Invariants for Lagrangian tori

We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from knot surgery using the trefoil knot in [Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori, J. Diff. Geom. (to appear)].Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper25.abs.htm

Subspace Clustering via Optimal Direction Search

This letter presents a new spectral-clustering-based approach to the subspace clustering problem. Underpinning the proposed method is a convex program for optimal direction search, which for each data point d finds an optimal direction in the span of the data that has minimum projection on the other data points and non-vanishing projection on d. The obtained directions are subsequently leveraged to identify a neighborhood set for each data point. An alternating direction method of multipliers framework is provided to efficiently solve for the optimal directions. The proposed method is shown to notably outperform the existing subspace clustering methods, particularly for unwieldy scenarios involving high levels of noise and close subspaces, and yields the state-of-the-art results for the problem of face clustering using subspace segmentation

Surgery on Nullhomologous Tori

By studying the example of smooth structures on CP^2#3(-CP^2) we illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold

The plastikstufe - a generalization of the overtwisted disk to higher dimensions

In this article, we give a first prototype-definition of overtwistedness in higher dimensions. According to this definition, a contact manifold is called "overtwisted" if it contains a "plastikstufe", a submanifold foliated by the contact structure in a certain way. In three dimensions the definition of the plastikstufe is identical to the one of the overtwisted disk. The main justification for this definition lies in the fact that the existence of a plastikstufe implies that the contact manifold does not have a (semipositive) symplectic filling.Comment: This is the version published by Algebraic & Geometric Topology on 15 December 200