An untwisted cube of resolutions for knot Floer homology

Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its E_1 page is isomorphic to the HOMFLY-PT chain complex of Khovanov and Rozansky. At the level of each E_1 summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.Comment: 28 pages, 10 figures; a few minor change

Floer theory and its topological applications

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.Comment: Notes based on the 14th Takagi Lectures at the University of Tokyo; to appear in the Japanese Journal of Mathematic

Lectures on the triangulation conjecture

We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.Comment: 33 pages. Notes prepared with the help of Eylem Zeliha Yildiz; to appear in Proceedings of the 22nd Gokova Geometry/Topology Conference. The arxiv version has a corrected statement on p.

An introduction to knot Floer homology

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three- and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology.Comment: 31 pages; final version, to appear in Proceedings of the 2013 SMS summer school on Homology theories of knots and link

DIMENSIONS OF POST-CRISIS COMPETITIVE MONETARY POLICY

This communication wants to highlight, synthetically, the need to reconsider the monetary policy pursued in post-crisis period, driven by the imperatives of necessity, the requirements increase its competitiveness in a global economy, integrated, computerized, subject, becoming more global governance multiform. In this respect, communication and defining attributes are revealed circumscribed managing corporate governance dimensions of competitive monetary policy, promoted by the central bank in post-crisis period, finally giving it a summary schedule determined relations between attributes and dimensions of size co-determinative scheme could offer possible opening for formalization and modeling. Dimensions listed in the paper confined field improvement and adaptation potential of monetary policy in a period of rebuilding and restructuring the global economy, world of regionalism and integration, the polarization of the world economy, the assertion of national economies in a new perspective, that of network economies networked and distributed market.monetary policy; competitiveness; attribute; coordinated dimensional size; decision; tool