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

- Laureate Prof. Jon Borwein (The University of Newcastle, Australia).
- Associate Prof. Shaun Cooper (Massey University, New Zealand).
- Prof. Karl Dilcher (Dalhousie University, Canada).
- Dr. Alan Haynes (The University of York, England).
- Prof. Mourad Ismail (The University of Central Florida, USA).
- Prof. Sinai Robins (Brown University, USA).
- Prof. Ole Warnaar (The University of Queensland, Australia).

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

Abstract: Following Pearson in

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

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.

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).

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 orderedâ€™ 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 orderedâ€™ quasicrystals, and explain how this line of thought quickly leads to a question which turns out to be equivalent to the Littlewood conjecture.

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.

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.

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.

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

## Friday 18th September | |

12:00-12:45 | Registration, refreshment and discussions (V215) |

12:45-13:00 | Welcome (VG01) |

13:00-14:00 | Jon Borwein "Moments and densities of short walks in higher dimensions" |

14:00-14:30 | Questions/coffee |

14:30-15:30 | Shaun Cooper "The Rogers-Ramanujan continued fraction" |

15:30-16:00 | Questions/coffee |

16:00-17:00 | Ole Warnaar "Virtual Koornwinder integrals and Rogers-Ramanujan identities" |

19:00- | Dinner at Customs House Hotel |

## Saturday 19th September | |

9:00-9:30 | Coffee |

9:30-10:30 | Alan Haynes "Perfectly ordered quasicrystals and the Littlewood conjecture" |

10:30-11:00 | Questions/coffee |

11:00-12:00 | Sinai Robins "Asymptotics of cone theta functions, Gauss sums over parallelepipeds, and generalized Gram relations for polyhedra" |

12:00-13:30 | Poster session and lunch |

13:30-14:30 | Mourad Ismail "Bessel Functions and Rogers-Ramanujan Identities" |

14:30-15:00 | Questions/coffee |

15:00-16:00 | Karl 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.

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.

- Novotel Newcastle Beach
- Noah's on the Beach
- YHA Newcastle Beach (backpackers, great location and nice place. discount for HI members)
- Terraces for Tourists (good choice if coming with family etc)

- Crowne Plaza
- Chifley (two locations — Honeysuckle and next to the Newcastle train station)
- Ibis (on Hunter St, can walk or take a bus along Hunter St to the conference venue)

The website wotif.com can sometimes have good deals.

Another backpacker option (with a pool) is Backpackers Newcastle on Denison St (Hamilton).