Page 63 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 63
INVITED SPEAKERS
discretization of the surface, invoking a local one-dimensional root-finding procedure.
Numerical examples are given to illustrate the performance of the quadrature error esti-
mates. The estimates for integration over curves are in many cases remarkably precise, and the
estimates for curved surfaces in R3 are also sufficiently precise, with sufficiently low computa-
tional cost, to be practically useful.
AAA-least squares rational approximation and solution of Laplace
problems
Nick Trefethen, trefethen@maths.ox.ac.uk
Oxford University, United Kingdom
In the past five years, computation with rational functions has advanced greatly with the in-
troduction of the AAA algorithm for barycentric rational approximation and of lightning least-
squares solvers for Laplace, Stokes, and Helmholtz problems. Here we combine these methods
into a two-step method for solving planar Laplace problems. First, complex rational approxi-
mations to the boundary data are determined by AAA approximation, locally near each corner
or other singularity. The poles of these approximations outside the problem domain are then
collected and used for a global least-squares fit to the solution. Typical problems are solved in a
second of laptop time to 8-digit accuracy, all the way up to the corners. This is joint work with
Stefano Costa.
Structure and classification of simple amenable C∗-algebras
Stuart White, stuart.white@maths.ox.ac.uk
University of Oxford, United Kingdom
In this talk I will give an overview of recent progress in the structure theory of simple amenable
C∗-algebras and classification results. C∗-algebras are norm closed self-adjoint subalgebras of
the bounded operators on a Hilbert space, with examples arising naturally from unitary repre-
sentations of groups, and topological dynamics. They have a topological flavour, seen through
the commutative algebras of continuous functions on locally compact Hausdorff spaces.
The classification of C∗-algebras has its spiritual origins in the powerful structure and classi-
fication theorems for von Neumann algebras of Connes in the ’70s. However, in the topological
setting of C∗-algebras, higher dimensional phenomena can obstruct classification in general.
Progress over the last decade has seen the identification of abstract structural conditions which
give the maximal family of algebras which can be classified by K-theory and traces. These
conditions now have equivalent formulations of very different natures, which can be used to
bring naturally occurring examples within the scope of classification.
The talk is based in part on joint works with Castillejos, Carrión, Evington, Gabe,
Schafhauser, Tikuisis, and Winter.
61
discretization of the surface, invoking a local one-dimensional root-finding procedure.
Numerical examples are given to illustrate the performance of the quadrature error esti-
mates. The estimates for integration over curves are in many cases remarkably precise, and the
estimates for curved surfaces in R3 are also sufficiently precise, with sufficiently low computa-
tional cost, to be practically useful.
AAA-least squares rational approximation and solution of Laplace
problems
Nick Trefethen, trefethen@maths.ox.ac.uk
Oxford University, United Kingdom
In the past five years, computation with rational functions has advanced greatly with the in-
troduction of the AAA algorithm for barycentric rational approximation and of lightning least-
squares solvers for Laplace, Stokes, and Helmholtz problems. Here we combine these methods
into a two-step method for solving planar Laplace problems. First, complex rational approxi-
mations to the boundary data are determined by AAA approximation, locally near each corner
or other singularity. The poles of these approximations outside the problem domain are then
collected and used for a global least-squares fit to the solution. Typical problems are solved in a
second of laptop time to 8-digit accuracy, all the way up to the corners. This is joint work with
Stefano Costa.
Structure and classification of simple amenable C∗-algebras
Stuart White, stuart.white@maths.ox.ac.uk
University of Oxford, United Kingdom
In this talk I will give an overview of recent progress in the structure theory of simple amenable
C∗-algebras and classification results. C∗-algebras are norm closed self-adjoint subalgebras of
the bounded operators on a Hilbert space, with examples arising naturally from unitary repre-
sentations of groups, and topological dynamics. They have a topological flavour, seen through
the commutative algebras of continuous functions on locally compact Hausdorff spaces.
The classification of C∗-algebras has its spiritual origins in the powerful structure and classi-
fication theorems for von Neumann algebras of Connes in the ’70s. However, in the topological
setting of C∗-algebras, higher dimensional phenomena can obstruct classification in general.
Progress over the last decade has seen the identification of abstract structural conditions which
give the maximal family of algebras which can be classified by K-theory and traces. These
conditions now have equivalent formulations of very different natures, which can be used to
bring naturally occurring examples within the scope of classification.
The talk is based in part on joint works with Castillejos, Carrión, Evington, Gabe,
Schafhauser, Tikuisis, and Winter.
61
