Integrable Geometry: Smooth and Discrete

David Wood

  •  21 Jul 2025 - 1 Aug 2025
     8:00 am - 5:00 pm

Organisers:
Emma Carberry ( The University of Sydney)
Lyn Heller (Beijing Inst of Math Sciences and Applications (CN)
Udo Hertrich-Jeromin (TU Wien and The University of Sydney)
Franz Pedit (University of Massachussetts)
Wolfgang Schief (Univ of New South Wales)

Program Description:This program explores the interplay between smooth and discrete geometry in the presence of integrability in low dimensions, which has been highly successful in the study of global properties of surfaces arising from variational problems, including constant mean curvature, Willmore surfaces and harmonic maps in general. Smooth theories, rooted in geometric analysis/topology/algebraic geometry, utilize highly technical tools (e.g., global existence results of non-linear PDEs; existence of spectral curves with certain period properties) making it difficult to calculate explicit examples. Integrable discretisations of these theories often clarify their conceptual backdrop and at the same time may lead to fast and robust numerical algorithms. This enables researchers to carry out mathematical experiments which, on the one hand, collect evidence for predictions of the smooth theories; on the other hand, lead to new conjectures not detectable in the smooth setting. This feedback loop has proven to be extraordinarily successful in the deeper understanding|and resolution of conjectures|of integrable surfaces in 3-space.

This research program brings together key international researchers of both camps, discrete and continuous, for the first time. Even though there have been interactions between the two groups on various individual bases, the time seems ripe to have a more stuctured interaction among proponents of the two groups. Often, problems, conjectures, methods, and results developed in the discrete setting have independently found analogs in the smooth setting (a striking example of this is the very recent resolution of the Bonnet pair problem by Bobenko, Hoffmann, Sageman-Furnas). We believe that by now there exists a large enough body of work from both camps which justifies closer interactions and collaborations between those researchers. This also includes aspects of training and exposing PhD students and postdocs to these areas. In addition to applications in mathematical physics, discrete surface geometry is being used successfully in architectural design, construction engineering, and material science.

This research program, with its emphasis on few lectures, ample free time for collaborations, ad hoc presentations and discussions of the most recent and exciting results, provides the ideal venue to foster deeper connections and collaborations between those two camps.

Program Structure: TBD

Registration:

  • Deadline: TBD
  • Registration is by invitation only. If you are interested to participate in this program, please contact one of the organisers with your CV and research background.
  • Arrival date: 20 July 2025

ASSOCIATED EVENTS
MATRIX Wine and Cheese Afternoon 22 July 2025
On the first Tuesday of each program, MATRIX provides a pre-dinner wine and cheese afternoon. Produce is locally-sourced to showcase delicacies from the region.