Это --- crash-course (2 пары в неделю). Первая лекция --- в пятницу 13 апреля; курс будет проходить в апреле и начале мая. Расписание:
Лекция 5 (38K)|Лекция 6 (41K)|Лекция 7 (61K)|Лекция 8a (74K)|Лекция 9 (74K)]
Лекция 5 (17K)|Лекция 6 (18K)|Лекция 7 (24K)|Лекция 8a (28K)|Лекция 9 (29K)]
Why attempts to solve explicitly differential equations usually fail? The first rigorous answer was obtained in the 1840's by Liouville. Another approach to the problem of solvability of linear differential equations by quadrature was developed by Picard. Picard generalized Galois theory to the case of linear differential equations.
In this course I will describe a new third approach to the problem: I will construct a topological variant of Galois theory, in which the Riemann surface plays the role of algebraic field and the monodromy group plays the role of Galois group. It turns out that there are topological restrictions on the way the Riemann surface of a function represented by quadratures covers complex plane. I also will describe a multidimension variant of the theory which is very recent development. Topological argument explain also why algebraic equation which coefficient are rational functions in many complex variables is unsolvable in radicals if its Galois group is unsolvable.
This approach has the following advantage, beside its geometric clarity. The topological prohibitions concern the character of the multivaluedness of the function. They are valid not only for functions that are representable by quadratures, but also for a much wider class of functions. One obtains this class if one adds the meromorphic functions to the class of functions representable by quadratures, as well as all functions representable by formulas containing the above. Because of this, the topological results on nonrepresentability of functions by quadratures are stronger than the algebraic results. This is because composition of functions is not an algebraic operation.
I do not assume any knowledge of Galois theory, but acquaintance with analysis in one complex variable will be useful.