19,412 research outputs found
Exponential stabilization without geometric control
We present examples of exponential stabilization for the damped wave equation
on a compact manifold in situations where the geometric control condition is
not satisfied. This follows from a dynamical argument involving a topological
pressure on a suitable uncontrolled set
Spectral morphisms, K-theory, and stable ranks
We give a brief account of the interplay between spectral morphisms,
K-theory, and stable ranks in the context of Banach algebras.Comment: 12 pages; to appear in the Proceedings of the Workshop on
Noncommutative Geometry (Fields Institute, Toronto 2008
Seeking the Ground State of String Theory
Recently, a number of authors have challenged the conventional assumption
that the string scale, Planck mass, and unification scale are roughly
comparable. It has been suggested that the string scale could be as low as a
TeV. The greatest obstacle to developing a string phenomenology is our lack of
understanding of the ground state. We explain why the dynamics which determines
this state is not likely to be accessible to any systematic approximation. We
note that the racetrack scheme, often cited as a counterexample, suffers from
similar difficulties. We stress that the weakness of the gauge couplings, the
gauge hierarchy, and coupling unification suggest that it may be possible to
extract some information in a systematic approximation. We review the ideas of
Kahler stabilization, an attempt to reconcile these facts. We consider whether
the system is likely to sit at extremes of the moduli space, as in recent
proposals for a low string scale. Finally we discuss the idea of Maximally
Enhanced Symmetry, a hypothesis which is technically natural, compatible with
basic facts about cosmology, and potentially predictive.Comment: 22 pp. latex. Invited talk presented at Yukawa-Nishinomiya symposium.
Minor correction
Racetrack inflation and assisted moduli stabilisation
We present a model of inflation based on a racetrack model without flux
stabilization. The initial conditions are set automatically through topological
inflation. This ensures that the dilaton is not swept to weak coupling through
either thermal effects or fast roll. Including the effect of non-dilaton fields
we find that moduli provide natural candidates for the inflaton. The resulting
potential generates slow-roll inflation without the need to fine tune
parameters. The energy scale of inflation must be near the GUT scale and the
scalar density perturbation generated has a spectrum consistent with WMAP data.Comment: 17 pages, 6 figures (Latex); Error in v.1 eliminated and improved
example of modular inflation presente
Defect-Mediated Emulsification in Two Dimensions
We consider two dimensional dispersions of droplets of isotropic phase in a
liquid with an XY-like order parameter, tilt, nematic, and hexatic symmetries
being included. Strong anchoring boundary conditions are assumed. Textures for
a single droplet and a pair of droplets are calculated and a universal
droplet-droplet pair potential is obtained. The interaction of dispersed
droplets via the ordered phase is attractive at large distances and repulsive
at short distances, which results in a well defined preferred separation for
two droplets and topological stabilization of the emulsion. This interaction
also drives self-assembly into chains. Preferred separations and energy
barriers to coalescence are calculated, and effects of thermal fluctuations and
film thickness are discussed.Comment: revtex4, 13 pages, 12 figure
Braids, knots and contact structures
These notes were prepared to supplement the talk that I gave on Feb 19, 2004,
at the First East Asian School of Knots and Related Topics, Seoul, South Korea.
In this article I review aspects of the interconnections between braids, knots
and contact structures on Euclidean 3-space. I discuss my recent work with
William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the
latter we prove that there are distinct transversal knot types in contact
3-space having the same topological knot type and the same Bennequin invariant.Comment: 10 pages, 5 figure
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