7,324 research outputs found
On odd-periodic orbits in complex planar billiards
The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard
with infinitely-smooth boundary in a Euclidean space the set of periodic orbits
has measure zero}. In the present paper we study the complex version of Ivrii's
conjecture for odd-periodic orbits in planar billiards, with reflections from
complex analytic curves. We prove positive answer in the following cases: 1)
triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are
algebraic curves avoiding two special points at infinity, the so-called
isotropic points. We provide immediate applications to the real
piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility
theory
Regular homotopy and total curvature
We consider properties of the total absolute geodesic curvature functional on
circle immersions into a Riemann surface. In particular, we study its behavior
under regular homotopies, its infima in regular homotopy classes, and the
homotopy types of spaces of its local minima.
We consider properties of the total curvature functional on the space of
2-sphere immersions into 3-space. We show that the infimum over all sphere
eversions of the maximum of the total curvature during an eversion is at most
8\pi and we establish a non-injectivity result for local minima.Comment: This is the version published by Algebraic & Geometric Topology on 23
March 2006. arXiv admin note: this version concatenates two articles
published in Algebraic & Geometric Topolog
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
On polynomially integrable Birkhoff billiards on surfaces of constant curvature
We present a solution of the algebraic version of Birkhoff Conjecture on
integrable billiards. Namely we show that every polynomially integrable real
bounded convex planar billiard with smooth boundary is an ellipse. We extend
this result to billiards with piecewise-smooth and not necessarily convex
boundary on arbitrary two-dimensional surface of constant curvature: plane,
sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane
or a (pseudo-) sphere in equipped with appropriate quadratic
form. Namely, we show that a billiard is polynomially integrable, if and only
if its boundary is a union of confocal conical arcs and appropriate geodesic
segments. We also present a complexification of these results. These are joint
results of Mikhail Bialy, Andrey Mironov and the author. The proof is split
into two parts. The first part is given by Bialy and Mironov in their two joint
papers. They considered the tautological projection of the boundary to
and studied its orthogonal-polar dual curve, which is piecewise
algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's
theorem, it suffices to show that each non-linear complex irreducible component
of the dual curve is a conic. They have proved that all its singularities and
inflection points (if any) lie in the projectivized zero locus of the
corresponding quadratic form on . The present paper provides the
second part of the proof: we show that each above irreducible component is a
conic and finish the solution of the Algebraic Birkhoff Conjecture in constant
curvature.Comment: To appear in the Journal of the European Mathematical Society (JEMS),
69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and
misprint correction
Amoebas of algebraic varieties and tropical geometry
This survey consists of two parts. Part 1 is devoted to amoebas. These are
images of algebraic subvarieties in the complex torus under the logarithmic
moment map. The amoebas have essentially piecewise-linear shape if viewed at
large. Furthermore, they degenerate to certain piecewise-linear objects called
tropical varieties whose behavior is governed by algebraic geometry over the
so-called tropical semifield. Geometric aspects of tropical algebraic geometry
are the content of Part 2. We pay special attention to tropical curves. Both
parts also include relevant applications of the theories. Part 1 of this survey
is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in
Geometry
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