# А.Г.Хованский

## Топологический вариант теории Галуа.

Это --- crash-course (2 пары в неделю). Первая лекция ---
в пятницу 13 апреля; курс будет проходить в апреле и начале
мая. Расписание:

- Пятница, 17.30, ауд. 309.
- Среда, 17.30, ауд. 308.

## Записки лекций (Lecture notes)

### Postscript

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### Zipped postscript

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## Программа

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.