# The 3rd summer school on geometry of differential equations

September 8–12, 2014

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 Tromsø, Tromsø, 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 Wilczyński invariants), contact differential invariants for Monge–Ampère 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–Schröodinger 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 ${\displaystyle \mathbb {C} P^{2}}$.
• Curvilinear orthogonal coordinate systems in ${\displaystyle \mathbb {R} ^{n}}$ 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.

## Organization

The summer school will take place in Malenovice, hotel Petr Bezruč, 49°33'59.360"N 18°25'32.670"E in the foothill of Lysá Hora, the highest mountain (1 323 m above the sea level) of the Moravian–Silesian Beskydy in the Eastern part of the Czech Republic. The school will last for five days with a total of 36 academic hours of lectures. The teaching will be in English.

The participants who have completed the courses in their entirety will receive certificates.

## Costs

International participants will pay subsistence costs (accommodation and meals) 50€ per person per day on their own. There is no tuition fee.

Doctoral students and academics from Czech universities (excluding the Prague region) are eligible for full support from the European Social Fund under project CZ.1.07/2.3.00/20.0002. Please contact school-gde@math.slu.cz.

## Registration

Please contact the organizers at school-gde@math.slu.cz as soon as possible, because of limited capacity of the school.

## Travel

The arrival day is Sunday, September 7, the departure day is Saturday, September 13. Visit Czech railways timetable to plan the journey between the nearest international airports (Prague, Vienna, Katowice) and Frýdlant nad Ostravicí, the nearest railway station (7 km) from the hotel. For bus service between the Vienna and Katowice airports and Ostrava or Frýdek--Místek see Tiger express.

On Sundays, the bus service No. 860345 from Frýdlant nad Ostravicí to the hotel departs at 14:50 and 18:50. Taxi is available all time.