Васильев Виктор Анатольевич

Homology of spaces of knots in any dimensions.

2001 · ARTICLE · en

I shall describe the recent progress in the study of cohomology rings of spaces of knots in R^n, H^∗({knots in R^n}), with arbitrary n>23. ‘Any dimensions’ in the title can be read as dimensions n of spaces R^n, as dimensions i of the cohomology groups H^i, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero-dimensional cohomology classes of the space of knots in R^3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero-dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. There- fore, I shall deal almost completely with results in higher (or arbitrary) dimensions.

Лагранжевы и лежандровы характеристические классы

2000 · BOOK · ru

Ветвящиеся интегралы

2000 · BOOK · ru

Как вычислять гомологии пространств неособых алгебраических проективных гиперповерхностей

1999 · ARTICLE · ru

Описан общий метод вычисления групп когомологий пространства неособых алгебраичес­ких гиперповерхностей степени d в СР^n. С помощью этого метода вычислены группы рацио­нальных когомологий таких пространств с n = 2, d ≤ 4 и n = 3 = d, а также пространства невырожденных квадратичных векторных полей в С^3.

Topological order complexes and resolutions of discriminant sets

1999 · ARTICLE · en

If elements of a partially ordered set run over a topological space, then the corresponding order complex admits a natural topology, providing that similar interior points of simplices with close vertices are close to one another. Such topological order complexes appear naturally in the conical resolutions of many singular algebraic varieties, especially of discriminant varieties, i.e. the spaces of singular geometric objects. These resolutions generalize the simplicial resolutions to the case of non-normal varieties. Using these order complexes, we study the cohomology rings of many spaces of nonsingular geometrical objects, including the spaces of nondegenerate linear operators in R^n, C^n or H^n, of homogeneous functions R^2 --> R^1 without roots of high multiplicity in RP^1, of nonsingular hypersurfaces of fixed degree in CP^n, and of Hermitian matrices with simple spectra.

Об одной задаче М.Э.Казаряна

1999 · ARTICLE · ru

On finite order invariants of triple point free plane curves.

1999 · ARTICLE · en

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves S^1 → R^2. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as the calculation of knot invariants is based on the study of chord diagrams and connected graphs. E.g., the simplest such invariant is of degree 4 and corresponds to the diagram consisting of two triangles with alternating vertices in a circle in the same way as the simplest knot invariant (of degree 2) corresponds to the 2-chord diagram ⨁. Also, following V.I.Arnold and other authors we consider invariants of {\em immersed} triple points free curves and describe similar techniques also for this problem, and, more generally, for the calculation of homology groups of the space of immersed plane curves without points of multiplicity ≥k for any k≥3.

Topology of two-connected graphs and homology of spaces of knots.

1999 · ARTICLE · en

We propose a new method of computing cohomology groups of spaces of knots in R^n, n≥3, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order ≤3. As a byproduct we define the higher indices, which invariants of knots in R^3 define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.

On r-neighbourly submanifolds in R^N

1998 · ARTICLE · en

A submanifold M ⊂ R^N is r-neighborly if for any r points in M there is a hyperplane, supporting M and touching it at exactly these r points. We prove that the minimal dimension Δ(k,r) of the Euclidean space, containing a stably r-neighborly submanifold, is asymptotically not smaller than 2kr−k

Гомологии пространств однородных полиномов в R^2 без многократных нулей

1998 · ARTICLE · ru

Для любых натуральных d≥k≥2 мы вычисляем гомологии пространства однородных полиномов R^2 —> R степени d, не обращающихся в 0 с кратностью ≥k на вещественных прямых. При k = 2 эта задача дает простейшую ситуацию, когда инварианты "конечного порядка" неособых объектов не являются полной системой инвариантов.