# The 2nd summer school on geometry of differential equations

The summer school will take place 9-13 September 2013 and will be held in the Institute of Mathematics of Silesian University in Opava, Czech Republic.

The school will be hosted in the Hotel Dlouhé Stráně.

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: Conservation laws - theory and computation

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

Syllabus:

• Definition of jet space and its Cartan distribution
• Differential equations and their Cartan distribution; symmetries
• Horizontal forms, horizontal differential, horizontal de Rham cohomology on jet spaces
• Geometric theory of conservation laws
• Correspondence of a conservation law with a uniquely defined vector function, the generating function of the conservation law: theoretical aspects
• Examples of computations
• Elements of calculus of variations with differential forms; Noether’s theorem
• Variational equations and correspondence between symmetries and conservation laws
• Integration of ODEs by quadratures using symmetries and conservation laws
• The method of characteristics and integration of first-order PDEs
• Computation of conservation laws, also by the use of specialized symbolic software

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

Syllabus:

• Symplectic and Poisson manifolds
• The Schouten bracket
• Finite-dimensional Hamiltonian formalism. Integrability
• Infinite jets and infinite prolongations of PDEs
• Cartan distribution and symmetries
• Horizontal and Cartan forms. Conservation laws
• $\mathcal{C}$-differential operators. Adjont operators. The Green formula
• The Euler operator. Cosymmetries
• Variational bivectors. Variational Poisson structures
• The variational Schouten bracket
• Compatible Poisson structures. The Magri scheme
• Coverings. Nonlocal symmetries and shadows. Bäcklund transformations. Tangent and cotangent coverings
• Poisson structures as shadows of symmetries in the cotangent covering
• Variational forms. Variational symplectic structures
• Symplectic structures as shadows of cosymmetries in the tangent covering
• Variational Poisson and symplectic structures as Bäcklund transformations between tangent and cotangent coverings
• Examples
• Infinitesimal deformations of Poisson structures and compatibility

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 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 36 academic hours of morning lectures and afternoon training sessions.

The teaching will be in English.

In the course of the training sessions the participants will solve (sets of) problems and submit their solutions. The instructor will provide advice and feedback on these if need be. Included will be short tutorials to use the software for the computer-aided calculations.

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 Silesian university in Opava can, under certain conditions, offer some financial support to the participants from the Czech Republic under the European Social Fund project CZ.1.07/2.3.00/20.0002 and those potentially eligible should contact school-gde@math.slu.cz for details.

This School is the second 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. The previous school held in 2012. The next one will take place in 2014.