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Victor A. Vassiliev

Publications and source records attributed to Victor A. Vassiliev.

At least 19 recordsLinked to original sources
arXiv2026

Complements of discriminants of real singularities of type $X_{10}$

A conjecturally complete list of connected components of complements of discriminant varieties (aka wave fronts) of smooth function singularities of type $X_{10}^3$ and $X_{10}^1$ is presented; it are the first examples of not semi-quasihomogeneous plane function singularities. It is shown that the complements of discriminants of singularities of classes $X_9^{\pm}$ and $X_{10}^1$ have non-trivial 1-dimensional homology groups, in contrast to all simple singularity classes.

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arXiv2023

Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions

For any finite-dimensional vector space ${\mathcal F}$ of continuous functions $f:{\mathbb R}^1 \to {\mathbb R}^1$ consider subspaces in ${\mathcal F}$ defined by systems of {\em equality conditions} $f(a_i) = f(b_i)$, where $(a_i, b_i)$, $i=1, \dots, n$, are some pairs of points in ${\mathbb R}^1$. It is proved that if $\dim {\mathcal F} < 2n-I(n)$, where $I(n)$ is the number of ones in the binary notation of $n$, then there necessarily are independent systems of $n$ equality conditions defining the subspaces of codimension greater than $n$ in ${\mathcal F}$. We also prove lower estimates of the sizes of the inevitable drops of the codimensions of these subspaces. These estimates are then applied to knot theory (in which systems of equality conditions are known as {\em chord diagrams}). The inevitable presence of complicated non-stable terms in sequences of spectral sequences calculating the cohomology groups of spaces of knots is proved. Keywords: Chord diagram, configuration space, characteristic class

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arXiv2020

New Examples of Irreducible Local Diffusion of Hyperbolic PDE's

Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.

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arXiv2019

New examples of locally algebraically integrable bodies

Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even and is very rarely algebraic if $N$ is odd: all known examples (found essentially by Archimedes \cite{arch} for $N=3$) of bodies defining algebraic volume functions are these of ellipsoids. We demonstrate a large new series of {\em locally} algebraically integrable bodies with algebraic boundaries in the spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones at least in some open domains of the space of hyperplanes intersecting the body. Further, we discuss (also following Archimedes) the extension of this problem to non-compact bodies

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arXiv2018

Lacunas and local algebraicity of volume functions

The volume cut off by a hyperplane from a bounded body with smooth boundary in $R^{2k}$ never is an algebraic function on the space of hyperplanes: for k=1 it is the famous lemma XXVIII from Newton's Principia. Following an analogy of these volume functions with the solutions of hyperbolic PDE's, we study the local version of the same problem: can such a volume function coincide with an algebraic one at least in some domains of the space of hyperplanes, intersecting the body? We prove some homological and geometric obstructions to this integrability property. Based on these restrictions, we find a family of examples of such "locally integrable" bodies in Euclidean spaces

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arXiv2017

Multiplicities of bifurcation sets of Pham singularities

The local multiplicities of the Maxwell sets in the spaces of versal deformations of Pham holomorphic function singularities are calculated. A similar calculation for some other bifurcation sets (generalized Stokes' sets) defined by more complicated relations between the critical values is given. Aplications to the complexity of algorithms enumerating topologically distinct morsifications of complicated real function singularities are discussed.

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arXiv2016

Local Petrovskii lacunas at parabolic singular points of wavefronts of strictly hyperbolic PDE's

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.

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arXiv2016

Linking numbers in non-orientable 3-manifolds

The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.

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arXiv2014

Monodromy of complete intersections and surface potentials

Following Newton, Ivory and Arnold, we study the Newtonian potentials of algebraic hypersurfaces in $R^n$. The ramification of (analytic continuations of) these potential depends on a monodromy group, which can be considered as a proper subgroup of the local monodromy group of a complete intersection (acting on a {\em twisted} vanishing homology group if $n$ is odd). Studying this monodromy group we prove, in particular, that the attraction force of a hyperbolic layer of degree $d$ in $R^n$ coincides with appropriate algebraic vector-functions everywhere outside the attracting surface if $n=2$ or $d=2$, and is non-algebraic in all domains other than the hyperbolicity domain if the surface is generic and $d\ge 3$ and $n\ge 3$ and $n+d \ge 8$. (Later, W. Ebeling has removed the last restriction $d+n \ge 8$).

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arXiv2014

There are no algebraically integrable ovals in even-dimensional spaces

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in $R^2$, this is Lemma XXVIII from Newton's "Principia".

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arXiv2014

On Finite Order Invariants of Triple Points Free Plane Curves

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. 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 $\bigoplus$. 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 $\ge k$ for any $k \ge 3.$

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arXiv2014

Homology of spaces of homogeneous polynomials in $R^2$ without multiple zeros

For any natural $d \ge k \ge 2$ we calculate the cohomology groups of the space of homogeneous polynomials $R^2 \to R$ of degree $d$, which do not vanish with multiplicity $\ge k$ on real lines. For $k=2$ this problem provides the simplest example of the situation, when the "finite degree" invariants of nonsingular objects are not a complete system of invariants.

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arXiv2014

Stratified Picard-Lefschetz Theory with Twisted Coefficients

The monodromy action in the homology (generally with twisted coefficients) of complements of stratified complex analytic varieties depending on parameters is studied. For a wide class of local degenerations of such families (stratified Morse singularities) local monodromy and variation operators are reduced to similar operators acting in the transversal slices of corresponding strata. In particular, these results imply a main part of F. Pham's generalized Picard--Lefschetz formulae.

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