Report to the 2002 General Assembly for 19992002, Berlin, Germany October 712, 2002 Activities The mandate of C18 covers the mathematical studies of problems originating in or relevant to physics, including mathematical models of physical systems, mathematical aspects of physical theories, and computational techniques.The current Commission has met twice during IUPAP sponsored conferences, once in London (2000) and once in Paris (2002). Approximately half the membership was present at each meeting. However, most of the discussions among members have been by email and any recommendations or motions have been put to an email vote. The primary objective of the Commission, during the past three years, has been to find useful ways to serve the community of mathematical physicists and represent its interests to IUPAP. A particular concern has been to promote events in which mathematicians and physicists are able to communicate effectively. With such objectives in mind, we have established strong interactions with the International Association of Mathematical Physicists (IAMP); for example, the President of IAMP was appointed as an Associate Member of C18 and we have frequently consulted with current and past members of the IAMP Executive Committee. As a service to the mathematical physics community, we have developed a C18 web site with links to the various bodies and research institutions in mathematical physics. The site also contains a listing of conferences in mathematical physics. Our hope is that the site will provide a useful resource and help maintain awareness of the many activities in the field.The C18 Commission has traditionally given its highest priority to supporting and cosponsoring the triennial International Congress of Mathematical Physics (ICMP) organized by the IAMP. The most recent ICMP was held in London at Imperial College from 1722 July 2000. It took the opportunity to herald in the new millennium with a number of innovations, including the inclusion in the program of parallel sessions aimed at serving the needs of young researchers. These sessions were well attended and much appreciated by many seasoned researchers interested in learning about a new subfield of mathematical physics as well as by the young researchers for whom they were designed. The practice is highly recommended for other large conferences. The tradition of cosponsoring the biannual International Colloquium on Group Theoretical Methods in Physics was continued with the 23rd conference in the series held in Dubna, Russia from 31 July to 5 August 2000. During the current year, C18 joined other IUPAP commissions in cosponsoring an International Conference on Theoretical Physics held in Paris from 2227 July 2002. The conference was held in the UNESCO building, which was provided free of charge so that the monies saved could be used to enable many scientists from developing countries to participate. The next ICMP conference is scheduled to take place at the University of Lisbon from July 28 to August 2, 2003. This conference will be preceded by a twoday symposium for young researchers and several satellite conferences are planned (cf. http://icmp2003.net/). The next International Colloquium on Group Theoretical Methods in Physics will be held in Cocoyoc, Mexico, from 26 August 2004. A topical conference on the foundations of quantum theory is planned for the centenary of von Neumann's birth in October 2003 in Budapest. D.J. Rowe, Chair RECENT DEVELOPMENTS Introduction Theory of Radiation The main results of this work are: proofs of the existence of the ground state, of exponential localization of the particle system in this ground state, and of instability of excited states of the particle system and their bifurcation into resonances when the coupling between the particles and radiation is taken into account. The mathematical theory developed provides an effective method for computation of radiative corrections and, in particular, Lamb shifts, and lifetimes of metastable states. This is done through application of a novel renormalization group technique. The latter uses a renormalization flow that acts directly on a space of quantum equations of motion or quantum Hamiltonians rather than on Green's or correlation functions. Although the BFS work covers many areas of physical interest, it assumes extremely small values for the fine structure constant and considers only nonrelativistic systems. In a related development Lieb, Loss, et al. obtained several results on the selfenergy and ground state valid for physically realistic values of the fine structure constant and some models of relativistic mechanics. In the last three years, there has been an explosion of activity, with crucial extensions and improvements of some of the results described above as well as important progress on other aspects of the problem of interaction of quantum radiation with nonrelativistic matter. Field and String Theory In particular departure from conformal invariance is still manageable for integrable perturbations of the quantum theory. Other backgrounds than anti de Sitter are under active study. One of the frontiers is to dispense with supersymmetry by breaking it for instance so as to handle realistic 4dimensional theories. Another important physical problem is the regularization of spacetime singularities (orbifold singularities) by quantum and string effects. The picture of a web of dualities some of them exchanging weak and strong couplings, as in the SineGordon Thirring equivalence, is emerging as a key to understanding nonperturbative processes. These dualities are intimately related to properties and the existence of certain arithmetic groups and modular forms that provide exact forms of partition functions and other amplitudes. The key to understanding weakstrong coupling dualities is the presence of certain extended solitonic objects, the socalled Dbranes. From a spacetime point of view, Dbranes arise by imposing Dirichlettype boundary conditions on the open string fields. Various Ktheories have been introduced to handle charges of these Dbranes. From a worldsheet point of view, Dbranes are described by boundary conformal field theories, and the use of techniques from integrable systems has proven to be quite successful. Dynamical processes involving Dbranes, such as creation and annihilation of Dbrane antiDbrane pairs due to tachyon condensation, have led to a renewed interest and remarkable progress in the investigation and reformulation of string field theories. Mirror symmetry is one other instance of (perturbative) duality. The mathematical study of mirror symmetry, and algebraic geometry in general, Algebraic geometry is expected to be even more useful and to benefit enormously from the supergravitystring theoryMtheory perspective and profusion of concrete and unexpected features. Finally noncommutative geometry is taking a prominent place in the field. It arises, for instance, in the description of Dbranes in the presence of a background twoform gauge field (Bfield). In general, noncommutative geometry is expected to be crucial in Quantum Gravity as the notion of a spacetime point looses its relevance at short distances. In fact, even the number of dimensions of spacetime itself is an artifact of a particular classical limit. Aside from String Theory, noncommutative geometry and its close cousin, quantum groups, are still very active fields of study. For instance, Hopf algebras have recently been used to handle perturbative quantum field theory expansions and renormalisation (Kreimer and Connes). Twodimensional conformal invariance. More recently, it has been realized that many twodimensional systems should have a much greater symmetry in the continuum limit. For certain models of geometric probability, such as percolation and selfavoiding walks in twodimensional lattices, conformal invariance has proved to be a useful tool. Furthermore, new universal geometrical properties emerge. It is possible to predict not only critical exponents, but also various universal probability distributions. Some of these properties are described in elementary terms by the stochastic Löwner evolution with parameter k, introduced in 2000 by Schramm. The evolution is a random process of growth of a set. It may be constructed from a Bessel process corresponding to dimension parameter d. When d is a natural number, this process is the radial component of a ddimensional Brownian motion. Moreover, these ideas can be extended to positive real d, and even to complex d. Since k = 4/(d 1), the most regular regime k < = 4 corresponds to 2 £•d when the Bessel process is repelled from the origin. The picture is that different models correspond to different values of k. Thus a selfavoiding walk should be described by k = 8/3, while percolation should be described by k = 6. Recently the conformal invariance of critical site percolation on the triangular lattice in the continuum lattice was proved by Smirnov. His work also provides a rigorous proof that k = 6 describes this system, and a verification of Cardy's 1992 formula for crossing probabilities as well as Aizenman's conjecture of their conformal invariance. This is a remarkable vindication of the ideas of conformal invariance, and it applies to probability models that are rigorously defined and intuitively understandable. Quantum Information Theory. Historically, proposals for quantum computation grew out of questions about irreversibility and the realization in the 1980's that in quantum mechanics gates would quite naturally be associated with unitary operators that are reversible. The potential power then comes from that fact that the action of a unitary operator on a superposition gives an effective parallelism. It was therefore rather surprising that in 2001 several groups found new methods for universal quantum computation in which the gate action is the result of irreversible measurements performed on the computer in a highly entangled state. In addition to being significant in their own right, these developments raise many challenging new mathematical questions in the theory of computation and classification of entanglement. Quantum information theory also provides new methods for cryptography, communication and such important new applications as digital signatures, which are much more tractable experimentally. The extension of Shannon's classical information theory to this noncommutative arena is nontrivial because of the much richer variety of resources and protocols, leading to new mathematical questions. Many of the necessary tools, such as the strong subadditivity of quantum mechanical entropy and the notion of completely positive maps, were developed by mathematical physicists studying the rigorous foundations of quantum statistical mechanics and the theory of open quantum systems in the 1970's. Among noteworthy recent developments are: the discovery of a simple closedform expression for the entanglementassisted classical capacity of a noisy quantum channel, King's use of maximal Lp norms to establish additivity of the minimal entropy and Holevo capacity for a large class of channels, and the discovery of a connection between the superadditivity of Holevo capacity and the entanglement of formation. Group representations, geometric quantization, and coherent states. Dilute Bose Gases. Chaos in the Outer Solar System. M.B. Ruskai, ViceChair 


