Branching volumes and groups of reflections

The task of the dependence of the volume, cut off by the plane on a limited body, is considered to be the archimed and Newton, on this plane. In particular, we will prove the hypothesis of V. I. Arnold that this volume cannot algebraically depend on the body with a smooth boundary in the even -ry space on the coefficients of the equation of the plane, and we give geometric obstacles to such algebrainity in the odd -free case. The book tells about the history of the issue and methods that allow solving such and similar problems (including the tasks of solving equations in radicals): the theory of monodrome, analytical continuation, groups of transformations generated by reflections, and topology of complex diversity. The book is based on lecture courses read at the LCSM in 2013 and 2014. For high school students and junior students