455 research outputs found
Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
We consider the suspension operation on Lefschetz fibrations, which takes
p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant,
and changes the category of the fibre (or more precisely, the subcategory
consisting of a basis of vanishing cycles) in a specific way. As an
application, we prove part of Homological Mirror Symmetry for the total spaces
of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio
Contour Detection and Completion for Inpainting and Segmentation Based on Topological Gradient and Fast Marching Algorithms
We combine in this paper the topological gradient, which is a powerful method for edge detection in image processing, and a variant of the minimal path method in order to find connected contours. The topological gradient provides a more global analysis of the image than the standard gradient and identifies the main edges of an image. Several image processing problems (e.g., inpainting and segmentation) require continuous contours. For this purpose, we consider the fast marching algorithm in order to find minimal paths in the topological gradient image. This coupled algorithm quickly provides accurate and connected contours. We present then two numerical applications, to image inpainting and segmentation, of this hybrid algorithm
Special lagrangian fibrations on flag variety
One constructs lagrangian fibrations on the flag variety and proves
that the fibrations are special.Comment: 19 page
Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
In this paper we establish an equivalence between the category of graded
D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W
and the triangulated category of singularities of the fiber of W over zero. The
main result is a theorem that shows that the graded triangulated category of
singularities of the cone over a projective variety is connected via a fully
faithful functor to the bounded derived category of coherent sheaves on the
base of the cone. This implies that the category of graded D-branes of type B
in Landau-Ginzburg models with homogeneous superpotential W is connected via a
fully faithful functor to the derived category of coherent sheaves on the
projective variety defined by the equation W=0.Comment: 26 pp., LaTe
A beginner's introduction to Fukaya categories
The goal of these notes is to give a short introduction to Fukaya categories
and some of their applications. The first half of the text is devoted to a
brief review of Lagrangian Floer (co)homology and product structures. Then we
introduce the Fukaya category (informally and without a lot of the necessary
technical detail), and briefly discuss algebraic concepts such as exact
triangles and generators. Finally, we mention wrapped Fukaya categories and
outline a few applications to symplectic topology, mirror symmetry and
low-dimensional topology. This text is based on a series of lectures given at a
Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in
June 2011.Comment: 42 pages, 13 figure
Brieskorn manifolds as contact branched covers of spheres
We show that Brieskorn manifolds with their standard contact structures are
contact branched coverings of spheres. This covering maps a contact open book
decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur
Symplectic cohomology and q-intersection numbers
Given a symplectic cohomology class of degree 1, we define the notion of an
equivariant Lagrangian submanifold. The Floer cohomology of equivariant
Lagrangian submanifolds has a natural endomorphism, which induces a grading by
generalized eigenspaces. Taking Euler characteristics with respect to the
induced grading yields a deformation of the intersection number. Dehn twists
act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz
fibrations give fully computable examples. A key step in computations is to
impose the "dilation" condition stipulating that the BV operator applied to the
symplectic cohomology class gives the identity. Equivariant Lagrangians mirror
equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example
7.5, added discussion of sign
Recovering the observable part of the initial data of an infinite-dimensional linear system with skew-adjoint generator
We consider the problem of recovering the initial data (or initial state) of infinite-dimensional linear systems with unitary semigroups. It is well-known that this inverse problem is well posed if the system is exactly observable, but this assumption may be very restrictive in some applications. In this paper we are interested in systems which are not exactly observable, and in particular, where we cannot expect a full reconstruction. We propose to use the algorithm studied by Ramdani et al. in (Automatica 46:1616–1625, 2010) and prove that it always converges towards the observable part of the initial state. We give necessary and sufficient condition to have an exponential rate of convergence. Numerical simulations are presented to illustratethe theoretical results
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