Optimal Filling of Shapes

We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls. For polygons, we provide a heuristic for finding solutions of maximal discs. We consider the properties of ideal distributions of N discs as N approaches infinity. We note an analogy with energy landscapes.Comment: 5 page

Motion Planning for Unlabeled Discs with Optimality Guarantees

We study the problem of path planning for unlabeled (indistinguishable) unit-disc robots in a planar environment cluttered with polygonal obstacles. We introduce an algorithm which minimizes the total path length, i.e., the sum of lengths of the individual paths. Our algorithm is guaranteed to find a solution if one exists, or report that none exists otherwise. It runs in time $\tilde{O}(m^4+m^2n^2)$, where $m$ is the number of robots and $n$ is the total complexity of the workspace. Moreover, the total length of the returned solution is at most $\text{OPT}+4m$, where OPT is the optimal solution cost. To the best of our knowledge this is the first algorithm for the problem that has such guarantees. The algorithm has been implemented in an exact manner and we present experimental results that attest to its efficiency

Generalized SYZ mirror transformation

Strominger-Yau-Zaslow proposed that mirror symmetry can be understood by torus duality. In this article we explain how it fits into a bigger framework, where tori are replaced by general Lagrangian immersions. The generalized construction is applicable to a wider class of geometries. We also give a brief introduction to our ongoing work on gluing local mirrors into global geometries

On the Fukaya Categories of Higher Genus Surfaces

We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.Comment: 40 pages, 17 figures. Final Versio

Combinatorial cohomology of the space of long knots

The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology define algebraic intersections with some "geometrically simple" strata in the space of knots. Such strata are endowed with explicit co-orientations, that are canonical in some sense. The combinatorial tools involved are natural generalisations (degeneracies) of usual methods using arrow diagrams.Comment: 20p. 9 fig