arXiv subjects
Publications and source records attributed to V. A. Vassiliev.
According to a theorem by F.~Dyson, every continuous function $f: S^2 \to {\mathbb R}$ takes the same value at the vertices of a square inscribed into a great circle of $S^2$. We give a proof of this statement based on the theory of characteristic classes and indicate other potential applications of this approach.
We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all 71 possible values of these invariants for the case of degree 4, and realize them by concrete Morse polynomials. Also we count all 45460 classes of {\em strictly} Morse polynomials of degree four with the maximal possible number (nine) of real critical points. Keywords: real algebraic geometry, Morse function, Milnor fiber, Coxeter-Dynkin graph, vanishing cycle, topological invariant, surgery, Lyashko--Looijenga map.
Singular knots are smooth maps $S^1 \to {\mathbb R}^3$ that have self-intersections or points at which the derivative vanishes. We describe an algebraic classification of their singularities, and study the topological properties of the corresponding stratification of the discriminant subset $Σ\subset C^\infty (S^1, {\mathbb R}^3)$ consisting of all singular knots. This classification is defined by a system of special subalgebras of the function space $C^\infty(S^1, {\mathbb R})$. These are either defined by the chord diagrams, that is, by finite sets of conditions of the form $f(x_i) = f(\tilde x_i)$, $\{x_i, \tilde x_i\} \subset S^1$, or are the limit positions of such subalgebras in the space of all subspaces of a fixed codimension in $C^\infty(S^1, {\mathbb R})$. These limits arise at various collisions of the points $x_i, \tilde x_i$ that define the chord diagrams. For each natural number $k$, the set of such codimension-$k$ subalgebras is a $2k$-dimensional compact semialgebraic variety with a canonical $k$-dimensional vector bundle on it. We describe the natural stratification of these varieties for $k \leq 3$ and compute their cohomology rings and characteristic classes of canonical vector bundles. We also find many cohomology classes of sets of these subalgebras of arbitrary codimensions, and prove geometric corollaries concerning the topology of corresponding discriminant strata, in particular on their intersections with the finite-dimensional approximating subspaces of the knot space.
We list all connected components of sets of non-discriminant functions near all {\em parabolic} function singularities (which are the second most important family of singularity classes of smooth functions after {\em simple} singularities). Thus, we prove (and improve in one particular case) all the corresponding conjectures from the previous work \cite{para} with the same title. As an application, we enumerate all {\em local Petrovskii lacunas} near arbitrary parabolic singularities of wavefronts of hyperbolic PDEs. We also show that the complements of the discriminant varieties of the versal deformations of $X_9^{\pm}$ and $P_8^1$ singularities have nontrivial one-dimensional homology groups, in contrast to all simple singularities. These results are applications of a general method for investigating and separating non-singular perturbations of real function singularities. An important part of this method is a computer program that formalizes local Picard--Lefschetz theory and surgeries of Morse functions.
We enumerate all isotopy classes of degree three Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of {\em strictly} Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ of degree three with the maximal possible number (eight) of real critical points. A main tool in this classification is a combinatorial computer program that formalizes Morse surgeries, local monodromy and Picard-Lefschetz theory.
The complete list of connected components of the set of Morse functions in the deformations of function singularities of class $J_{10}$ is given. Thus, the isotopy classification of Morse perturbations of parabolic real function singularities is finished.
A {\em $k$-trinitary algebra} is any subalgebra of the space of smooth functions $f: M \to {\mathbb R}$ that is distinguished in this space by $k$ independent conditions of the form $f(x_i) = f(\tilde x_i) = f(\hat x_i)$, where $x_i, \tilde x_i,$ and $ \hat x_i $ are distinct points in $ M$, $i=1, \dots, k$, or is approximated by such subalgebras. Trinitary algebras naturally arise in the study of {\em discriminant varieties,} that is, the spaces of singular geometric objects, when the property of being singular is formulated in terms of the simultaneous behavior at three distinct points. The simplest singular objects of this kind are the plane curves with triple self-intersections, see \cite{A}, \cite{MD}. The spaces of all $k$-trinitary algebras in $C^\infty(M, {\mathbb R})$ are analogous to the spaces of all ideals of finite codimension, which play the same role in the study of discriminants defined in the terms of a single singular point. These spaces are also analogous to the spaces of {\em equilevel algebras} (see \cite{EA}), which arise in the study of discriminants defined by binary singularities. We classify the trinitary algebras up to the codimension four in $C^\infty(S^1, {\mathbb R})$, compute the cohomology rings of their varieties and find the Stiefel--Whitney classes of their canonical normal bundles. We also present a series of $(2k-2)$-dimensional cohomology classes of the spaces of trinitary algebras of codimension $2k$ for any natural $k$.
We study the space of codimension two subalgebras in $C^\infty(S^1, {\mathbb R})$ defined by pairs of conditions $f(φ)=f(ψ)$, $φ\neq ψ\in S^1$, or by their limits. We compute the mod 2 cohomology ring of this space, and also the Stiefel--Whitney classes of the tautological vector bundle on it.
For any natural numbers $k \leq n$, the rational cohomology ring of the space of continuous maps $S^{2k-1} \to S^{2n-1}$ (respectively, $S^{4k-1} \to S^{4n-1}$) equivariant under the Hopf action of the circle (respectively, of the group $S^3$ of unit quaternions) is naturally isomorphic to that of the Stiefel manifold $V_k({\mathbb C}^n)$ (respectively, $V_k({\mathbb H}^n)$). The natural maps of integral cohomology groups of these spaces of equivariant maps to cohomology of Stiefel manifolds are surjective but not injective.
We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in ${\mathbb C}^k$, $k \geq 3$, and carry out these calculations to obtain these groups of low dimensions.
A (conjecturally complete) list of components of complements of discriminant varieties of parabolic singularities of smooth real functions is given. We also promote a combinatorial program that enumerates possible topological types of non-discriminant morsifications of isolated real function singularities and provides a strong invariant of components of complements of discriminant varieties.
We study the topology of complements of caustics of function singularities of low codimensions, in particular 1) complete the enumeration of connected components of the complements of caustics of {\em simple} (in the sense of V.Arnold) singularities, in particular find the numbers of these components for the last two classes, $E_7$ and $E_8$, remaining unknown after the works of R.Thom, V.Arnold and V.Sedykh; 2) realize all these components for simple singularities by explicitly constructed functions, and realize their one-dimensional homology and cohomology groups by cycles and cocycles; 3) prove that (in contrast to the case of simple singularities) for some parabolic singularities the two-dimensional homology groups of the complements of their caustics are nontrivial.
The average value of the number of intersection points of two plane curves given by trigonometrical polynomial maps of degree $N$ and bounded $L_2$- or $W_2^r$- norms is calculated
All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm enumerating topological types of morsifications of real function singularities is promoted.
V. Arnold's problem 1987-14 asks whether there exist smooth hypersurfaces in $R^N$ (other than the conics in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body bounded by such a hypersurface is an algebraic function of the hyperplane. We desribe very realistic candidates for the role of such new hypersurfaces: in particular, it are examples (additional to Archimedes' conics) of such hypersurfaces, for which the analytic continuation of this volume function is finitely valued.
A problem list in singularity theory. Most of these problems are related with the algorithmic enumeration of possible topological types of non-discriminant Morsifications of real function singularities, and/or with the Picard--Lefschetz theory.
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory, combinatorics, and singularity theory. The multidimensional analogues of this complex are indicated, which arise naturally in the homotopy theory, higher Chern-Simons theory and complexity theory.
The resultant veriety in the space of systems of homogeneous polynomials of given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials $R^2 \to R$, and also the rational cohomology groups of similar systems in $C^2$.