Event info

Intensive Lectures by Professor Schief (UNSW, Australia)

Lecturer: Professor Wolfgang Karl Schief (University of New South Wales, Australia)

Title: Discrete Differential Geometry: Application to Classical Integrable Shell Membrane Theory

Schedule:

1: June 14 (Mon.) 8:40am - 10:10am (JST)
2: June 15 (Tue.) 8:40am - 10:10am (JST)
3: June 16 (Wed.) 8:40am - 10:10am (JST)
4: June 17 (Thurs.)8:40am - 10:10am (JST)
5: June 18 (Fri.) 8:40am - 10:10am (JST)

Language: English

Registration is required on Zoom at

https://zoom.us/meeting/register/tJEpfumvqzIjHNOXr4jyPDA3V0CBseb05W-V

The details of the Zoom connection to this course will be sent upon registration.

Outline:

This course is based on the work of the lecturer and his collaborator Professor C. Rogers which followed their discovery in 2003 that a well-known system of classical shell theorydescriptive of membranes in equilibrium is, in fact, integrable if certain natural assumptions are made. Here, integrability of partial differential equations refers to their amenability to the powerful techniques of soliton theory. The corresponding shell membranes are shown to have geometries within the integrable class of so-called O surfaces. The membrane O surfaces include inter alia minimal, constant mean curvature, constant Gaussian curvature and, more generally, linear Weingarten surfaces, as well as canal surfaces and Dupin cyclides.

It turns out that the classical theory may be discretized in such a manner that integrability is preserved. From a geometric point of view, this fact is not surprising since an integrability-preserving discretisation of O surface theory is known. However, it turns out that, a priori, a physical interpretation of this discretisation procedure is not available and, hence, a different (physical) route has to be taken to develop a discrete model which makes sense in the context of physics. Remarkably, that route also leads to discrete O surface theory, but in an unexpected manner. Moreover, a practical application of the fundamental concept of multi-dimensional consistency will be discussed.

The course is meant to be self-contained to a large extent so that both the classical and discrete theories are explained and/or developed from scratch. Discrete versions of various notions and objects of classical differential geometry will be introduced and, thereby, relevant fundamental ideas of discrete differential geometry reviewed. Now and then, we will also come across curious classical objects of elementary geometry and their generalisations such as Euler lines and nine-point circles which make an appearance in a physical context. If time permits, we may also briefly digress to mention related areas such as nonlinear elastic shell systems in liquid crystal theory and their relation to generalised Willmore surfaces and the integrability of infinitesimal and finite deformations of polyhedral surfaces.

References:

1. C. Rogers and W.K. Schief, B?cklund and Darboux
Transformations. Geometry and Modern Applications in Soliton
Theory, Cambridge Texts in Applied Mathematics, Cambridge
University Press (2002).

2. W.K. Schief and B.G. Konopelchenko, On the unification of
classical and novel integrable surfaces: I. Differential
geometry, Proc. R. Soc. London A 459 (2003) 67-84.

3. W.K. Schief, On the unification of classical and novel
integrable surfaces: II. Difference geometry,
Proc. R. Soc. London A 459 (2003) 373-391.

4. C. Rogers and W.K. Schief, On the equilibrium of shell
membranes under normal loading. Hidden integrability,
Proc. R. Soc. London A 459 (2003) 2449-2462.

5. W.K. Schief, Integrable discrete differential geometry of
‘plated’ membranes in equilibrium, Proc. R. Soc. London A
461 (2005) 3213-3229.

6. W.K. Schief, M. Kl?man and C. Rogers, On a nonlinear elastic
shell system in liquid crystal theory: generalised Willmore
surfaces and Dupin cyclides, Proc. R. Soc. London A 461 (2005)
2817-2837.

7. W.K. Schief, On a maximum principle for minimal surfaces and
their integrable discrete counterparts, J. Geom. Phys. 56 (2006)
1484-1495.

8. W.K. Schief, A.I. Bobenko and T. Hoffmann, On the
integrability of infinitesimal and finite deformations of
polyhedral surfaces, in A.I. Bobenko, P. Schr?der, J.M. Sullivan
and G.M. Ziegler, eds, Discrete Differential Geometry,
Oberwolfach Seminars 38, Birkh?user (2008) 67-93.

9. W.K. Schief, Integrable structure in discrete shell membrane
theory, Proc. R. Soc. London A 470 (2014) 20130757 (22 pp).