The 1st summer school on geometry of differential equations

The summer school will take place 17-21 September 2012 and will be held in the Institute  of  Mathematics  of Silesian University in Opava, Czech Republic.

Applications should be sent to: school-gde@math.slu.cz

There is no formal deadline for applications, but please contact the organizers as soon as possible, because of limited capacity of the school.

There will be two parallel courses:

Basic Course: Symmetry and conservation laws

(Raffaele Vitolo, University of Salento, and Giovanni Moreno, Silesian University in Opava)

Syllabus:

1. The geometry of jet spaces: Cartan distribution, prolongation of diffeomorphisms and vector fields.
2. Differential equations as submanifolds of jet spaces. # Symmetries as distinguished vector fields on differential equations.
3. Computation of symmetries, also by the use of specialized symbolic computation software.
4. The geometry of the infinite order jet space and infinitely prolonged equations.
5. Higher symmetries as distinguished vector fields on infinitely prolonged differential equations.
6. Applications of symmetries to the solution of ODEs and PDEs, with computations.
7. Conservation laws as distinguished differential forms on differential equations. Computation of conservation laws, also by the use of specialized symbolic software.

Advanced Course: Recursion operators

(Joseph Krasil'shchik, Silesian University in Opava and Moscow Independent University, Michal Marvan, Silesian University in Opava, and Alexander Verbovetsky, Moscow Independent University)

Syllabus:

1. Introduction: finite and infinite jets, the Cartan distribution, equations and prolongations. Solutions.
2. Symmetries. Their description via generating functions. The Jacobi bracket. Linearization (the Frechet derivative) and defining equation for symmetries.
3. Examples: the Burgers and Korteweg-de Vries (KdV) equations.
4. An application: 1-solitons as invariant solutions of the KdV equation.
5. Back to Item 3 how to finish the computation of symmetries for the KdV equation. Lenard's recursion operator.
6. What happens when applying Lenard's recursion operator to $(x,t)$-dependent symmetries? A naïve way to introduce nonlocal variables. Other examples of nonlocal constructions.
7. Geometrization: coverings. More examples. Relation to conservation laws. Zero-curvature representations.
8. The Wahlquist-Estabrook algebra.
9. Examples of computations: the Burgers and KdV equations. The Cole-Hopf and Miura transformations.
10. An application: Bäcklund transformations. Multi-soliton solutions of the KdV equation. Multi-kink solutions of the sine-Gordon equation. Bianchi's permutability theorem.
11. Back to Item 6 what did we obtain using Lenard's operator? Nonlocal symmetries and shadows.
12. The reconstruction theorem. The universal Abelian covering. Master-symmetries and the commutator problem.
13. Back to recursion operators: The tangent covering $\tau$. Symmetries as its holonomic sections. Geometric viewpoint on recursion operators: auto-Bäcklund transformations of $\tau$. Examples.
14. An alternative viewpoint: shadows of symmetries in $\tau$. Examples. "Canonical" conservation laws in the tangent covering.
15. Commutativity of hierarchies: hereditary recursion operators and the Nijenhuis torsion.
16. Perspectives.
17. Recursion operators for a class of multidimensional dispersionless PDEs: an explicit construction (a lecture by Artur Sergyeyev)

The basic course is aimed at the beginners, with the pace and style of presentation to match. The advanced course is aimed at the students who are already familiar with the contents of the basic course.

The courses will provide students with a comprehensive presentation of the respective subjects, and introduce them to the basic motivations, methods and results of the relevant field of study. The participants will also be informed about the open problems in the field.

The courses will include training sessions, in the course of which the participants will learn to use the software for the computer-aided calculation of symmetries and conservation laws, resp. of coverings and recursion operators.

The Summer School will take place in the North-Eastern part of the Czech Republic (the exact location will be announced later) and will last for five days with a total of 40 academic hours of lectures and training sessions.

The teaching will be in English and will take place in the form of lectures in the morning and training sessions in the afternoon.

In the course of the training sessions the participants will solve (sets of) problems given to them and submit their solutions to the instructor who will provide advice and feedback on these if need be.

The successful participants will receive a certificate; the latter will be awarded on the basis of performance at the training sessions.

Participation in the school is free of charge.

Travel and subsistence costs are to be covered by the participants themselves.

The school will be hosted in Davidův mlýn, see the [http://maps.google.cz/maps?ll=49.82756,17.682238&spn=0.010825,0.017467&gl=c z&brcurrent=5,0,0&t=h&z=16 location on Google Maps]

This School is the first of three summer schools on the geometry of differential equations supported by the European Social Fund under the project CZ.1.07/2.3.00/20.0002.

Two further summer schools on similar subjects will take place in 2013 and 2014.