Current Workshop  Overview  Back  Home  Search   
Noncommutative Algebraic Geometry and its Applications to Physics

Topic: the workshop is about the
interaction and/or unification of various flavors of noncommutative
geometry (such as algebraic, categorical, differentialgeometric), in relation
to their applications in theoretical physics, notably quantum field theory and
string theory. Description
and aim There are many incarnations of the notion of noncommutative
geometry. After the work of A. Connes, noncommutative geometry emerged as a new and powerful
evolution of modern differential geometry. Among other things, it allows for a
natural description of singular spaces like leaf spaces of foliations and its
applications range in such disparate fields as the standard model of particle
physics and the geometric interpretation of formulas of number theory. There is another variant, known as “projective noncommutative geometry”, that appeared in the mid 80’s,
owing to work of M. Artin, J. Tate and M. van den
Bergh. Roughly speaking, it deals with noncommutative
analogues of algebraic varieties. It is based on the possibility to somehow
twist the usual homogeneous coordinate ring of a projective variety to obtain noncommutative analogues of various geometric objects of
projective geometry. These noncommutative analogues
are usually described in terms of an abelian and/or
triangulated category, possibly endowed with additional structure. The link
with ordinary algebraic geometry stems from the possibility of reconstructing,
under some conditions, a scheme its derived category of quasicoherent sheaves. There are other ways to describe noncommutativity
in the realm of algebraic geometry (and associated to such names as A.
Rosenberg, M. Kontsevich, V. Ginzburg, ...). All these
approaches have an interesting and intriguing common point, that could be
called a “noncommutative hamiltonian
formalism” (Kontsevich necklace brackets, Le Bruyn and CrawleyBoevey’s noncommutative symplectic geometry,
double Poisson structures of M.Van den Bergh, et
cetera), and which includes, among other things, noncommutative
Poisson geometry, CalabiYau algebras, preprojective algebras, quiver representations. We believe that now is exactly the time where all these
geometries have ripened to a stage where they are welldeveloped enough to
start interacting — whence the timeliness of the workshop. Much of this research is inspired and motivated by questions of
modern theoretical physics and, reciprocally, some important mathematical
discoveries have generated new directions in modern quantum field theory. Thus Kontsevich’s results, like deformation quantization of
Poisson phase spaces and formality, inspired a number of noncommutative
quantum field theory models (SeibergWitten, A.
Schwarz, M. Douglas, D. Gross, N. Nekrasov
and others). Moreover, it appears that the “integrable
sector” in these examples has deep relations with the abovementioned noncommutative algebrogeometric
constructions and has produced results such as noncommutative
instantons, twisted Hilbert schemes of points on
complex plane, or CalogeroMoser spaces. The aim of the workshop to provide a unified view of all these
different aspects of noncommutative geometry, and
help the “physics oriented” young researchers to understand the possible
interactions and respective advantages of different approaches visavis the challenges and the
demands of modern quantum field theory. The Workshop will have an
interdisciplinary nature, not only because of the participation of scientists
both from the physics and mathematics communities, but also for the diversity
of topics that will be touched (algebra, algebraic geometry, differential
geometry, category theory, functional analysis in mathematics, quantum
mechanics, quantum field theory, string theory, integrable
systems in physics). This topic lies in the intersection of some very “hot” areas of
research in presentday pure mathematics and mathematical physics, such as noncommutative geometry, the theory of geometric
invariants, enumerative geometry, string theory, integrable
systems. We believe that for this reason this workshop will attract
considerable interest from the mathematics and physics communities. [Back] 