9 марта 2017 г. ПОМИ, Фонтанка, 27, Мраморный зал, 17 час.

Совместный коллоквиум ПОМИ РАН, Лаборатории Чебышева СПбГУ и
Санкт-Петербургского математического общества

академик ВИКТОР ВАСИЛЬЕВ (МИАН, ВШЭ)
“МНОГОМЕРНЫЙ ВАРИАНТ ЛЕММЫ НЬЮТОНА ОБ ИНТЕГРИРУЕМЫХ ОБЛАСТЯХ
И ТЕОРИЯ МОНОДРОМИИ”

A bounded domain in a Euclidean space defines a (two-valued) function on
the space of all affine hyperplanes in it: the volumes cut by the
hyperplanes from our domain. A domain is called algebraically integrable if
this function is algebraic. The famous Lemma XXVIII from Newton’s
“Principia” says that there are no integrable domains with smooth boundary
in the plane. We show that the same holds for the domains in any
even-dimensional space (while for the case of odd dimensions we have the
Archimedes’ counterexample). The proof is based on the (Picard-Lefschetz)
monodromy theory of complex algebraic varieties, and the theory of finite
reflection groups. This integrability problem is a sample of numerous
problems of mathematics and physics related with inte gral representations,
in which the methods of Picard-Lefschetz theory give us crucial information
on analytical properties (such as existence, ramification, number of, etc)
of the functions given by such representations.