3,526 research outputs found
Enumerative aspects of the Gross-Siebert program
We present enumerative aspects of the Gross-Siebert program in this
introductory survey. After sketching the program's main themes and goals, we
review the basic definitions and results of logarithmic and tropical geometry.
We give examples and a proof for counting algebraic curves via tropical curves.
To illustrate an application of tropical geometry and the Gross-Siebert program
to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming
Fields Institute volume. 81 page
Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
We discuss how the shape of a special Cosserat rod can be represented as a
path in the special Euclidean algebra. By shape we mean all those geometric
features that are invariant under isometries of the three-dimensional ambient
space. The representation of the shape as a path in the special Euclidean
algebra is intrinsic to the description of the mechanical properties of a rod,
since it is given directly in terms of the strain fields that stimulate the
elastic response of special Cosserat rods. Moreover, such a representation
leads naturally to discretization schemes that avoid the need for the expensive
reconstruction of the strains from the discretized placement and for
interpolation procedures which introduce some arbitrariness in popular
numerical schemes. Given the shape of a rod and the positioning of one of its
cross sections, the full placement in the ambient space can be uniquely
reconstructed and described by means of a base curve endowed with a material
frame. By viewing a geometric curve as a rod with degenerate point-like cross
sections, we highlight the essential difference between rods and framed curves,
and clarify why the family of relatively parallel adapted frames is not
suitable for describing the mechanics of rods but is the appropriate tool for
dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure
Dehn Twists in Heegaard Floer Homology
We derive a new exact sequence in the hat-version of Heegaard Floer homology.
As a consequence we see a functorial connection between the invariant of
Legendrian knots and the contact element. As an application we derive two
vanishing results of the contact element making it possible to easily read off
its vanishing out of a surgery presentation in suitable situations.Comment: 61 pages, 39 figures; added details to several proofs. Version
published by Algebr. Geom. Topol. 10 (2010), 465--52
Topological and geometric decomposition of nematic textures
Directional media, such as nematic liquid crystals and ferromagnets, are
characterized by their topologically stabilized defects in directional order.
In nematics, boundary conditions and surface-treated inclusions often create
complex structures, which are difficult to classify. Topological charge of
point defects in nematics has ambiguously defined sign and its additivity
cannot be ensured when defects are observed separately. We demonstrate how the
topological charge of complex defect structures can be determined by
identifying and counting parts of the texture that satisfy simple geometric
rules. We introduce a parameter called the defect rank and show that it
corresponds to what is intuitively perceived as a point charge based on the
properties of the director field. Finally, we discuss the role of free energy
constraints in validity of the classification with the defect rank.Comment: 16 pages, 5 figure
Positioning systems in Minkowski space-time: Bifurcation problem and observational data
In the framework of relativistic positioning systems in Minkowski space-time,
the determination of the inertial coordinates of a user involves the {\em
bifurcation problem} (which is the indeterminate location of a pair of
different events receiving the same emission coordinates). To solve it, in
addition to the user emission coordinates and the emitter positions in inertial
coordinates, it may happen that the user needs to know {\em independently} the
orientation of its emission coordinates. Assuming that the user may observe the
relative positions of the four emitters on its celestial sphere, an
observational rule to determine this orientation is presented. The bifurcation
problem is thus solved by applying this observational rule, and consequently,
{\em all} of the parameters in the general expression of the coordinate
transformation from emission coordinates to inertial ones may be computed from
the data received by the user of the relativistic positioning system.Comment: 10 pages, 7 figures. The version published in PRD contains a misprint
in the caption of Figure 3, which is here amende
Reconfigurable knots and links in chiral nematic colloids
Tying knots and linking microscopic loops of polymers, macromolecules, or
defect lines in complex materials is a challenging task for material
scientists. We demonstrate the knotting of microscopic topological defect lines
in chiral nematic liquid crystal colloids into knots and links of arbitrary
complexity by using laser tweezers as a micromanipulation tool. All knots and
links with up to six crossings, including the Hopf link, the Star of David and
the Borromean rings are demonstrated, stabilizing colloidal particles into an
unusual soft matter. The knots in chiral nematic colloids are classified by the
quantized self-linking number, a direct measure of the geometric, or Berry's,
phase. Forming arbitrary microscopic knots and links in chiral nematic colloids
is a demonstration of how relevant the topology can be for the material
engineering of soft matter.Comment: 6 pages, 3 figure
- …