Mathematics

Minicourse (approx. 5 lectures): In these series of lectures we show how conformal structures of various dimensions and signatures appear in the theory of ordinary differential equations. In 1905 Karl Wuenschmann and, recently, Ezra T. Newman observed that certain contact equivalent classes of 3rd order ordinary differential equations are in one-to-one correspondence with 3-dimensional Lorentzian conformal geometries. In the lecture we describe the Wuenschmann-Newman result and show other examples of differential equations considered modulo various transformations of variables which are equivalent to conformal geometries. In particular, we discuss: 1) a relation between 3rd order ODEs considered modulo point transformations of variables and 3-dimensional Lorentzian Weyl and Einstein-Weyl geometries 2) a relation between 2nd order ODEs considered modulo point transformations and 4-dimensional split-signature conformal geometries with Cartan normal conformal connection reducible to an sl(3,R) connection 3) a relation between 2nd order Monge equations and (+,+,-,-,-) signature conformal geometries with Cartan normal conformal connection reducible to a connection with values in the Lie algebra of the noncompact form of the exceptional group G_2 The Cartan method of equivalence, which is the main technique to obtain the above results, will be discussed at the first lecture, so that a person not familiar with this approach be able to follow the course. Following lectures in February 4, 18, 21 and 23 (14:30)