26,216 research outputs found
Quantum Hall Effect on the Hyperbolic Plane
In this paper, we study both the continuous model and the discrete model of
the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is
identified as a geometric invariant associated to an imprimitivity algebra of
observables. We define a twisted analogue of the Kasparov map, which enables us
to use the pairing between -theory and cyclic cohomology theory, to identify
this geometric invariant with a topological index, thereby proving the
integrality of the Hall conductivity in this case.Comment: AMS-LaTeX, 28 page
Twisted higher index theory on good orbifolds and fractional quantum numbers
The twisted Connes-Moscovici higher index theorem is generalized to the case
of good orbifolds. The higher index is shown to be a rational number, and in
fact non-integer in specific examples of 2-orbifolds. This results in a
non-commutative geometry model that predicts the occurrence of fractional
quantum numbers in the Hall effect on the hyperbolic plane.Comment: 47 pages, Late
The notion of dimension in geometry and algebra
This talk reviews some mathematical and physical ideas related to the notion
of dimension. After a brief historical introduction, various modern
constructions from fractal geometry, noncommutative geometry, and theoretical
physics are invoked and compared.Comment: 29 pages, a revie
Multivariable Hodge theoretical invariants of germs of plane curves
We describe methods for calculation of polytopes of quasiadjunction for plane
curve singularities which are invariants giving a Hodge theoretical refinement
of the zero sets of multivariable Alexander polynomials. In particular we
identify some hyperplanes on which all polynomials in multivariable Bernstein
ideal vanish
Noncommutative knot theory
The classical abelian invariants of a knot are the Alexander module, which is
the first homology group of the the unique infinite cyclic covering space of
S^3-K, considered as a module over the (commutative) Laurent polynomial ring,
and the Blanchfield linking pairing defined on this module. From the
perspective of the knot group, G, these invariants reflect the structure of
G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the
derived series of G). Hence any phenomenon associated to G^(2) is invisible to
abelian invariants. This paper begins the systematic study of invariants
associated to solvable covering spaces of knot exteriors, in particular the
study of what we call the n-th higher-order Alexander module, G^(n+1)/G^(n+2),
considered as a Z[G/G^(n+1)$-module. We show that these modules share almost
all of the properties of the classical Alexander module. They are torsion
modules with higher-order Alexander polynomials whose degrees give lower bounds
for the knot genus. The modules have presentation matrices derived either from
a group presentation or from a Seifert surface. They admit higher-order linking
forms exhibiting self-duality. There are applications to estimating knot genus
and to detecting fibered, prime and alternating knots. There are also
surprising applications to detecting symplectic structures on 4-manifolds.
These modules are similar to but different from those considered by the author,
Kent Orr and Peter Teichner and are special cases of the modules considered
subsequently by Shelly Harvey for arbitrary 3-manifolds.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-19.abs.htm
A Quasi Curtis-Tits-Phan theorem for the symplectic group
We obtain the symplectic group \SP(V) as the universal completion of an
amalgam of low rank subgroups akin to Levi components. We let \SP(V) act
flag-transitively on the geometry of maximal rank subspaces of . We show
that this geometry and its rank residues are simply connected with few
exceptions. The main exceptional residue is described in some detail. The
amalgamation result is then obtained by applying Tits' lemma. This provides a
new way of recognizing the symplectic groups from a small collection of small
subgroups
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