On Gradings in Khovanov homology and sutured Floer homology

We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) interpretations, which we also review.Comment: 17 pages, 5 figures, to be submitted to Proceedings of Jaco's 70th Birthday Conference, 201

On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz. In the present work we prove the naturality of the spectral sequence under certain elementary TQFT operations, using a generalization of Juhasz's surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.Comment: 36 pages, 13 figure

Generation and Breakup of Worthington Jets After Cavity Collapse

Helped by the careful analysis of their experimental data, Worthington (1897) described roughly the mechanism underlying the formation of high-speed jets ejected after the impact of an axisymmetric solid on a liquid-air interface. In this work we combine detailed boundary-integral simulations with analytical modeling to describe the formation and break-up of such Worthington jets in two common physical systems: the impact of a circular disc on a liquid surface and the release of air bubbles from an underwater nozzle. We first show that the jet base dynamics can be predicted for both systems using our earlier model in Gekle, Gordillo, van der Meer and Lohse. Phys. Rev. Lett. 102 (2009). Nevertheless, our main point here is to present a model which allows us to accurately predict the shape of the entire jet. Good agreement with numerics and some experimental data is found. Moreover, we find that, contrarily to the capillary breakup of liquid cylinders in vacuum studied by Rayleigh, the breakup of stretched liquid jets at high values of both Weber and Reynolds numbers is not triggered by the growth of perturbations coming from an external source of noise. Instead, the jet breaks up due to the capillary deceleration of the liquid at the tip which produces a corrugation to the jet shape. This perturbation, which is self-induced by the flow, will grow in time promoted by a capillary mechanism. We are able to predict the exact shape evolution of Worthington jets ejected after the impact of a solid object - including the size of small droplets ejected from the tip due to a surface-tension driven instability - using as the single input parameters the minimum radius of the cavity and the flow field before the jet emerges

Almost-Commutative Geometries Beyond the Standard Model III: Vector Doublets

We will present a new extension of the standard model of particle physics in its almostcommutative formulation. This extension has as its basis the algebra of the standard model with four summands [11], and enlarges only the particle content by an arbitrary number of generations of left-right symmetric doublets which couple vectorially to the U(1)_YxSU(2)_w subgroup of the standard model. As in the model presented in [8], which introduced particles with a new colour, grand unification is no longer required by the spectral action. The new model may also possess a candidate for dark matter in the hundred TeV mass range with neutrino-like cross section

Almost-Commutative Geometries Beyond the Standard Model II: New Colours

We will present an extension of the standard model of particle physics in its almost-commutative formulation. This extension is guided by the minimal approach to almost-commutative geometries employed in [13], although the model presented here is not minimal itself. The corresponding almost-commutative geometry leads to a Yang-Mills-Higgs model which consists of the standard model and two new fermions of opposite electro-magnetic charge which may possess a new colour like gauge group. As a new phenomenon, grand unification is no longer required by the spectral action.Comment: Revised version for publication in J.Phys.A with corrected Higgs masse

Almost-Commutative Geometries Beyond the Standard Model

In [7-9] and [10] the conjecture is presented that almost-commutative geometries, with respect to sensible physical constraints, allow only the standard model of particle physics and electro-strong models as Yang-Mills-Higgs theories. In this publication a counter example will be given. The corresponding almost-commutative geometry leads to a Yang-Mills-Higgs model which consists of the standard model of particle physics and two new fermions of opposite electro-magnetic charge. This is the second Yang-Mills-Higgs model within noncommutative geometry, after the standard model, which could be compatible with experiments. Combined to a hydrogen-like composite particle these new particles provide a novel dark matter candidate