# Number Theory Down Under

## The University of Newcastle

It is our great pleasure to announce that the Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) will be hosting a two-day number theory workshop Number Theory Down Under (NTDU) on 18th and 19th September 2015, at the University of Newcastle. The talks will be in the lecture theatre VG01. Links for the previous meetings are available here NTDU13 and NTDU14 and a bunch of photographs.

Details about registration, confirmed participants, accommodation, travel, and anything else needed can be found below.

If you have any questions, please email Mumtaz Hussain at Mumtaz.Hussain@newcastle.edu.au

## Speakers

• Laureate Prof. Jon Borwein (The University of Newcastle, Australia).

• Title: Moments and densities of short walks in higher dimensions.

Abstract: Following Pearson in 1905, we first study the expected distance and density of a two-dimensional walk in the plane with $n$ unit steps in random directions --- what Pearson called a random walk. We finish by presenting our recent work in higher dimensions. We present recent arithmetic results on the densities, $p_n$, of $n$-step random uniform random walks in the plane ($d:=2\nu+2=2$). For $n \ge 7$ asymptotic formulas first developed by Raleigh are largely sufficient to describe the density. For $2 \le n \le 6$ this is far from true, as first investigated by Pearson. We shall see remarkable new hypergeometric closed forms for $p_3, p_4$ and precise analytic information for larger $n$. Heavy use is made of analytic continuation of the integral (also of modern special functions (e.g., Meijer-G) and computer algebra (CAS)).

This is joint work with Armin Straub Christophe Vignat, James Wan, and Wadim Zudlin

• Associate Prof. Shaun Cooper (Massey University, New Zealand).

• Title: The Rogers-Ramanujan continued fraction

Abstract: The Rogers-Ramanujan continued fraction was first studied by L. J. Rogers and then made famous by S. Ramanujan in letters to G. H. Hardy. Hardy said that Ramanujan's results defeated me completely; I had never seen anything in the least like them before'' and concluded A single look at them is enough to show that they could only be written down by a mathematician of the highest class''.

This talk will outline the basic properties of the Rogers-Ramanujan continued fraction and explain how Ramanujan's results fit in a wider setting.

• Prof. Karl Dilcher (Dalhousie University, Canada).

• Title: Applications of generalized Stern polynomials

Abstract: Two different concepts of Stern polynomials were introduced in recent years, both reducing to the classical Stern diatomic sequence. In this talk I will present two closely related classes of generalized Stern polynomials, both depending on a positive integer parameter. This not only shows that the two original polynomial sequences are more closely related than originally thought, but they also have the following applications: (1) The complete characterization of all hyperbinary expansions of a given integer n; (2) the characterization of certain tilings and of lattice paths related to some specific weighted Delannoy numbers; (3) the evaluation of certain finite and infinite continued fractions. (Joint work with Larry Ericksen). >

• Dr. Alan Haynes (The University of York, England).

• Title: Perfectly ordered quasicrystals and the Littlewood conjecture.

Abstract: This talk is about connections between Diophantine approximation and patterns in quasicrystals, as modelled by cut and project sets. Cut and project sets are point patterns in Euclidean space defined by a dynamical construction- they can be thought of as collections of return times of linear R^d actions on tori, to some target regions. A cut and project set Y is called linearly repetitive if there exists a constant C>0 such that every point pattern of size r which occurs in Y, occurs in every ball of diameter Cr in R^d. Linearly repetitive cut and project sets were presented by Lagarias and Pleasants as a model for perfectly ordered’ quasicrystals. We will present a characterization of all such sets, involving an algebraic condition and a Diophantine approximation condition. Furthermore, we will explore the possible existence of super perfectly ordered’ quasicrystals, and explain how this line of thought quickly leads to a question which turns out to be equivalent to the Littlewood conjecture.

• Prof. Mourad Ismail (The University of Central Florida, USA).

• Title: Bessel Functions and Rogers-Ramanujan Identities.

Abstract: We indicate several new identities of Rogers-Ramanujan type involving q-Bessel functions and orthogonal polynomials. We also identify generalizations of the Schur polynomials and the Andrews finite versions of the Rogers-Ramanujan identities.

• Prof. Sinai Robins (Brown University, USA).

• Title: Asymptotics of cone theta functions, Gauss sums over parallelepipeds, and generalized Gram relations for polyhedra.

Abstract. We begin with cone theta functions, which are cousins of theta functions, but are defined as a natural discretization of the volume of a spherical polytope. We obtain precise asymptotics of the cone theta function attached to any simplicial cone, near a rational "cusp", and use these asymptotics to give new extensions of the Gram-relations for the solid angles of faces of a simple rational polytope.

• Prof. Ole Warnaar (The University of Queensland, Australia).

• Title: Virtual Koornwinder integrals and Rogers-Ramanujan identities

Abstract: The Koornwinder polynomials are a generalisation of the famous Askey-Wilson polynomials to the (non-reduced) root system BC_n. In this talk I will explain how the computation of virtual Koornwinder integrals, which are certain naturally occurring integrals in the BC_n theory, leads to Rogers-Ramanujan identities for characters of affine Lie algebras.

## Poster Session

There will be a poster session which will provide an opportunity to present/discuss your work with your peers in an informal settings. All the participants are encouraged to present a poster.

## Registration

Registration is free, but please register at Eventbrite. Any problems please send an email to Juliane Turner at Juliane.Turner@newcastle.edu.au.

## Workshop Programme

All the refreshments (coffee/tea/lunch) will be in the Room V215 and all the lectures will be in the room VG01.

## Friday 18th September

12:00-12:45Registration, refreshment and discussions (V215)
12:45-13:00Welcome (VG01)
13:00-14:00Jon Borwein
"Moments and densities of short walks in higher dimensions"
14:00-14:30Questions/coffee
14:30-15:30Shaun Cooper
"The Rogers-Ramanujan continued fraction"
15:30-16:00Questions/coffee
16:00-17:00Ole Warnaar
"Virtual Koornwinder integrals and Rogers-Ramanujan identities"
19:00-Dinner at Customs House Hotel

## Saturday 19th September

9:00-9:30Coffee
9:30-10:30Alan Haynes
"Perfectly ordered quasicrystals and the Littlewood conjecture"
10:30-11:00Questions/coffee
11:00-12:00Sinai Robins
"Asymptotics of cone theta functions, Gauss sums over parallelepipeds, and generalized Gram relations for polyhedra"
12:00-13:30Poster session and lunch
"Bessel Functions and Rogers-Ramanujan Identities"
14:30-15:00Questions/coffee
15:00-16:00Karl Dilcher
"Applications of generalized Stern polynomials"
16:00-Questions/coffee/discussions

The dinner will be at The Customs House Hotel, Newcastle http://www.customshouse.net.au/. Everything is fully subsidised by CARMA.

## Travel

The University of Newcastle is easily accessible via train or driving from Sydney; a train leaves every hour from Central Station to Newcastle ending at Newcastle Station, from where you can catch bus number 100, 225 or 226. There are also direct flights available from Brisbane, Canberra, Melbourne and Sydney.

Newcastle Airport is a small domestic airport with frequent daily flights to Melbourne, Brisbane and a few other places. There is a local bus (Port Stephens Coaches) from the airport to Newcastle town centre (\$4.60). A taxi from the airport to the city costs around \$60.

## Accommodation

There are two main areas with hotels and backpacker accommodation — Newcastle East and Honeysuckle, both in walking distance to each other, and close to train stations. Honeysuckle is a popular area with lots of restaurants with great views across the harbour.