# International Symposium on Mathematical Sciences

## Abstracts for the talks

### Incompressibility

**Stuart S. Antman** *(University of Maryland, College Park)*

Thursday, October 06, 2005

A material body is incompressible if every deformation of it locally
preserves its volume, in particular, if the Jacobian determinant of every
continuously differentiable deformation of it is identically 1. (Rubber
and much living tissue (which is composed mostly of water) are examples of
incompressible materials.) Since the nonlinear PDEs of evolution for such
3-dimensional bodies have largely resisted analysis, it is useful to have
effective theories for slender bodies governed by equations with but one
independent spatial variable. This lecture shows that the actual
construction of one such very attractive theory requires the solutions of
a sequence of first-order PDEs (by the method of characteristics).
Although the resulting equations are more complicated than those for
bodies not subject to the constraint of incompressibility, they admit some
tricky a priori bounds and they have novel regularity properties not
enjoyed by the latter. The governing equations for an elastic body can be
characterized by Hamilton's Principle. The ODEs governing travelling
waves for these equations can also be characterized by Hamilton's
Principle, but the kinetic and potential energies for these ODEs do not
correspond to those of the PDEs. These ODEs, which have a nonstandard
structure, admit, under favorable assumptions, periodic travelling waves
with wave speeds that are are supersonic with respect to some modes of
motion and subsonic with respect to others.

### Linear hyperbolic equations in a rotating black hole geometry

**Felix Finster** *(Universität Regensburg, Regensburg)*

Saturday, October 08, 2005

Linear hyperbolic equations describe the dynamics of quantum mechanical particles (Dirac equation) and of classical waves (equations for scalar, electromagnetic or gravitational waves). After a brief review of relativity and black holes, we consider the Cauchy problem for a linear hyperbolic equation in the Kerr geometry, the mathematical model of a rotating black hole. For the Dirac equation, an integral representation of the propagator is obtained, which yields pointwise decay and allows to develop the complete scattering theory. We also outline the analysis for the scalar wave equation, which is considerably harder due to the ergosphere, an annular region around the black hole where the classical energy density may be negative. We mention recent results for pointwise decay and discuss the phenomenon of superradiance. We finally give an outlook on electromagnetic and gravitational waves, and to the problem of linear stability of rotating black holes.

### Loop Quantum Gravity

**Christian Fleischhack** *(MPI MIS, Leipzig)*

Friday, October 07, 2005

Loop quantum gravity is one of the most prominent approaches to quantum
gravity. Starting with its origins, we outline the gauge-theoretic and
quantum-geometric foundations. Then the main achievements are summarized.
Here, we will focus on black holes and quantum cosmology. Finally, we are
going to discuss perspectives, limitations and open problems of the
theory.

### Turbulent transport and integrable chaos

**Krzysztof Gawedzki** *(CNRS ENS, Lyon)*

Wednesday, October 05, 2005

Many basic features of transport by turbulent
flows may be captured by simple models where turbulent
velocities are described by an imposed random ensemble and
the transported matter (pollutant) or fields (temperature,
magnetic field) are assumed to be carried by the flow
without influencing it. From the mathematical point of
view, such models are random dynamical systems where one
studies the flow equations
with a random right hand side. Such systems come in two
sorts: the more standard one, with typical velocities
smooth in space, used to model flows at moderate Reynolds
numbers, and a less standard one, with spatially rough
velocities, that applies to high Reynolds number flows.
In the simplest model proposed by Robert Kraichnan almost
40 years ago, the velocities are assumed to form a white
noise in time and many of the interesting questions, some
traditionally asked for dynamical systems, some going
beyond, find analytic answers. As an illustration, I shall
sketch how some known integrable models of quantum mechanics
provide the control of large deviations of finite-time
Lyapunov exponents of the Kraichnan model. Such large
deviations determine the decay of temperature fluctuations
and the growth of pollutant or magnetic field inhomogeneities
as well as the multifractal properties of particle suspensions.

### Global Charge, Gauss Law and Superselection Rules for QCD on the Lattice

**Jerzy Kijowski** *(Polish Academy of Sciences, Warsaw)*

Friday, October 07, 2005

Quantum chromodynamics (QCD) on a finite lattice
in the Hamiltonian approach is analyzed. First, we present the field
algebra as comprising a gluonic part,
with basic building block being the crossed product -algebra
, and a fermionic (CAR-algebra) part
generated by quark fields. By classical arguments, has a unique (up to unitary equivalence) irreducible
representation. Next, the algebra of
internal observables is defined as the algebra of gauge invariant
fields, satisfying the Gauss law. In order to take into account
correlations of field degrees of freedom inside with the
``rest of the world', we have to extend by tensorizing with the algebra of external gauge invariant
operators. This way we construct the full observable
algebra . We prove that its
irreducible representations are labelled by -valued
boundary flux distributions. Then, it is shown that there exist
unitary operators (charge carrying fields), which intertwine
between irreducible sectors leading to a classification of
irreducible representations in terms of the -valued
global boundary flux. By the global Gauss law, these 3
inequivalent charge superselection sectors can be labeled in terms
of the global colour charge (triality) carried by quark fields.
Finally, is discussed in terms of
generators and relations.

### Asymptotic Completeness and Celestial Mechanics

**Andreas Knauf** *(Universität Erlangen-Nürnberg, Erlangen)*

Friday, October 07, 2005

We discuss the notions of asymptotic completeness of
classical and quantum n-body scattering, and present
results obtained in the nineties by different authors
for the quantum case and, for smooth potentials,
in classical mechanics.
Then we explain pitfalls and partial results regarding unbounded
two-body interactions, including the case of gravitational attraction.

### A new mathematical foundation for contact interactions in continuum physics

**Friedemann Schuricht** *(Universität zu Köln, Köln)*

Friday, October 07, 2005

The investigation of contact interactions, such as traction and heat flux, that are exerted from contiguous bodies across the common boundary is a fundamental issue in continuum physics. However, the traditional theory of stress established by Cauchy and extended by Noll and his successors is insufficient for needs in modern physics where one has to handle lack of regularity that is present in shocks, corners, and contact of deformable bodies. The talk provides a new mathematical foundation to the treatment of contact interactions. Based on mild physically motivated postulates, that essentially differ from those used before, the existence of a corresponding interaction tensor is verified. While in former treatments contact interactions are basically defined on surfaces, here contact interactions are rigorously considered as maps on pairs of subbodies. This allows to define the interaction exerted on a subbody not only, as usual, for sets with a sufficiently regular boundary but for any Borel set (which includes all open and closed sets). In addition to the classical representation of such interactions by means of integrals on smooth surfaces, a general representation using the distributional divergence of the tensor is derived. In the case where concentrations occur the new approach allows a more precise description of contact phenomena than before.

### Quantum fields in curved spacetime, a grand tour

**Rainer Verch** *(MPI MIS, Leipzig)*

Friday, October 07, 2005

I will review the development of quantum field theory in curved spacetime
from its beginnings towards the new developments of current research.
Emphasis will be put on the modern mathematical ond conceptual
developments, among them operator algebraic and microlocal methods,
the microlocal spectrum condition, quantum energy inequalities [on which
there are also talks in the workshop program of Oct 3rd-4th by some of
the leading specialists in the field] and generally convariant quantum
field theory. I will review some of the recent results of these
developments, and give an outlook on the lines of future research.

### Random continuous planar systems

**Wendelin Werner** *(Université Paris-Sud, Orsay)*

Wednesday, October 05, 2005

I will describe some aspects of the recent progress made by
mathematicians in the understanding of the (possible) conformally invariant
scaling limits of two-dimensional physical systems, their relation to complex
analysis, representation theory and the new light that they shed on conformal
field theory.

### Three levels in the theory of quantum groups

**Stanislaw Lech Woronowicz** *(Warsaw University, Warsaw)*

Wednesday, October 05, 2005

Quantum groups may be considered on three levels. On the Hopf algebra (or Hopf *-algebra)
level we deal with polynomial functions on the group. Quantum groups appears as deformations
of classical algebraic groups. One the other side we have C*-level. On this level we work
with quantum versions of locally compact topological groups and the concepts and methods of
functional analysis are intensively used. In between we have Hilbert space level, where we deal
with closed operators acting on a Hilbert space which are interpreted as coordinates on quantum
groups.

With a number of examples we shall discuss the characteristic features of each level and the
relations between the levels. In particular the speaker hopes to attract the listeners attention
to the Hilbert space level.

## Date and Location

**October 05 - 08, 2005**

Universität Leipzig

Linnéstr. 5

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Jürgen Jost**

Max Planck Institute for Mathematics in the Sciences

Leipzig

**Stefan Müller**

Max Planck Institute for Mathematics in the Sciences

Leipzig

**Klaus Sibold**

Universität Leipzig

Institut für Theoretische Physik (ITP)

Leipzig

## Administrative Contact

**Katja Bieling**

Max Planck Institute for Mathematics in the Sciences

Contact by Email

**Antje Vandenberg**

Max Planck Institute for Mathematics in the Sciences

Contact by Email