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ICMP 2003 > Plenary lectures |
These were one hour lectures. No other Congress activities took place during these lectures.
The invited speakers were:
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On the relation between the Master equation and the Boltzmann Equation in Kinetic Theory
This talk is based in recent joint work with Maria Carvalho and Michael Loss. It has its origins in an old proposal of Kac to study relaxation to equilibrium in Kinetic Theory problems directly in terms of an evolution equation -- the so-called "Master equation" for a stochastic system of N randomly colliding particles. Kac proved a relation between the solutions of the Master equation, and the solutions of a corresponding Boltzmann equation. He conjectured that the latter could be studied in terms of the former in a quantitative way. Specifically, he conjectured that the N dependence in the spectral gap associated to the master equation was such that it would imply a relaxation rate for the Boltzmann equation.
Solving this problem requires a method for precisely controlling the N dependence of quantities like the spectral gap in the generator for a stochastic evolution. For a one dimensional model of collision in which energy is conserved but not momentum, this was done by Janvress in 2000 using Yau's martingale method. Our approach exploits a permutation invariance to reveal a very simple relation between the relaxation rate for N particles and for N+1. This relation does not involve any details of the dynamics, and is in a strong sense purely kinematical. This has the remarkable consequence that the details of the dynamics enter the problem only at the level of 2 particles.
This method is robust and direct enough that it can be applied to physically realistic collisions that conserve energy and momentum. It can be uses to bound, and in some cases compute, the spectral gap for a number systems of many particles evolving under the influence a random collision mechanism. We will also discuss recent work on entropic correlation bounds needed in this framework to compute rates of relaxation in entropic terms.
Symmetries and "simple" solutions of the classical n-body problem
The Lagrangian of the classical n-body problem has well known symmetries: isometries of the ambiant euclidean space (translations, rotations, reflexions) and changes of scale coming from the homogeneity of the potential. To these symmetries are associated "simple" solutions of the problem, the so-called homographic motions, but also the classical subproblems (planar, isosceles). These symmetries of the Lagrangian imply symmetries of the action functional, defined on the space of regular enough loops of a given period in the configuration space of the problem. Minimization of the action under well-chosen symmetry constraints leads to remarkable "simple" solutions which could share with the homographic ones the role of organizing centers in the global dynamics of the problem.
Relativistic models in atomic and molecular physics
In this talk I will present recent results on various models describing stationnary states for atoms and molecules in which the relativistic effects are not neglected.
All these models are based on the Dirac operator and are commonly used in computational Atomic Physics and Quantum Chemistry. The main topics to be addressed are:
Locally covariant quantum field theory
A principle of local covariance is imposed on quantum field theories on curved spacetimes. The implications of this principle for renormalization, PCT symmetry and the physical interpretation of the theory are discussed.
Simple models of turbulent transport
Turbulent transport of pollutants, heat, chemical agents or magnetic fields is a nonequilibrium phenomenon playing an important role in environmental issues, meteorology, engineering or astrophysics. The simplest mathematical models of turbulent transport study advection of scalar or vector quantities by random velocities with prescribed statistics. Such reduced systems show many expected features of the full-fledged nonlinear models of developed hydrodynamical turbulence, including cascades of conserved quantities and intermittency. Mathematically, they provide examples of random nondifferentiable dynamical systems. Analysis of the simplified models of advection has allowed to relate the transport properties of flows to nonstandard behaviors of fluid particles that is made possible by the lack of regularity of turbulent velocities. The models exhibit a new robust mechanism of intermittency involving hidden statistically conserved quantities. Its understanding has led to a perturbative calculation of the intermittency exponents.
Algebraic versus Liouville integrability of the soliton systems
Since the beginning of the 1970s the complete Liouville integrability has been considered as common characteristic features of all the soliton systems constructed in the framework of the inverse spectral transform method. Bi-Hamiltonian formalism, classical r-matrix approach or representation of these systems, as a result of the Hamiltonian reduction of "free" systems, are often considered as the defining starting points of the Hamiltonian theory of the soliton equations. Although each of these approaches has its own advantages, none of them is applicable to all known integrable systems.
More universal is the "definition" of the soliton equations as non-linear (ordinary or partial) differential equations that are equivalent to compatibility conditions of over-determined systems of linear equations. In this approach the direct and the inverse spectral transforms "solve" the non-linear equations with no use of the Hamiltonian theory. In particular, for finite-dimensional systems that admit the Lax representation ∂tL=[M,L], where L(t,z) and M(t,z) are rational matrix functions of the spectral parameter z, the algebro-geometric scheme based on the concept of the Baker-Akhiezer functions, gives an explicit solution of the equations in terms of the Riemann theta-functions. This scheme identifies the phase space of the systems with the Jacobian bundle over the space of the spectral curves. The Abel transform linearizes the motion along the fibers of the bundle. Although in this description the corresponding soliton system clearly exhibits dynamics of the completely integrable system, until the end of the 1990s there was no universal answer to the question: why the Lax equations are Hamiltonian.
The new approach to the Hamiltonian theory of the soliton equations developed in the works of D.H. Phong and the author is based on the discovery of some universal two-form defined on a space of meromorphic matrix-functions. A direct and simple corollary of the definition of this form is that its contraction by a vector-field defined by the Lax or the zero-curvature equations is an exact one form. Therefore, whenever the form is non-degenerate the corresponding system is the Hamiltonian system.
This talk presents the extension of our approach on the Lax and the zero-curvature equations with the spectral parameter on an algebraic genus g>0 curve will be presented. The corresponding systems can be regarded as infinite-dimensional field analogs of the famous Hitchin systems.
The main goal of the talk is to introduce new integrable systems associated with discrete version of the zero-curvature equations with the spectral parameter on an algebraic curve Γ. The phase space of these systems is the space of meromorphic homomorphisms Ln∈ H0(Hom(Vn, Vn+1(D+))), where Vn is a sequence of stable rank r>1 holomorphic vector bundles on Γ, and D+ is a fixed divisor. It is assumed that Ln is almost everywhere invertible and the inverse homomorphism has a fixed divisor of poles D-, i.e. Ln-1∈ H0(Hom(Vn+1, Vn(D-))). We show that a hierarchy of commuting difference-differential equations of the periodic Toda-lattice type on this space is algebraically integrable. For periodic chains Ln=Ln+N the equations are linearized by the spectral transform and can be explicitly solved in terms of the Riemann theta-function of the spectral curve defined by the monodromy operator T=LNLN-1...L1.
In certain sense, the corresponding systems on algebraic curves of genus g>1 can be seen as super integrable, because the dimension of the Jacobians of the spectral curves is bigger than the dimension of a space the integrals. On the other hand, these systems can not be treated as conventional Hamiltonian systems. Any two-form vanishing along the Jacobians is degenerate. Instead we find a family of compatible two-forms, which when contracted by the Lax vector-field are exact one-forms. Although each of the forms is degenerate the family as a whole is non-degenerate.
Long-range order and diffraction in mathematical quasicrystals
Physical quasicrystals are more or less characterized as non-crystalline materials that have very pronounced Bragg spectra (many point-like bright spots). Their existence has given new relevance to the question of determining which types of point sets are actually pure point diffractive.
One of the well known ways to construct pure point diffractive sets is by use of the cut and project formalism. In recent years considerable progress has been made in reversing the roles and producing the cut and project formalism out of pure pointedness.
In this talk we discuss the background and some of the recent developments of the sujbect. Key roles are played by dynamical hulls, almost periodic measures and their Fourier transforms, and the interweaving of a pair of topologies that see the systems from very different points of view.
Critical percolation and conformal invariance
Many 2D latttice models (critical percolation, self avoiding random walk, Ising model at critical temperature, ...) are believed to have conformally invariant scaling limits. This belief allowed physicists to predict (unrigorously) many of their properties, including exact values of various dimensions and scaling exponents.
We will describe some of the recent progress in the mathematical understanding of these models and will discuss open problems and perspectives. We will start with the proofs of several predictions for the critical site percolation on triangular lattice in the plane, including the Cardy's formula (conformal field theory prediction for the probabilities of crossings) and its application to the construction of the conformally invariant continuum scaling limit. We will also describe connection with the recent work of Lawler, Schramm, and Werner on the Schramm-Loewner Evolution.
The energy of charged matter
In this talk I will discuss some of the techniques that have been developed over the past 35 years to estimate the energy of charged matter. These techniques have been used to solve stability of (Fermionic) matter in different contexts, and to control the instability of charged Bosonic matter. The final goal will be to indicate how these techniques with certain improvements can be used to prove Dyson's 1967 conjecture for the energy of a charged Bose gas---the sharp N7/5 law.
Strings through the microscope
Over the last few years, string theory has changed profoundly. Most importantly, novel duality relations which involve open strings attached to branes on one side and various closed string backgrounds on the other, have blurred the traditional distinction between gauge- and string theory. In my lecture, I will introduce the fundamental ingredients of modern string theory and explain how they are modeled through 2D (boundary) conformal field theory. Recently, this so-called `microscopic description' of strings and branes has been a field of active research producing many new results, ranging from the classification and construction of boundary conditions to studies of 2D renormalization group flows. I will present a few highlights of these developments in the second part before concluding with an extensive outlook on some present and future research that is motivated by current problems in string theory. This includes, in particular, investigations of non-rational and non-unitary conformal field theories.
Entropy production and convergence to equilibrium for the Boltzmann equation
Even in a classical setting, it is still a major open problem of nonequilibrium statistical physics to prove that solutions of the Boltzmann equation, starting far from equilibrium, converge towards the maximum entropy state as time becomes large. To solve this problem in a satisfactory way, one would like to obtain constructive, quantitative estimates about the speed of convergence, because the Boltzmann approximation of particle dynamics breaks down in very large time. It was recently shown by the author in collaboration with Desvillettes that such estimates can be derived as soon as one has some a priori estimates about the smoothness, positivity and large-velocity decay of the solutions of the Boltzmann equation. The proof involves the resolution of a longstanding conjecture by Cercignani about Boltzmann's entropy production functional, with the help of some tools from information theory and hypercontractivity. A new method was developed to overcome the major difficulty caused by the degeneracy of the Boltzmann collision operator with respect to the position variable. Surprisingly, a new inequality a la Korn was also needed. In my talk, I will review these recent developments and the ideas underlying the proofs.
Aspects of free probability
Free probability theory provides a probabilistic framework for quantities with the highest degree of noncommutativity. This brings out the ties among von Neumann algebras of free groups, the large N limit of random multimatrix models, the operators of the Boltzmann Fock space and combinatorics of noncrossing partitions.
Many concepts of classical probability theory have free probability counterparts. In particular the free analogues of entropy and of Fisher's information will be one of the main focuses of the talk.
The analysis of the free variables involved, relies on free difference quotient derivations, which give rise to bialgebras in the class with derivation comultiplication, which appears to be selfdual. Duality for such bialgebras underlies the free entropy and analytic aspects of the free Markov property.