Description: The aim of the seminars is to bring together people who are interested in integrable systems and related topics.  We welcome anyone interested in participating.  The seminars are planned to be informal and at an introductory level, e.g., understandable for honours/PhD students and beyond, with questions and discussions encouraged.  The seminars will be held on Fridays and alternated weekly between UNSW and USYD.  Our first topic will be elliptic difference equations, with possible future topics depending on the interest of participants. 

If you have any questions or wish to be added to the mailing list please contact the organisers: Andrew Kels or Pieter Roffelsen (contact details below).

Time: Fridays 14:00-16:00 AEST (2x40 mins with 10 min break and 30 mins questions/discussion)

Place: We will alternate the seminars between UNSW and USYD:

• UNSW, Anita B Lawrence (H13) Room 3085 (google maps) (room location)

• USYD, SMRI Seminar Room at Macleay Building (A12) Room 301 (google maps)

Schedule: See below

Contact:  a.kels@unsw.edu.au and pieter.roffelsen@usyd.edu.au 

Elliptic Difference Equations

Special functions are ubiquitous in pure and applied mathematics and have been studied for over 300 years. During the mid-nineteenth century, it became clear that many of them have natural one-parameter generalisations involving a parameter q, giving rise to the theory of q-special functions and corresponding q-difference equations. Over the past few decades, a picture has started to emerge that this theory is but a stepping stone to a potentially even richer theory of elliptic special functions and corresponding elliptic difference equations.

In this seminar series, we delve into the fascinating new world of elliptic difference equations. We start by covering the essential mathematical concepts that are required to understand the main special functions involved,  elliptic hypergeometric integrals.  Once we get a handle on these, we proceed to study a special class of elliptic difference equations introduced by Rains.  In particular, we focus on Rains' construction of explicit solutions for some of these equations in terms of elliptic analogs of orthogonal polynomials. We then turn our attention to Krichever's general analytic theory of elliptic difference equations.

References:

• G. Gasper and M. Rahman - Basic Hypergeometric Series (2nd ed), Cambridge University Press, 2004  (link) 

• G. Heckman - Hypergeometric Functions, lecture notes, 2015 (link)

• G. Andrews, R. Askey, and R. Roy - Special Functions, Cambridge University Press, 1999 (link)

• I.M. Krichever - Analytic theory of difference equations with rational and elliptic coefficients and the Riemann–Hilbert problem,  RMS, 2004 (link)

• V.P. Spiridonov - Essays on the theory of elliptic hypergeometric functions,  RMS, 2008 (link) 

• E.M. Rains - An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations),  SIGMA, 2011 (link)

• Chris M. Ormerod and E.M. Rains - An Elliptic Garnier System, CMP, 2017 (link)

• M. Noumi - Remarks on τ-functions for the difference Painlevé equations of type E8, Adv. Stud. Pure Math , 2018 (link)

Schedule:

• 1. August 30, UNSW                  Speaker: Pieter Roffelsen       Topics: classical hypergeometric functions, ODEs

• 2. September 6, USYD              Speaker: Pieter Roffelsen       Topics: integral representations and monodromy

• 3. September 13, USYD           Speaker: Andrew Kels               Topics: beta function, Barnes integrals, q-hypergeometric functions

• 4. September 20, UNSW         Not held 

• 5. September 27, UNSW         Speaker: Andrew Kels                Topics: elliptic gamma function, elliptic hypergeometric functions

• 6. October 4,  USYD                   Speaker: Andrew Kels                Topics: elliptic beta integral                             

• 7. October 11, UNSW               Speaker: Pieter Roffelsen        Topics: discrete linear systems and p-theta functions

• 8. October 18, USYD                 Speaker: Pieter Roffelsen        Topics: an elliptic analog of the Gauss hypergeometric function                                      

• 9. October 25, UNSW               Speaker: Pieter Roffelsen        Topics: elliptic difference equations and isomonodromy

• 10. November 1, USYD            Speaker: Andrew Kels                Topics: W(E7) transformation and theta functions

• November 8, UNSW                  Speaker: Andrew Kels                Topics: BC1-symmetric theta functions and elliptic hypergeometric integrals

• November 15, USYD                 Speaker: Andrew Kels                Topics: Rains's elliptic difference equations