ICMP 2003 > Young Researchers Symposium > YRS plenary lectures | |
These were one hour lectures. No other activities took place during these lectures.
M. Aizenman | Princeton |
J. Baez | Riverside |
T. Gannon | Alberta |
V.F.R. Jones | Berkeley |
M. Loss | Atlanta |
M. Mariño | CERN |
Y.G. Sinaï | Princeton |
H. Araki | Kyoto |
E. Lieb | Princeton |
O. Schramm | Microsoft |
The SK spin-glass puzzle
The statistical mechanics of spin-glass models is characterized by the existence of a diverse collection of competing states, slow relaxation of the quenched dynamics and a rather involved picture of the equilibrium state. A great deal of insight on systems with such characteristics has been produced through the study of the Sherrington-Kirkpatrick model, in particular through the "ultrametric", or "hierarchal", ansatz proposed by G. Parisi. The talk will present some recent developments with implications on the question of the validity of this solution-facilitating ansatz, an issue which is still unresolved but on which a new perspective has recently emerged.
Equilibrium statistical mechanics of quantum lattice
systems
[Henri Poincaré Lecture]
Recent work on equilibrium statistical mechnics of quantum lattice systems will be reviewed. A new feature of the problem is the mutual non-commutativity of operators on different lattice sites. A new method is a commuting system of C*-conditional expectations which is used for the unique association of the standard potential to a given dynamics or more precisely to the time derivative (at time 0) of strictly local operators, and for estimates without any assumption on the norm of individual potentials. Under more or less minimal assumptions on the dynamics, the equivalence of various characterization of equilibrium states, such as the KMS condition, the variational principle (for translation invariant states) and the local thermaodynamical stability, follows.
Struggles with the continuum
Many of the most important theories of physics are still lacking a completely rigorous foundation, due to problems associated with the assumption that spacetime is a continuum. Regardless of whether spacetime is really a continuum, solving these problems --- or proving they cannot be solved --- is an enormous challenge for mathematical physics. Here we tour some important theories and the basic open questions associated with them. The best known examples concern the difficulty in rigorously constructing interacting quantum fields in 4-dimensional spacetime, due largely to ``ultraviolet divergences'' arising from arbitrarily small distances between points. Less widely known is that there is still no completely satisfactory formulation of the classical electrodynamics of point particles --- or even the Newtonian mechanics of point particles interacting by an attractive inverse square force law!
Moonshine beyond the Monster
In this talk we present a few snapshots of Moonshine: a bridge linking various algebraic structures (such as lattices, Kac-Moody algebras, and the Monster finite group), to number theory (such as theta functions, Jacobi forms, and Hauptmoduls). We will see how this leads us to string theory, vertex operator algebras, and knots.
In and around the origin of quantum groups
I will present several topics related to the development of quantum groups including solvable lattice models, braid groups, subfactors and the McKay correspondence. The latter will lead to the notion of ``finite subgroups'' of quantum groups and their graphs, as first pursued by Zuber and di Francesco. Subfactors will provide a framework where all these ideas and structures exist simultaneously.
A second look at the Second Law of Thermodynamics
[Henri Poincaré Lecture]
The essence of the second law of thermodynamics is the statement that all adiabatic processes (slow or violent, reversible or not) can be quantified by a unique entropy function, S, on the equilibrium states of all macroscopic systems, whose increase is a necessary and sufficient condition for such a process to occur. It is one of the few really fundamental physical laws in the sense that no deviation, however tiny, is permitted and its consequences are far reaching. Since the entropy principle is independent of any statistical mechanical model, it ought to be derivable from a few logical principles without recourse to Carnot cycles, ideal gases and other assumptions about such things as 'heat', 'hot' and 'cold', 'temperature', 'reversible processes', etc. Indeed, temperature is a consequence of entropy rather than the other way around. In this lecture on joint work with Jakob Yngvason, the foundations of the subject and the construction of entropy from a few simple, physical principles will be presented.
References:
Stability of matter and its implications for quantum electrodynamics
We review the stability of matter problem and its consequences for quantum electrodynamics. The subject started with the result of Dyson and Lenard that non relativistic matter is stable. Since then similar results have been obtained, mainly by Lieb and collaborators, for models that include relativistic features and interaction with fields, classical and quantized. We shall present these results and end with a discussion of the relativistic "no pair" model. This model has the unusual feature that for stability to hold, one is forced to consider a Hilbert space in which the electrons and photons are inextricably linked.
Knot invariants and string theory
In the last years we have witnessed a rather remarkable connection between knot invariants obtained from quantum groups, and a certain class of string theories. One consequence of this relation is that knot invariants like the HOMFLY polynomial and its generalizations can be written in terms of enumerative invariants (more precisely, open Gromov-Witten invariants). Conversely, complicated enumerative problems can be completely solved for a certain class of algebraic varities by using quantum group invariants. In this talk I will review these developments and point out various avenues for further research.
Scaling limits of random 2d processes
[Henri Poincaré Lecture]
We will survey a recent theory describing precisely the scaling limits of many random systems in two dimensions. Random paths associated with each of these systems are believed to converge to a path among a one-parameter family of curves called stochastic Loewner evolution (or SLE). Several instances of this statement have been proven, for example, percolation and loop-erased random walks, while others are still conjectural, e.g., the Ising and Potts models and the self-avoiding walk. The theory is useful, mainly because the SLE description facilitates explicit calculations of properties of the scaling limits.
Navier-Stokes equation as a dynamical system
Navier-Stokes equation is one of the most important equations in hydrodynamics and mathematical physics. The properties are quite different in 2D case and 3D case. I shall make a short review of 2D hydrodynamics, also with random forcing. The main emphasis will be on the 3D case.