The Arnold-Gelfand Mathematical Seminars

435 Pages·1996·5.27 MB·English

by  Francesca Aicardi (auth.)

The Arnold-Gelfand
Mathematical Seminars
y.1. Arnold
I. M. Gelfand
Y. S. Retakh
M. Smimov
Editors
Birkhauser
Boston • Basel • Berlin
V.I. Arnold I. M. Gelfand
Ceremade Department of Mathematics
Universite Paris-Dauphine Rutgers University
75775 Paris Cedex 16 New Brunswick, NJ 08903
France
V. S. Retakh M. Smirnov
Department of Mathematics Department of Mathematics
Harvard University Columbia University
Cambridge, MA 02139 New York, NY 10027
Printed on acid-free paper cr»«>
© 1997 Birkhauser Boston Birkhiiuser HI»>
ISBN-13: 978-1-4612-8663-9 e-ISBN-13: 978-1-4612-4122-5
DOl: 10.1007/978-1-4612-4122-5
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ISBN 0-8176-3883-0
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Contents
Preface ............................................................... vii
Discriminants and Local Invariants of Planar Fronts
Francesca Aicardi ...................................................... 1
Crofton Densities, Symplectic Geometry and Hilbert's Fourth Problem
J.C. Alvarez, 1.M. Gelfand, and M. Smirnov ........................... 77
Projective Convex Curves
S. Anisov ............................................................. 93
Topological Classification of Real 'Irigonometric Polynomials
and Cyclic Serpents Polyhedron
V. Arnold ........................................................... 101
Singularities of Short Linear Waves on the Plane
Ria A. Bogaevski ..................................................... 107
New Generalizations of Poincare's Geometric Theorem
Yu. V. Chekanov .................................................... 113
Explicit Formulas for Arnold's Generic Curve Invariants
S. Chmutov and S. Duzhin ........................................... 123
Nonlinear Integrable Equations and Nonlinear Fourier 'Iransform
A.S. Fokas, 1.M. Gelfand, and M. V. Zyskin .......................... 139
Elliptic Solutions of the Yang-Baxter Equation and Modular
Hypergeometric Functions
Igor B. Frenkel and Vladimir G. Turaev .............................. 171
Combinatorics of Hypergeometric Functions
Associated with Positive Roots
Israel M. Gelfand, Mark 1. Graev, and Alexander Postnikov .......... 205
Local Invariants of Mappings of Surfaces into Three-Space
Victor V. Goryunov ................................................. 223
Theorem on Six Vertices of a Plane Curve Via Sturm Theory
L. Guieu, E. Mourre, and V. Yu. Ovsienko .......................... 257
vi Contents
The Arf-Invariant and the Arnold Invariants of Plane Curves
S.M. Gusein-Zade and S.M. Natanzon ............................... 267
Produit cyclique d'espaces et operations de Steenrod
Max Karoubi ........................................................ 281
Characteristic Classes of Singularity Theory
M.E. Kazarian ...................................................... 325
Value of Generalized Hypergeometric Function at Unity
A. Kasarnovski-Krol ................................................. 341
Harish-Chandra Decomposition for Zonal Spherical Function of type An
A. Kazarnovski-Krol ................................................. 347
Positive Paths in the Linear Symplectic Group
Franfois Lalonde and Dusa McDuff .................................. 361
Invariants of Submanifolds in Euclidean Space
V.D. Sedykh ......................................................... 389
On Combinatorics and Topology of Pairwise Intersections
of Schubert Cells in SLn/ B
Boris Shapiro, Michael Shapiro, and Alek Vainshtein ................. 397
Preface
It is very tempting but a little bit dangerous to compare the style of two
great mathematicians or of their schools. I think that it would be better to
compare papers from both schools dedicated to one area, geometry and to
leave conclusions to a reader of this volume.
The collaboration of these two schools is not new. One of the best
mathematics journals Functional Analysis and its Applications had I.M.
Gelfand as its chief editor and V.I. Arnold as vice-chief editor. Appearances
in one issue of the journal presenting remarkable papers from seminars of
Arnold and Gelfand always left a strong impact on all of mathematics.
We hope that this volume will have a similar impact. Papers from
Arnold's seminar are devoted to three important directions developed by
his school: Symplectic Geometry (F. Lalonde and D. McDuff), Theory of
Singularities and its applications (F. Aicardi, I. Bogaevski, M. Kazarian),
Geometry of Curves and Manifolds (S. Anisov, V. Chekanov, L. Guieu, E.
Mourre and V. Ovsienko, S. Gusein-Zade and S. Natanzon). A little bit
outside of these areas is a very interesting paper by M. Karoubi Produit
cyclique d'espaces et operations de Steenrod.
Papers from Gelfand's seminar are complimentary to a recent Gelfand
seminar volume and are more or less related by the notion of integral trans
forms. An application of Radon transforms to a solution of Hilbert's fourth
problem is given by J.C. Alvarez, I. Gelfand and M. Smirnov, nonlinear
integrable equations and nonlinear Fourier transform are considered by A.
Fokas, I. Gelfand and M. Zyzkin. Two papers (I. Gelfand, M. Graev and A.
Postnikov, A. Kazarnovski-Krol) are devoted to hypergeometric functions
- a cornerstone of modern theory of such functions developed by Gelfand's
school - is also a Radon transform of homogenous functions.
The Arnold-Gelfand
Mathematical Seminars
Discriminants and local invariants
of planar fronts
Francesca Aicardi
Introduction
The aim of this study is the description of all the local additive invari
ants of the plane wave fronts. A generic wave front is a curve whose
only singularities are the transversal self-intersections on the semicubi
cal cusps (see fig. la,b). The invariants that we shall find are "dual"
to different strata of the discriminant formed by the nongeneric wave
fronts.
For instance, our results imply the following minoration of the num
bers of different events of nongenericity on any path connecting the two
curves W3,O and K3,o shown in fig. Ib:
i) the number of self-tangencies with parallel coorientations is at least
2;
ii) the number of self-tangencies with antiparallel coorientations is at
least one;
iii) the number of cusp crossings is at least 4, provided that the triple
points are avoided;
iv) the number of triple points is at least 2, provided that the cusp
crossings are avoided.
QQ K3,u W3,O
(a) (b)
Figure 1. Oriented and cooriented planar fronts
The proofs are based on the introduction of some topological in
variants of the generic wave fronts, whose jumps at the crossing of the
corresponding nongeneric events are fixed. The calculation of the val-
2 F. Aicardi
ues of these invariants on the curves W3,O and K3,o provides the above
minoration.
There are five basic invariants (independent of the orientations of
the front and plane as well as of the co orientation of the front: n
(the number of double points), A (the number of cusp points), J+
(responsible for the parallel co orientations tangencies), J- (responsible
for the antiparallel co orientations tangencies), Sp (responsible of the
triple points and cusp crossing).
The invariants J+, J- and Sp of wave fronts reduce to the invari
ants J+, J- and St for those wave fronts which have no cusps, i.e., to
the Arnold invariants of smooth closed plane curves.
The number of the independent local additive invariants of generic
oriented and cooriented wave fronts on the oriented plane (counting
both those invariants, like J±, which are independent of the orienta
tions and those which depend on the orientations of the front and plane
as well as of the coorientation of the front) is equal to 10. One of these
invariants is the Maslov index f1. We introduce 6 new basic invariants
(1+, f-, Xf, Al, pT, pl).
The invariant f+ (1-) is counting the double points where two pos
itive (negative) branches of the front do intersect (the branch is positive
if its orientation and coorientation define the positive orientation of the
plane).
The invariant AT (Al) is counting the number of cusps at which
the orientation of the plane defined by the two branches of the oriented
curve leaving the cusp point (first along the coming branch, second
along the outgoing branch) is positive (negative).
The invariants p T, pl are responsible for the cusp crossings taking
some sign rule into account (see §3). These six new invariants verify
the relations:
hence only 4 of them are really new.
A planar front is the projection to R2 (with coordinates x,y) of a
legendrian curve. A legendrian curve is the image of a Cl-immersion
of SI in the space M3 (with coordinates x, y, <p(mod27r) for cooriented
fronts, (mod7r) for noncooriented fronts) of the contact elements of the
plane, with its natural contact structure
+
(cos<p)dx (sin<p)dy = O.
We call such an immersion of Sl into M3 an L-immersion.
Discriminants and local invariants of planar fronts 3
The front is cooriented, if the contact element is cooriented, i.e.,
if a choice of one of two half-planes into which it divides the tangent
plane is made. We shall now consider the cooriented fronts, and M3
will be the space of the co oriented contact elements.
The front is oriented if the preimage circle SI is oriented.
A front class is a class of L-immersions having generic fronts up to
diffeomorphisms of the plane and preimage circle.
In the space of L-immersions the immersions having nongeneric
fronts form a hypersurface (a subvariety of codimension one) called the
discriminant.
A coorientation on a smooth hypersurface in a functional space is
a choice of one of the two parts separated by this hypersurface in a
neighborhood of any of its points. This part is called positive. The
discriminant (or a connected component of it) is called coorientable if
the coorientations of its parts separated by strata of codimension 1 on
it (of co dimension 2 in the ambient space) are consistent. This means
that the intersection index of any generic small oriented closed curve
with the cooriented hypersurface should vanish.
In the functional space the connected components of the comple
ment to the discriminant are the front classes. A local invariant of a
front class is a function constant over each of such components. Local
means that its increment under a generic crossing of the discriminant
depends only on the behavior of the family of immersions in the neigh
borhoods of the points of the preimage circle sent to the points involved
in the singularity.
A local invariant is thus defined up to an additive constant in terms
of its jump at the crossing of the discriminant, provided that such
discriminant is coorientable and that it defines in the ambient space a
trivial 1-cocycle.
In fact every cooriented stratum of codimension 1, provided with a
jump at its crossing in the positive direction (Le., from the negative to
the positive side defined by the coorientation), gives a 1-cocycle. A 1-
co cycle is a function defined on generic curves in the ambient space: the
value of a 1-cocycle on a generic curve is defined by the sum of the jumps
(with signs) on all the crossings of the curve with the corresponding
stratum.
The study using this approach of the immersion of a circle in the
plane [2] leads to the definition of three basic invariants. One proves
that the space of immersions with a given index is connected. In this
space the discriminant has three different components, all coorientable,
corresponding to three different types of degenerations (see fig.2):
1) self-tangency point with parallel orientations,