 22 June 2007, 03:30 PM » 04:30 PM
Title: Diophntine equations from an algorithmical point of view
Speaker: Yuri Matiyasevich (St-Petersburg)
Time and Place:  3.30pm    Rm G/010

Spring Term 2007

There is no general theme this term. Instead a number of seminar talks will be given in which current research directions will be discussed -- mainly by PhD students. The seminars are absolutely informal with no time boundary.


Date: 16 March 2007
Title: The quantitative Duffin-Schaeffer conjecture for lacunary sequences - revisited
Speaker: Cem Salp
Time and Place:  2pm    Rm G/010


Date: 9 March 2007
Title:  Simultaneous Diophantine approximation on Topological Products of curves. 
Speaker:  Jason Levesley 
Time and Place:  2pm    Rm G/010


Date: 2 March 2007
Title:  Lacunary Sequences and Khintchines Theorem.  
Speaker:  Mumtaz Hussain
Time and Place:  2pm    Rm G/010


Date: 23th February 2007
Title: The general Hausdorff measure law for multiplicative approximation on planar curves.  
Speaker: Dzmitry Badziahin
Time and Place:  2pm    Rm G/010

Winter Term 2006

General theme: Ubiquity and the Mass Transference Principle

The aim of this terms study group is to understand the relationship between the recent developments of ubiquity and the recent introduction of the Mass Transference Principle to the theory of metric number theory.

Everyone is more than welcome to participate. The starting `level' of this study group is elementary.

Unless announced otherwise the group meets on Mondays in Rm G010 at 2.15 till 4.15.

Summer Term 2006

General theme: Multiplicative aspects of Metric Number Theory (Continued)

The aim is specific:  to obtain the `harder' multiplicative results mentioned and discussed in last terms study group. The first talk by Sanju Velani will summarize the events of last term and lay down the concrete statements.

Unless announced otherwise the group meets on Mondays in Rm G010 at 2.15 till 4.15.
 

Spring Term 2006

General theme: Multiplicative aspects of Metric Number Theory

Everyone is more than welcome to participate. The starting `level' of this study group is elementary.

Unless announced otherwise the group meets on Mondays in Rm G010 at 2.15 till 4.15.
 
Date: 13th February 2006
Title: On the `manifolds' dimension results of R.C. Baker
Speaker: Dzmitry Badziahin 

Date: 6th February 2006
Title: The quantitative Duffin-Schaeffer conjecture for lacunary sequences
Speaker: Cem Salp

Date: 30th January 2006
Title: On a theorem of Gallagher
Speaker: Dzmitry Badziahin

Date: 23rd January 2006
Title: Multiplicative Simultaneous Diophantine Approximation on Planar Curves
Speaker: Victor Beresnevich

Winter Term 2005

General theme: Normality and Bad
 
This term there is a study group under the vague heading: Normality and Bad. Everyone is more than welcome to participate. The starting `level' of this study group is elementary with the hope that we will understand something rather deep by the end -- possibly leading to new research. The first talk is:

Date: Wednesday 2nd November
Title: An introduction and motivation
 Speaker: Sanju Velani 


After this Jason Levesley and Cem Salp will lead the study group by explaining the relationship between `thermodynamic formalism' and `fourier transforms of measures' -- all great stuff !!

The goal is to understand and rip to pieces the papers of:

Mark Pollicott: Fourier Transforms of measures on nonlinear Cantor sets.

Martine Queffelec and Olivier Ramare: Analyse de Fourier des fractions continues a quotients restreints.


Talks outside of general theme and/or by other speakers:

Date: Wednesday 26th October
Title: Rational Points near Planar Curves
 Speaker: Robert Vaughan (Penn. State)


Date: Wednesday 2nd November
Title: Good & Bad Vibes: Applications of Diophantine Approximation
Speaker: Maurice Dodson

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