The 3rd summer school on geometry of differential equations

Organized by the Mathematical Institute of Silesian University in Opava, the school is the third in a series supported by the European Social Fund under the project CZ.1.07/2.3.00/20.0002.

Scientific Programme

Two courses will take place during the summer school.

Differential Invariants

V.V. Lychagin (University of Troms\o, Troms\o, Norway)

• Invariants in Algebra, Geometry and Analysis.
• Actions of algebraic Lie groups and Lie algebras. Polynomial and rational invariants. Geometrical and categorical factors. Hilbert and Rosenlicht theorems.
• Geometry of jet spaces and PDEs. Algebraic structures on PDEs. Algebraic PDEs.
• Jets of diffeomorphisms and Lie algebraic pseudogroups, their actions on PDEs.
• Polynomial and rational differential invariants. Lie--Tresse theorem.
• Differential syzygies and factor equations.
• Various methods for computation differential invariants: computer, moving frames, connections.
• Applications:
  • Algebra: differential contra algebraic invariants; differential invariants in algebraic problems.
  • Geometry: Klein geometries and invariants of geometrical quantities, Riemann and Einstein manifolds, etc.
  • Analysis: Differential invariants for ODEs (Tresse and Wilczy\'{n}ski invariants),

contact differential invariants for Monge--Amp\`ere equations, projective and conformal invariants in neurogeometry and image recognition.

Riemann Surfaces and Soliton Equations

A.E. Mironov (Sobolev Institute of Mathematics, Novosibirsk, Russia)

• Riemann surfaces. Holomorphic differentials. Periods of holomorphic differentials, Jacobi variety of the Riemann surface.
• Abel's theorem. Theta functions. The Riemann--Roch theorem. Baker--Akhiezer functions.
• Finite-gap solutions of the Korteweg--de Vries equation and Kadomtsev--Petviashvili equation.
• 2D--Schr\"odinger operators integrable on one energy level. Algebro-geometric solutions of the Novikov--Veselov equation.
• Algebro-geometric solutions of the Tzitzeica equation. Minimal Lagrangian tori in $\mathbb {C} P^{2}$.
• Curvilinear orthogonal coordinate systems in $\mathbb {R} ^{n}$ and algebro-geometric solutions of the WDVV equation.

The courses are aimed at the beginners, with the pace and style of presentation to match. The lecturers will provide students with a comprehensive presentation of the respective subjects, and introduce them to the basic motivations, methods and results of the relevant fields of study. The participants will also be informed about open problems in the field.