18 research outputs found
Rings with Topologies Induced by Spaces of Functions
The paper contains the proof, in dimension 2, of a conjecture of R. G.
Douglas and V. Paulsen concerning the characterization of the ideals of
polynomials which are closed in the relative topology induced by the Hardy
space of the polydisk.Comment: LATE
Compact Perturbations of Fredholm n-tuples
The paper gives a negative answer to the question whether one can perturb a
Fredholm pair of index 0 by compact operators to a pair with exact Koszul
complex.Comment: LATE
Topological quantum field theory with corners based on the Kauffman bracket
The paper contains the construction of a topological quantum field theory
with corners that underlies the smooth topological quantum field theory of
Lickorish. Among other things, a contraction formula for diagrams is proved,
the presence of the skein relation at the level of manifolds with boundary is
shown and an explanation for the surgery formula for invariants of closed
manifolds is given.Comment: Latex, 28 pages, 39 figure
Noncommutative trigonometry and the A-polynomial of the trefoil knot
The paper shows the computation of the noncommutative generalization of the
A-polynomial of the trefoil knot. The classical A-polynomial was introduced by
Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of
Kauffman bracket skein modules by the author in joint work with Frohman and
Lofaro. A major step in determining the noncommutative version of the
A-polynomial of the trefoil is the description of the action of the Kauffman
bracket skein algebra of the torus on the skein module of the knot complement.
As such, the computation reduces to operations with noncommutative
trigonometric functions.Comment: LATEX, 7 figure
On the relation between the A-polynomial and the Jones polynomial
In previous joint work with Frohman and Lofaro a noncommutative
generalization of the A-polynomial of a knot was introduced, consisting of a
finitely generated ideal of polynomials (the noncommutative A-ideal) in the
quantum plane. The present paper shows that the noncommutative A-ideal of a
knot, together with finitely many of its colored Jones polynomials determines
all other colored Jones polynomials. Also, for particular knots, such as the
unknot and the trefoil, the noncommutative A-ideal completely determines all
colored Jones polynomials.Comment: LATEX, 6 page
On the holomorphic point of view in the theory of quantum knot invariants
In this paper we describe progress made toward the construction of the
Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of
view. This is done in the perspective of a joint result of the author with A.
Uribe which relates the quantum group and the Weyl quantizations of the moduli
space of flat SU(2)-connections on the tours. Two results are emphasized: the
reconstruction from Weyl quantization of the restriction to the torus of the
modular functor, and a description of a basis of the space of quantum
observables on the torus in terms of colored curves, which answers a question
related to quantum computing
Topological Hilbert Nullstellensatz for Bergman Spaces
The results in the paper are related to the classification problem for
invariant subspaces of multiplication operators in several variables. The main
results consist of characterizations, in the two dimensional case, of ideals of
polynomials and analytic functions which are closed in the relative topology
induced by Bergman spaces on certain domains. This provides proofs of the
Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.Comment: LATE
A product-to-sum formula for the quantum group of SL(2,C)
The paper exhibits a product-to-sum formula for the observables of a certain
quantization of the moduli space of flat SU(2)-connections on the torus. This
quantization was defined using the topological quantum field theory that was
developed by Reshetikhin and Turaev from the quantum group of SL(2,C) at roots
of unity. As a corollary it is shown that the algebra of quantum observables is
a subalgebra of the noncommutative torus with rational rotation angle. The
proof uses topological quantum field theory with corners, and is based on the
description of the matrices of the observables in a canonical basis of the
Hilbert space of the quantization.Comment: Latex, 10 pages, 2 figure
The topological quantum field theory of Riemann's theta functions
In this paper we prove the existence and uniqueness of a topological quantum
field theory that incorporates, for all Riemann surfaces, the corresponding
spaces of theta functions and the actions of the Heisenberg groups and modular
groups on them.Comment: 23 pages, 11 figure
Classical theta functions from a quantum group perspective
In this paper we construct the quantum group, at roots of unity, of abelian
Chern-Simons theory. We then use it to model classical theta functions and the
actions of the Heisenberg and modular groups on them