Oberwolfach Reports
Full-Text PDF (626 KB) | Introduction as PDF |
Table of Contents | OWR summary
Deformations and Contractions in Mathematics and Physics
Organized by: Marc de Montigny (1), Alice Fialowski (2), Sergey Novikov (3) and Martin Schlichenmaier (4)
(1) Faculte Saint-Jean, Physics Dept., University of Alberta, 8406-91 Street, AB T6G 4G9, EDMONTON, CANADA(2) Department of Analysis, ELTE TTK, Pazmany Peter setany 1/c, H-1117, BUDAPEST, HUNGARY
(3) Institute for Physical Science and Technology, University of Maryland, MD 20742-2431, COLLEGE PARK, UNITED STATES
(4) Laboratoire de Mathématique, Université du Luxembourg, 162A, Avenue de la Faiencerie, 1511, LUXEMBOURG, LUXEMBOURG
Deformations of mathematical structures are not only important in most
parts of mathematics but also to a large extend in physics.
Contractions are in some respect dual to deformations.
The aim of the proposed workshop was to bring together world
experts in these complementary topics of deformations
and contractions of various algebraic structures.
Deformations and contractions have been investigated by researchers
who had different approaches and goals.
Tools such as cohomology, gradings, etc. which are utilized in
the study of one concept, are likely to be useful
for the other concept as well.
At this meeting there were mathematicians,
mathematical physicists and physicists as well.
The organizers hope that the
meeting was of benefit to all groups.
Because various fields in mathematics and physics exist
in which deformations are used,
it was necessary to focus the topic of the workshop.
The meeting mainly
considered deformations of algebras (in particular, of
Lie algebras), groups, and related algebraic structures,
the corresponding contractions, and their applications to
problems in physics.
Nevertheless, other fields with
strong relations to the
central topic were present too.
One such field, discussed in detail at the workshop,
with tight interaction was deformation
quantization. But also other topics like
quantum groups, deformation of Hopf algebras,
q-deformed physics, fuzzy spaces,
quantum systems as deformations of classical systems, etc
showed up.
As the workshop had an interdisciplinary character it was considered
to be useful to start with some introductory talks on
\begin{enumerate}
\item
{Deformations in mathematics and physics,}
\item
Contractions of Lie algebras in physics,
\item
Cohomology and deformations,
\item
Deformation quantization,
\end{enumerate}
with the aim to introduce the necessary concepts which were not
always well-known to all the different communities present.
For more details on the concepts, see the corresponding extended abstracts
in this Oberwolfach report.
The following is a (non-exhaustive) list of topics which were
discussed at the workshop.
\begin{enumerate}
\item
The concept of rigidity and deformations in its different versions;
relations to cohomology, moduli spaces of algebras and existence
of versal families; for formally rigid infinite dimensional algebras
there exist nevertheless
global deformations which are locally non-trivial;
the deformations of enveloping algebras.
\item
Contractions and its relations to deformations
considered from a mathematical point of
view; the different concepts of contractions, generalized In\"on\"u - Wigner
contractions, graded contractions, degenerations, orbit closure,
jump deformations, expansions; invariants of Lie algebra.
\item
Contractions and their physical implications;
macroscopic quantum systems, local current algebras, supergravity,
regularisations, symmetry of the hydrogen atom.
\item
Deformation quantization; its application to field theory,
algebraic varieties, superformality, unimodular vector fields.
\item
Deformations of vector field algebras; its relation to
geometric moduli spaces, algebras of Krichever-Novikov type,
supergeometry
\item
Non-commutative spaces and related algebraic
objects; differential geometry of noncommutative spaces, Hopf algebras,
operads, quantum groups, curvature, elliptic gamma functions and triptic
curves.
\end{enumerate}
The workshop was attended by 44 participants from all over the world.
The official program consisted of 24 lectures. Two evening sessions of
informal presentations were organised. Beside the official program, there
was ample time for the participants for further
activities, such as self-organised sessions and discussion
groups.
No keywords available for this article.