It is my distinct pleasure to briefly discuss Horng-Tzer Yau's research accomplishments on this Poincaré prize occasion. I do this as a friend, a colleague at the Courant Institute of Mathematical Sciences of New York University for eleven years and as a neighbor in several senses.

First, we are office neighbors --- both on the eleventh floor at Courant (it is Courant and not Current, despite what appeared in the conference list of participants under my name). Second, we are apartment neighbors --- both living on the fourteenth floor at number 2 Washington Square Village. Third, we are scientific neighbors; although both working largely in Statistical Mechanics, I have focussed primarily on equilibrium issues and Yau on nonequilibrium. Indeed, my presentation today is given more as a scientific neighbor than as a participating expert in the field. As such, I have consulted with several of Yau's collaborators and several of his publications. One that I can highly recommend (that is, one of his publications, not one of his collaborators) is his lecture notes published in ``Current Developments in Mathematics -- 1998'' (that is Current and not Courant) and available on his web page. Before discussing his research, let me apologize for not having prepared extensive biographical information, but the skeletal version is as follows: born in Taiwan in 1959, B.Sc. from Taiwan National University, Ph.D. from Princeton University under Elliott Lieb, Professor at the Courant Institute, married, with one son.

Let me now proceed with a general research overview. Yau's research activites have ranged over physical systems on many scales from microscopic to astronomical, but he has concentrated on reinterpreting descriptive models of macroscopic behavior within the context of statistical mechanics.

Early in his career, he focused on Coulomb-type systems in many-body quantum mechanics. Notable papers from this period include ``The N^7/5 law for charged bosons'' with Conlon and Lieb, and with Lieb: ``The Chandrasekhar theory of stellar collapse'' and ``The stability and instability of relativistic matter''.

Yau then shifted toward explaining the macroscopic properties of fluids based on microscopic models of their constituent particles --- that is, deriving hydrodynamic limits (such as the Euler and Navier-Stokes equations of classical fluid mechanics) for stochastic particle systems. Among the notable work here are: the introduction in a 1991 paper of the relative entropy method for proving hydrodynamic limits, the derivation of spectral gap estimates describing convergence to equilibrium for the very difficult case of Kawasaki dynamics (where, unlike in Glauber dynamics, the particle number is conserved), work with Olla and Varadhan applying the relative entropy method to obtain the compressible Euler equations for large Hamiltonian systems perturbed by small noise, and work with Esposito and Marra, with Landim and with Quastel on deriving the incompressible Navier-Stokes equations. More recently, Yau has worked with Erdös on deriving the Boltzmann equation as a weak coupling limit of a random Schrödinger equation and with Nachtergaele on the Euler equation limit of quantum systems.

Besides the relative entropy method, the analytic tools used in deriving various macroscopic equations include: perturbation theory, large deviation theory, logarithmic Sobolev inequalities, renormalization/coarse-graining methods and multiscale analysis. In addition to the physical applications, from a mathematical point of view, Yau's work has contributed to probability theory, nonlinear partial differential equations, spectral theory and dynamical systems.

In the latter part of my presentation, I will attempt to give a few more details about some of Yau's work on classical equations obtained as scaling limits of large microscopic systems. The underlying microscopic physics is described by either Newton's equations or the Schrödinger equation, but I will focus today on the former case. (Erdös' lecture later this morning and Yau's lecture on Saturday will focus on the latter quantum case.)

The limiting equations are either the Boltzmann equation, the Euler equations or the Navier-Stokes equations. The Boltzmann equation describes the evolution of the probability density for the position and velocity of a single particle; it should be valid in the Grad limit of low density when space and time scale proportionally so that the mean number of collisions per particle per unit time stays finite. In the Euler limit, space and time still scale proportionally, but the density is of order one. The Euler equations involve (slowly-varying) macroscopic quantities such as density, velocity and energy. In the Navier-Stokes limit, density is still of order one, but space scales like the square root of time --- the so-called diffusive scaling. Euler and Navier-Stokes are both examples of hydrodynamic limits and I will focus on these.

The heuristic derivations of these hydrodynamic limits involve the following main ingredients. The Euler limit requires that (i) the microscopic dynamics are locally in equilibrium and (ii) the Boltzmann-Gibbs principle that local equilibria are Gibbs distributions parameterized by the macroscopic variables: mean density, mean velocity and mean energy. The Navier-Stokes limit requires further that the system is diffusive and that the fluctuations of the currents about their means are Gaussian.

In the joint work with Olla and Varadhan, a small amount of noise is added to the Newtonian dynamics, and the Euler equations are derived in a limit where that noise is also scaled to zero. One important result in that paper is a partial verification of the Boltzmann-Gibbs principle, a principle known to be not true in general since there are non-Gibbsian singular stationary measures of the dynamics: they proved, however, that all stationary measures with finite specific entropy are Gibbsian provided there are no correlations in their velocity distribution. It remains an open problem to rule out such velocity correlations. As to the issue of local equilibration, a basic problem here is to show that this property is preserved by the dynamics (on the appropriate space and time scales). Showing this preservation requires a careful understanding of relaxation to equilibrium. That understanding was obtained in this same paper by means of finite specific entropy together with the idea from Yau's 1991 paper to use entropy as a measure of the distance between the time evolution of the system and a carefully chosen (time-dependent) local equilibrium state.

In the case of the Navier-Stokes limit, the basic technical work requires qualitative estimates on the correlations of particles in large dynamical systems. This was carried out by Yau with Esposito and Marra and with Landim for stochastic lattice gas dynamics in dimensions three or more. It is quite interesting that the work with Landim and with Quastel suggests that the diffusive/Gaussian fluctuation property is not valid for dimensions two or less and thus that the applicability of the incompressible Navier-Stokes equations in, say, two dimensions, cannot be taken for granted.

It is clear that, overall, Horng-Tzer Yau's research involves strong physical intuition, very hard computational skills and tough estimation. To paraphrase an old letter of recommendation I looked at while preparing these remarks, the awarding to Yau of a Poincaré prize is an outstanding acknowledgement, but only of his achievements to date --- there are certainly many good things to look forward to.