Let @math be a Laurent polynomial in @math variables, and let @math be a generic smooth fiber of @math . In RSTZ Ruddat-Sibilla-Treumann-Zaslow give a combinatorial recipe for a skeleton for @math . In this paper, we show that for a suitable exact symplectic structure on @math , the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.
The skeleton for @math dimensional pair-of-pants @math has been studied by Nadler @cite_5 , where a higher dimensional analog of Figure is constructued. A @math -symmetric skeleton for @math is constructed by Gammage-Nadler @cite_14 , where @math is the symmetric group. With the technique of tropical phase variety of Kerr-Zharkov @cite_6 , we hope to find other Lagrangian skeleta @math for @math , where @math indicating the number of dominant terms in the defining equation of @math .
@cite_5 @cite_14 @cite_6
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We prove a homological mirror symmetry equivalence between an @math -brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a @math -brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the @math -model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces. Abstract It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants . Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n -dimensional pair-of-pants is diffeomorphic to CP n minus n +2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n -dimensional polyhedral complex as its base and a real n -torus as its fiber. The base accommodates the geometric genus of a hypersurface V . Its homotopy type is a wedge of h n , o ( V ) spheres S n . Recent advances on human motion analysis have made the extraction of human skeleton structure feasible, even from single depth images. This structure has been proven quite informative for discriminating actions in a recognition scenario. In this context, we propose a local skeleton descriptor that encodes the relative position of joint quadruples. Such a coding implies a similarity normalisation transform that leads to a compact (6D) view-invariant skelet al feature, referred to as skelet al quad. Further, the use of a Fisher kernel representation is suggested to describe the skelet al quads contained in a (sub)action. A Gaussian mixture model is learnt from training data, so that the generation of any set of quads is encoded by its Fisher vector. Finally, a multi-level representation of Fisher vectors leads to an action description that roughly carries the order of sub-action within each action sequence. Efficient classification is here achieved by linear SVMs. The proposed action representation is tested on widely used datasets, MSRAction3D and HDM05. The experimental evaluation shows that the proposed method outperforms state-of-the-art algorithms that rely only on joints, while it competes with methods that combine joints with extra cues. Skeletonization algorithms typically decompose an object's silhouette into a set of symmetric parts, offering a powerful representation for shape categorization. However, having access to an object's silhouette assumes correct figure-ground segmentation, leading to a disconnect with the mainstream categorization community, which attempts to recognize objects from cluttered images. In this paper, we present a novel approach to recovering and grouping the symmetric parts of an object from a cluttered scene. We begin by using a multiresolution superpixel segmentation to generate medial point hypotheses, and use a learned affinity function to perceptually group nearby medial points likely to belong to the same medial branch. In the next stage, we learn higher granularity affinity functions to group the resulting medial branches likely to belong to the same object. The resulting framework yields a skelet al approximation that is free of many of the instabilities that occur with traditional skeletons. More importantly, it does not require a closed contour, enabling the application of skeleton-based categorization systems to more realistic imagery.