148 research outputs found
Measure Theory in Noncommutative Spaces
The integral in noncommutative geometry (NCG) involves a non-standard trace
called a Dixmier trace. The geometric origins of this integral are well known.
From a measure-theoretic view, however, the formulation contains several
difficulties. We review results concerning the technical features of the
integral in NCG and some outstanding problems in this area. The review is aimed
for the general user of NCG
Defect measures on graded lie groups
In this article, we define a generalisation of microlocal defect measures
(also known as H-measures) to the setting of graded nilpotent Lie groups. This
requires to develop the notions of homogeneous symbols and classical
pseudo-differential calculus adapted to this setting and defined via the
representations of the groups. Our method relies on the study of the C
*-algebra of 0-homogeneous symbols. Then, we compute microlocal defect measures
for concentrating and oscillating sequences, which also requires to investigate
the notion of oscillating sequences in graded Lie groups. Finally, we discuss
compacity compactness approaches in the context of graded nilpotent Lie groups
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
Local Operator Multipliers and Positivity
We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.Comment: 31 page
On the structure of -Harmonic maps
Let , . The PDE system
\label{1} A_\infty u \, :=\, \Big(H_P \otimes H_P + H [H_P]^\bot H_{PP}
\Big)(Du) : D^2 u\, = \, 0 \tag{1} arises as the ``Euler-Lagrange PDE" of
vectorial variational problems for the functional defined on maps . \eqref{1} first appeared in the author's recent
work \cite{K3}. The scalar case though has a long history initiated by Aronsson
in \cite{A1}. Herein we study the solutions of \eqref{1} with emphasis on the
case of with the Euclidean norm on ,
which we call the ``-Laplacian". By establishing a rigidity theorem for
rank-one maps of independent interest, we analyse a phenomenon of separation of
the solutions to phases with qualitatively different behaviour. As a corollary,
we extend to the Aronsson-Evans-Yu theorem regarding non-existence
of zeros of and prove a Maximum Principle. We further characterise all
for which \eqref{1} is elliptic and also study the initial value problem
for the ODE system arising for but with depending on all
the arguments.Comment: 30 pages, 10 figures, revised including referees' comments,
(Communications in PDE
- …