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.
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 $\ d{\bf R}={\bf v}(t,{\bf R})\ $ 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.
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.
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.
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.
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.
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.
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.
Quantum chromodynamics (QCD) on a finite lattice $\Lambda$ in the Hamiltonian approach is analyzed. First, we present the field algebra ${\mathfrak A}_{\Lambda}$ as comprising a gluonic part, with basic building block being the crossed product $C^*$-algebra $C(G) \otimes_{\alpha} G$, and a fermionic (CAR-algebra) part generated by quark fields. By classical arguments, ${\mathfrak A}_{\Lambda}$ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra ${\mathfrak O}^i_{\Lambda}$ 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 $\Lambda$ with the "rest of the world", we have to extend ${\mathfrak O}^i_{\Lambda}$ by tensorizing with the algebra of external gauge invariant operators. This way we construct the full observable algebra ${\mathfrak O}_{\Lambda}$. We prove that its irreducible representations are labelled by ${\mathbb Z}_3$-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 ${\mathbb Z}_3$-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, ${\mathfrak O}_{\Lambda}$ is discussed in terms of generators and relations.
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.