Workshop on Geometry of Lagrangian Grassmannians and nonlinear PDEs

IMPAN, Warsaw, 5–9 Sep. 2016.

(From left to right: Jan Gutt, Jarosław Buczyński, Witold Respondek, Andrey Krutov, Emilio Musso, Lorenzo Nicolodi, Yewon Jeong, Giovanni Moreno, Gary Jensen, Abraham David Smith, Gianni Manno, Francesco Russo, Benjamin McKay, Katja Sagerschnig, Hélder Larraguível, Katarzyna Karnas, Jun-Muk Hwang, Dennis The, Andriy Panasiuk. The group picture is available here at full resolution)

Invited speakers.

Gary Jensen, Washington University, USA,
Abraham Smith, University of Wisconsin-Stout, USA,
Benjamin McKay, University College Cork, IE,
Francesco Russo, University of Catania, IT,
Emilio Musso, Polithecnic of Turin, IT,
Jarosław Buczynski, IMPAN, Warsaw, PL,
Paweł Nurowski, CFT PAN, Warsaw, PL,
Dennis The, University of Tromso, Tromso, NO.

Aim.

A consistent geometric theory of nonlinear PDEs can obtained by using jet bundles and their natural structures. The so–obtained framework is part of a larger scheme, involving prolongations of contact manifolds, which allows to regard any PDE as a bundle of hypersurfaces in a isotropic Grassmannian manifold. In particular, second–order nonlinear PDEs correspond to hypersurfaces in the Lagrangian Grassmannian.

The workshop is based on the following three premises.

Lagrangian Grassmannians are interesting manifolds on their own, independently on their relationship with nonlinear PDEs: they display a rich intrinsic geometry, and their hypersurfaces can be studied through old and recent methods coming from unexpectedly distant areas.
There are important open problems concerning nonlinear PDEs, like those of obtaining (partial) classifications, or proving (non) equivalence theorems, which will clearly benefit from an appropriate geometric reformulation, not necessarily based on jet spaces or contact manifolds.
An important class of nonlinear PDEs, which occurs ubiquitously and transversally in very distant areas of Mathematics and Mathematical Physics, is that of integrable systems. In spite of their importance, is not yet clear what is the correct, general and intrinsic geometric counterpart to the notion of integrability.

The workshop is aimed at clarifying these three points, through the three–hours mini–courses listed below, to present some particular ongoing research in these directions, through four one–hour seminars (see below), and to bring together experts with the necessary complementary skills to laid down effective and realistic plans for attacking unsolved problems.

Minicourses

The Lie quadric (based on Chap. 14 of this book; lecture notes here), by Gary Jensen (9:30-10:30).
An introduction to exterior differential systems: the Cartan-Kahler theory (lecture notes here), by Benjamin McKay.
Advanced topics in EDS: rank-one varieties, Guillemin normal forms and hydrodynamic integrability (lecture notes, even in compactified form), by Abraham David Smith.
Projective duality, by Francesco Russo (16:00-17:00).

Seminars

Conformal geometry of timelike curves in the (1+2)-Einstein universe, by Emilio Musso
Paweł Nurowski, TBA
Homogeneous contact manifolds and their varieties of minimal rational tangents, by Jarosław Buczynki
Exceptionally simple PDE, by Dennis The

Contributing Talks

Two half-hour time slots for contributing talks are scheduled for Thursday afternoon. People wishing to contribute are kindly requested to contact the organisers, and their request will be taken into cosideration.

Contact geometry as a source of new integrable PDE systems in four independent variables, by Artur Sergyeyev (Silesian University in Opava, CZ).
Laguerre deformation and Lawson correspondence, by Lorenzo Nicolodi (University of Parma, IT).
Veronese webs and nonlinear PDEs, by Andriy Panasyuk (University of Warmia and Mazury, Olsztyn, PL) (joint work with Boris Kruglikov).

Discussions

After the seminars, a roundtable for putting forward the existing cooperations and forge new one will be organised.

Timetable

The workshop will take place at IMPAN, room 403, fourth floor.

Program

Available here.

Proceedings & Lecture Notes

The Institute of Mathematics of the Polish Academy of Sciences (IMPAN) has its own publishing house, which also accepts lecture notes (IMPAN Lecture Note irregular series) and proceedings (Banach Centre Publciations regular series) from courses and workshops held within its premises. Lecturers and speakers are warmly encorauged to put their notes in a submitteable form, so that, at the end of the meeting, the organisers will be able to propose a special volume to the editors.

Accommodation

The participation to the workshop is free. Nevertheless, people interested into taking part to the workshop are kindly invited to notify one of the organizers. The closest lodging facilities to the workshop venue are the IMPAN's own guestrooms (though it may be hard to find an unbooked room) and the nearby Hotel OSiR Polna (unfortunately the webpage is only in Polish).