Volume 3 (2003), Number 3. Abstracts

E. Bierstone and P. Milman. Desingularization algorithms I. Role of exceptional divisors [PDF]

The article is about a "desingularization principle" (Theorem 1.14) common to various canonical desingularization algorithms in characteristic zero, and the roles played by the exceptional divisors in the underlying local construction. We compare algorithms of the authors and of Villamayor and his collaborators, distinguishing between the fundamental effect of the way the exceptional divisors are used, and different theorems obtained because of flexibility allowed in the choice of "input data". We show how the meaning of "invariance" of the desingularization invariant, and the efficiency of the algorithm depend on the notion of "equivalence" of collections of local data used in the inductive construction.

Keywords. Resolution of singularities, desingularization invariant, blowing-up, exceptional divisor.

2000 Mathematics Subject Classification. Primary 14E15, 32S45; Secondary 32S15, 32S20.

I. Bogaevsky. New Singularities and Perestroikas of Fronts of Linear Waves [PDF]

The subject of the paper is the propagation of linear waves in plane and three-dimensional space. We describe some new (as compared with the ADE-classification) typical singularities and perestroikas of their fronts when the light hypersurface has conical singularities. Such singularities appear if the waves propagate in a non-homogeneous anisotropic medium and are controlled by a variational principle.

Keywords. Singularity, perestroika, front, contact structure, Legendre submanifold, Legendre fibration.

2000 Mathematics Subject Classification. 58K40, 74J05, 58J47.

Yu. Burman and M. Polyak. Geometry of Whitney-type Formulas [PDF]

The article contains a generalization of the classical Whitney formula for the number of double points of a plane curve. This formula is split into a series of equalities, and also extended to curves on a torus, to non-pointed curves, and to wave fronts. All the theorems are given geometric proofs employing logarithmic Gauss-type maps from suitable configuration spaces to C.

Keywords. Plane curves, Whitney formula, Gauss map, intersection index.

2000 Mathematics Subject Classification. Primary 57M25, 57N35.

H. Cendra, J. Marsden, S. Pekarsky, and T. Ratiu. Variational Principles for Lie—Poisson and Hamilton—Poincaré Equations [PDF]

As is well-known, there is a variational principle for the Euler—Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton's principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie—Poisson equations on g^*, the dual of g, and also to generalize this construction.

The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q→Q/G becomes a principal bundle. Starting with a Lagrangian system on TQ invariant under the tangent lifted action of G, the reduced equations on (TQ)/G, appropriately identified, are the Lagrange—Poincaré equations. Similarly, if we start with a Hamiltonian system on T^*Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T^*Q)/G are called the Hamilton—Poincaré equations.

Amongst our new results, we derive a variational structure for the Hamilton—Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors.

Keywords. Geometric mechanics, Euler—Lagrange, Lagrangian reduction, Euler—Poincaré, Lagrange—Poincaré, Hamilton—Poincaré.

2000 Mathematics Subject Classification. 37J15, 70H25

S. Chmutov. An integral generalization of the Gusein-Zade—Natanzon theorem [PDF]

Several years ago N. A'Campo invented a construction of a link from a real curve immersed into the disk. In the case of the curve obtained by the real morsification method of singularity theory the link is isotopic to the link of the corresponding singularity. S. M. Gusein-Zade and S. M. Natanzon proved that the Arf invariant of the obtained knot equals J⁻/2 (mod 2) of the corresponding curve. Here we describe the Casson invariant of A'Campo knots as a J^±-type invariant of the immersed curve. Thus we get an integral generalization of the Gusein-Zade—Natanzon theorem. It turns out that this J₂^±-invariant is a second order invariant of the mixed J⁺- and J⁻-types. To the best of my knowledge, nobody has yet tried to study the mixed J^±-type invariants. It seems that our invariant is one of the simplest such invariants.

Keywords. Knots, A'Campo's divides, immersed curves, Casson invariant, J^±-type invariants.

2000 Mathematics Subject Classification. 57M25.

S. Duzhin. Decomposable skew-symmetric functions [PDF]

A skew-symmetric function F in several variables is said to be decomposable if it can be represented as a determinant det(f_i(x_j)) where f_i are univariate functions. We give a criterion of the decomposability in terms of a Plücker-type identity imposed on the function F.

Keywords. Skew-symmetric function, determinant, decomposable, Plücker relation.

2000 Mathematics Subject Classification. 05E05, 13J07, 14M15, 15A15.

E. Ferrand. Apparent contours and their Legendrian deformations [PDF]

After reviewing the notion of apparent contours of a smooth map φ from a compact manifold N to another manifold M, we recall the construction of an associated Legendrian subvariety in the space of contact elements of the goal manifold M and we study various examples. The main result is that, in some sense, non-trivial Legendrian deformations of apparent contours do not exist: In the space of contact elements of a real projective space, the set of the Legendrian submanifolds obtained in this way is closed under Legendrian isotopy.

Keywords. Contact topology.

2000 Mathematics Subject Classification. Primary 53C15.

M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster algebras and Poisson geometry [PDF]

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over R.

Keywords. Cluster algebras, Poisson brackets, toric action, symplectic leaves, real Grassmannians, Sklyanin bracket.

2000 Mathematics Subject Classification. 53D17, 14M15, 05E15.

J. Guckenheimer and Y. Xiang. Defining Equations for Bifurcations and Singularities [PDF]

Singularity theory and bifurcation theory lead us to consider varieties in jet spaces of mappings. Explicit defining equations for these varieties are complex and sometimes difficult to compute numerically. This paper considers two examples: saddle-node bifurcation of periodic orbits and the Thom—Boardman stratification of singularity theory. Saddle-node bifurcation of periodic orbits is determined by their monodromy matrices. The bifurcation occurs when the difference between the monodromy matrix and the identity has a two dimensional nilpotent subspace. We discuss numerical methods for computing this nilpotency. The usual definitions of the Thom—Boardman stratification of a map involve computing the rank of the map restricted to submanifolds. Without explicit formulas for these submanifolds, determination of the rank is a difficult numerical problem. We reformulate the defining equations for the submanifolds of the stratification here, producing a minimal set of regular defining equations for each stratum.

Keywords. Singularity, bifurcation, periodic orbit, saddle-node, Thom—Boardman stratification

2000 Mathematics Subject Classification. 32S60, 37G15, 58K05

V. Kharlamov and F. Sottile. Maximally inflected real rational curves [PDF]

We begin the topological study of real rational plane curves all of whose inflection points are real. The existence of such curves is implied by the results of real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.

Keywords. Real plane curves, Schubert calculus.

2000 Mathematics Subject Classification. 14P25, 14N10, 14M15.

B. Khesin. Geometry of higher helicities [PDF]

We revisit an interpretation of higher-dimensional helicities and Hopf—Novikov invariants from the point of view of the Brownian ergodic theorem. We also survey various results related to Arnold's theorem on the asymptotic Hopf invariant on three-dimensional manifolds and recent work on linking of a vector field with a foliation, the asymptotic crossing number, short path systems, and relations with the Calabi invariant.

Keywords. Asymptotic Hopf invariant, linking number, linking form, measured foliation, ergodic theorems.

2000 Mathematics Subject Classification. 37A15, 55Q25, 76W05.

A. Khovanskii and D. Novikov. L-convex-concave sets in real projective space and L-duality [PDF]

We define a class of L-convex-concave subsets of RPⁿ, where L is a projective subspace of dimension l in RPⁿ. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets and prove that the L-dual to an L-convex-concave set is an L^*-convex-concave subset of (RPⁿ)^*. We discuss a version of Arnold's conjecture for these sets and prove that it is true (or false) for an L-convex-concave set and its L-dual simultaneously.

Keywords. Separability, duality, convex-concave set, nondegenerate projective hypersurfaces.

2000 Mathematics Subject Classification. 52A30, 52A35.

A. Neishtadt. On Averaging in Two-Frequency Systems with Small Hamiltonian and Much Smaller Non-Hamiltonian Perturbations [PDF]

A system which differs from an integrable Hamiltonian system with two degrees of freedom by a small Hamiltonian perturbation and much a smaller non-Hamiltonian perturbation is considered. The unperturbed system is isoenergetically nondegenerate. The averaging method is used for an approximate description of solutions of the exact system on a time interval inversely proportional to the amplitude of the non-Hamiltonian perturbation. The error of this description (averaged over initial conditions) is estimated from above by a value proportional to the square root of the amplitude of the Hamiltonian perturbation.

Keywords. Perturbation theory, averaging method.

2000 Mathematics Subject Classification. 70K65, 34C26.

S. Orevkov and E. Shustin. Pseudoholomorphic Algebraically Unrealizable Curves [PDF]

We show that there exists a real non-singular pseudoholomorphic sextic curve in the affine plane which is not isotopic to any real algebraic sextic curve. This result completes the isotopy classification of real algebraic affine M-curves of degree 6. Comparing this with the isotopy classification of real affine pseudoholomorphic sextic M-curves obtained earlier by the first author, we find three pseudoholomorphic isotopy types which are algebraically unrealizable. In a similar way, we find a real pseudoholomorphic, algebraically unrealizable (M−1)-curve of degree 8 on a quadratic cone arranged in a special way with respect to a generating line. The proofs are based on the Hilbert—Rohn—Gudkov approach developed by the second author and on the cubic resolvent method developed by the first author.

Keywords. Pseudoholomorphic curves, real algebraic curves, equisingular family, cubic resolvent.

2000 Mathematics Subject Classification. Primary 14P25, 57M25; Secondary 14H20, 53D99.

C. Peters and J. Steenbrink. Degeneration of the Leray spectral sequence for certain geometric quotients [PDF]

We prove that the Leray spectral sequence in rational cohomology for the quotient map U_n,d→U_n,d/G where U_n,d is the affine variety of equations for smooth hypersurfaces of degree d in Pⁿ(C) and G is the general linear group, degenerates at E₂.

Keywords. Geometric quotient, hypersurfaces, Leray spectral sequence.

2000 Mathematics Subject Classification. 14D20, 14L35, 14J70.

V. Sedykh. On the topology of singularities of Maxwell sets [PDF]

We determine new conditions for the coexistence of corank-one singularities of the Maxwell set of a generic family of smooth functions with respect to taking global minima (or maxima) in cases when this set does not have more complicated singularities. In particular, the Euler number of every odd-dimensional manifold of singularities of a given type is a linear combination of the Euler numbers of even-dimensional manifolds of singularities of higher codimensions. The coefficients of this combination are universal numbers (that is, they do not depend on the family and depend only on the classes of singularities).

We obtain these conditions as a corollary to the general coexistence conditions for corank 1 singularities of generic wave fronts which were found recently by the author. As an application, we obtain many-dimensional generalizations of the classical Bose formula relating the number of supporting curvature circles for a smooth closed convex generic plane curve to the number of supporting circles which are tangent to this curve at three points.

Keywords. Families of smooth functions, global minima and maxima, Maxwell sets, corank-one singularities of smooth functions, Euler number, convex curves, supporting hyperspheres.

2000 Mathematics Subject Classification. Primary 58C05, 58K30; Secondary 53A04.

M. Sevryuk. The Classical KAM Theory at the Dawn of the Twenty-First Century [PDF]

We survey several recent achievements in KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov's original theorem of 1954. They include weak non-degeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, "exponential condensation" of perturbed tori, destruction mechanisms of resonant unperturbed tori, excitation of the elliptic normal modes of the unperturbed tori, and "atropic" invariant tori (i.e., tori that are neither isotropic nor coisotropic). The exposition is informal and nontechnical, and, as a rule, the methods of proofs are not discussed.

Keywords. KAM theory, Hamiltonian systems, invariant tori, quasi-periodic motions.

2000 Mathematics Subject Classification. Primary 37J40; Secondary 26E10, 58A10, 70H08.

V. Vassiliev. Spaces of Hermitian operators with simple spectra and their finite-order cohomology [PDF]

V. I. Arnold studied the topology of spaces of Hermitian operators with non-simple spectra in Cⁿ in relation to the theory of adiabatic connections and the quantum Hall effect. (Important physical motivations of this problem were also suggested by S. P. Novikov.) The natural stratification of these spaces into the sets of operators with fixed numbers of eigenvalues defines a spectral sequence providing interesting combinatorial and homological information on this stratification.

We construct a different spectral sequence, also converging to homology groups of these spaces; it is based on the universal techniques of topological order complexes and conical resolutions of algebraic varieties, which generalizes the combinatorial inclusion-exclusion formula, and is similar to the construction of finite-order knot invariants.

This spectral sequence stabilizes at the term E₁, is (conjecturally) multiplicative, and it converges as n→∞ to a stable spectral sequence calculating the cohomology of the space of infinite Hermitian operators without multiple eigenvalues whose all terms E_r^p,q are finitely generated. This allows us to define the finite-order cohomology classes of this space and apply well-known facts and methods of the topological theory of flag manifolds to problems of geometric combinatorics, especially to those concerning continuous partially ordered sets of subspaces and flags.

Keywords. Hermitian operator, simple spectrum, simplicial resolution, continuous order complex, finite type cohomology, stable filtration.

2000 Mathematics Subject Classification. 06B35, 06F30, 15A57, 81V70.

Y. Yomdin. The Center Problem for the Abel Equation, Compositions of Functions, and Moment Conditions [PDF]

An Abel differential equation y'=p(x)y²+q(x)y³ is said to have a center at a pair of complex numbers (a,b) if y(a)=y(b) for every solution y(x) with the initial value y(a) small enough. This notion is closely related to the classical center-focus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have been related to the composition factorization of P=∫p and Q=∫q on the one hand and to vanishing conditions for the moments m_i,j=∫PⁱQ^jq on the other hand. We give a detailed review of the recent results in each of these directions.

Keywords. Poincaré center-focus problem, Abel differential equation, composition of functions, generalized moments.

2000 Mathematics Subject Classification. Primary: 30E99, 30C99; Secondary: 34C99.

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