Публикации

Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.

This is the second part of a 2-year course of abstract algebra for students beginning a professional study of higher mathematics.1 This textbook is based on courses given at the Independent University of Moscow and at the Faculty of Mathematics at the National Research University Higher School of Economics. In particular, it contains a large number of exercises that were discussed in class, some of which are provided with commentary and hints, as well as problems for independent solution that were assigned as homework.Working out the exercises is of crucial importance in understanding the subject matter of this book.

We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(D 8) τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III(D 8) from the special representations of the Virasoro algebra of the highest weight (n  +  1/4)2.

В статье чисто комбинаторным способом доказана топологическая рекурсия для задачи перечисления двухцветных карт (являющихся двойственными объектами к детским рисункам). Кроме этого, доказано уравнение квантовой спектральной кривой для данной задачи. Предложено обобщение вышеописанных результатов на случай четырехцветных карт.

In this paper we provide some affirmative results and some counterexamples for a solution of the splitting problem for n multivalued mappings, n>2.

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.