Some Aspects of Slow Relaxation to Equilibrium Gerard Ben Arous

Out of equilibrium systems sometimes have a real hard time relaxing to equlibrium. They tend to never find it for very long time scales, they get trapped on a very broad range of time and space scales, they age, they lose Markovianity (or gain memory), they can be rejuvenated etc. Good examples of these phenomena are seen in dynamics of glassy systems, or in diffusion on critical percolation clusters (the “ant in the labyrinth”). In this colloquium talk, I will try to survey some of the mathematics of the phenomena linked to slow relaxation for complex out of equilibrium statistical mechanical systems, using mainly the very simple ansatz introduced by the physicist J.P. Bouchaud. This simple “trap model” is now well understood. Its surprising relevance for harder questions is also slowly being proved. I will give some of the recent advances about the simple Bouchaud model and its scaling limit on various graphs. I will finally explain the relevance and the shortcomings of this ansatz for the more difficult examples mentioned above, as well as some of the numerous open questions. This is based on joint work with J. Cerny, T. Mountford, A. Bovier, V. Gayrard. Lecture Notes covering part of this talk can be found here: http://ima.epfl.ch/cmos/publications/benarous/benarous_67.pdf Thursday, September 28 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium MULTIPLICATIVE FUNCTIONS KANNAN SOUNDARARAJAN Stanford Abstract A multiplicative function f : N → C is a function satisfying f(mn) = f(m)f(n). Many naturally occuring functions in number theory are multiplicative. Over the last several years, Andrew Granville and I have been studying various features of multiplicative functions. I will discuss some aspects of this work. For example, I will answer the question of how many numbers up to a given number x are quadratic residues (you are free to choose the prime p so as to minimize the answer). As another example, I will discuss character sums and a recent improvement of a classical inequality of Polya and Vinogradov.A multiplicative function f : N → C is a function satisfying f(mn) = f(m)f(n). Many naturally occuring functions in number theory are multiplicative. Over the last several years, Andrew Granville and I have been studying various features of multiplicative functions. I will discuss some aspects of this work. For example, I will answer the question of how many numbers up to a given number x are quadratic residues (you are free to choose the prime p so as to minimize the answer). As another example, I will discuss character sums and a recent improvement of a classical inequality of Polya and Vinogradov. Thursday, October 5 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium SOLVABILITY AND THE NIRENBERG-TREVES CONJECTURE NILS DENCKER University of Lund Abstract In the 50’s, the consensus was that all linear partial differential equations were solvable. Therefore, it came as a surprise 1957 when Hans Lewy found a non-solvable complex vector field. The vector field is a natural one, it is the Cauchy-Riemann operator on the boundary of a strictly pseudo-convex domain. The fact is that almost all linear PDE’s are unsolvable, because of the Hrmander bracket condition. A rapid development in the 60’s lead to the conjecture by Nirenberg and Treves in 1969: condition (Ψ) is necessary and sufficient for the solvability of partial differential equations of principal type. This is a condition which involves only the sign changes of the imaginary part of the highest order term of the operator along the bicharacteristics of the real part. The Nirenberg-Treves conjecture has recently been proved, see Annals of Mathematics, 163:2, 2006. We shall present the background and the ideas of the proof.In the 50’s, the consensus was that all linear partial differential equations were solvable. Therefore, it came as a surprise 1957 when Hans Lewy found a non-solvable complex vector field. The vector field is a natural one, it is the Cauchy-Riemann operator on the boundary of a strictly pseudo-convex domain. The fact is that almost all linear PDE’s are unsolvable, because of the Hrmander bracket condition. A rapid development in the 60’s lead to the conjecture by Nirenberg and Treves in 1969: condition (Ψ) is necessary and sufficient for the solvability of partial differential equations of principal type. This is a condition which involves only the sign changes of the imaginary part of the highest order term of the operator along the bicharacteristics of the real part. The Nirenberg-Treves conjecture has recently been proved, see Annals of Mathematics, 163:2, 2006. We shall present the background and the ideas of the proof. Thursday, October 12 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium L COHOMOLOGY AND RIESZ TRANSFORM GILLES CARRON Univ. Nantes Abstract Part of this talk is a joint work with T. Coulhon and A. Hassell . I will describe some link between the boundness on L of the Riesz transform d∆ and map between L and L cohomology. This link gives some limitation about the range of p where the Riesz transform is bounded for connected sum of manifolds. I will explain why these bounds are sharp in some geometrical case.Part of this talk is a joint work with T. Coulhon and A. Hassell . I will describe some link between the boundness on L of the Riesz transform d∆ and map between L and L cohomology. This link gives some limitation about the range of p where the Riesz transform is bounded for connected sum of manifolds. I will explain why these bounds are sharp in some geometrical case. Thursday, November 2 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Special Colloquium SENSE-LESS BUT SMART! SIGNAL ACQUISITION BY RANDOM SENSING EMMANUEL CANDES Caltech Abstract One of the central tenets of signal processing is the Shannon/Nyquist sampling theory: the number of samples needed to reconstruct a signal without error is dictated by its bandwidththe length of the shortest interval which contains the support of the spectrum of the signal under study. Very recently, an alternative sampling or sensing theory has emerged which goes against this conventional wisdom. This theory allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer data bits than traditional methods use. Underlying this metholdology is a concrete protocol for sensing and compressing data simultaneously. This talk will present the key mathematical ideas underlying this new sampling or sensing theory, and will survey some of the most important results. We will argue that this is a robust mathematical theory; not only is it possible to recover signals accurately from just an incomplete set of measurements, but it is also possible to do so when the measurements are unreliable and corrupted by noise. We will see that the reconstruction algorithms are very concrete, stable (in the sense that they degrade smoothly as the noise level increases) and practical; in fact, they only involve solving very simple convex optimization programs. An interesting aspect of this theory is that it has bearings on some fields in the applied sciences and engineering such as statistics, information theory, coding theory, theoretical computer science, and others as well. If time allows, we will try to explain these connections via a few selected examples. Some of this work is joint with Terence Tao and Justin Romberg. Wednesday, November 15 4:15 p.m. Room 383-N (NOTE UNUSUAL TIME AND LOCATION!) http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium RANDOM MATRICES AND COMBINATORICS OF GRAPHS ALICE GUIONNET Ecole Normale Supérieure de Lyon Abstract Random matrices have been used in theoretical physics for more than thirthy years to enumerate combinatorial objects such as maps. ’t Hooft in the seventies indeed showed that free energies, given as the logarithm of laplace transforms of traces of random Gaussian matrices, expand formally into a generating function of (colored) maps. We shall show that this formal expansion can be strenghtened into a large-N expansion. Such a result in particular shows that the limit of the free energies are generating functions for planar maps. A natural question that we shall adress at the end of the talk is why the approach by random matrices can be useful to solve the combinatorial problem of enumerating planar maps.Random matrices have been used in theoretical physics for more than thirthy years to enumerate combinatorial objects such as maps. ’t Hooft in the seventies indeed showed that free energies, given as the logarithm of laplace transforms of traces of random Gaussian matrices, expand formally into a generating function of (colored) maps. We shall show that this formal expansion can be strenghtened into a large-N expansion. Such a result in particular shows that the limit of the free energies are generating functions for planar maps. A natural question that we shall adress at the end of the talk is why the approach by random matrices can be useful to solve the combinatorial problem of enumerating planar maps. Thursday, November 16 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium HIGHER GAUGE THEORY JOHN BAEZ University of California at Riverside Abstract Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of “higher gauge theory” that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must “categorify” concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity.Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of “higher gauge theory” that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must “categorify” concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity. Thursday, December 7 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium DUALITY IN GRAPH HOMOLOGY ALEXANDER VORONOV University of Minnesota Abstract The popularity of graph homology owes largely to the fact that the cohomology of two important groups in mathematics, the outer automorphism group of a free group and the mapping class group, even though generally incomputable, may be computed via a deceptively simple combinatorial construction, called graph homology. Such computations done by CullerVogtmann, Penner, and Kontsevich indicate that, in certain particular cases, Poincare-type duality on the level of spaces corresponds to Koszul duality of operads. In this talk, based on a joint work with A. Lazarev, we explain how Verdier duality on graph spaces turns into Koszul duality for operads, in general.The popularity of graph homology owes largely to the fact that the cohomology of two important groups in mathematics, the outer automorphism group of a free group and the mapping class group, even though generally incomputable, may be computed via a deceptively simple combinatorial construction, called graph homology. Such computations done by CullerVogtmann, Penner, and Kontsevich indicate that, in certain particular cases, Poincare-type duality on the level of spaces corresponds to Koszul duality of operads. In this talk, based on a joint work with A. Lazarev, we explain how Verdier duality on graph spaces turns into Koszul duality for operads, in general. Thursday, January 11 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium THE FONTAINE-MAZUR CONJECTURE AND MODULAR FORMS MARK KISIN University of Chicago Abstract The Fontaine-Mazur conjecture describes which p-adic representations of the Galois group of Q ought to arise from algebraic geometry. Combined with the conjectures of Langlands, it predicts these representations should arise from automorphic forms. I will explain the statement of the conjecture and report on progress in the two dimensional case.The Fontaine-Mazur conjecture describes which p-adic representations of the Galois group of Q ought to arise from algebraic geometry. Combined with the conjectures of Langlands, it predicts these representations should arise from automorphic forms. I will explain the statement of the conjecture and report on progress in the two dimensional case. Thursday, January 18 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium NUMERICAL ZOOM OLIVIER PIRONNEAU Jussieu Abstract We give here numerical techniques and error estimates for the numerical solution of problems with multiple scales when the small scale is confined to geometrically small regions such as jumps of coefficients on curves and surfaces or complex variations of coefficients in small regions where numerical zooms can be made. The method is illustrated on the numerical assessment of a nuclear waste repository site. http://www.ann.jussieu.fr/~pironneau/publi/publications/wesseling65.pdfWe give here numerical techniques and error estimates for the numerical solution of problems with multiple scales when the small scale is confined to geometrically small regions such as jumps of coefficients on curves and surfaces or complex variations of coefficients in small regions where numerical zooms can be made. The method is illustrated on the numerical assessment of a nuclear waste repository site. http://www.ann.jussieu.fr/~pironneau/publi/publications/wesseling65.pdf Thursday, January 25 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium INTEGER SOLUTIONS TO x + y = z BJORN POONEN University of California at Berkeley Abstract We explain some of the general principles for solving generalized Fermat equations, and in particular prove that the title equation has exactly 16 solutions in relatively prime integers. One such solution is (21063928,−76271, 17). This is joint work with Ed Schaefer and Michael Stoll.We explain some of the general principles for solving generalized Fermat equations, and in particular prove that the title equation has exactly 16 solutions in relatively prime integers. One such solution is (21063928,−76271, 17). This is joint work with Ed Schaefer and Michael Stoll. Thursday, February 1 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium REACTION-DIFFUSION EQUATIONS AND BIOLOGICAL INVASIONS HENRI BERESTYCKI EHESS (École des hautes études en sciences sociales), Paris Abstract This lecture, meant for a general audience, will describe some mathematical properties of reaction-diffusion equations as an approach to spatial propagation and diffusion. After illustrating the mechanism of reaction and diffusion on examples and reviewing some classical properties, I will present two recent works in the context of ecology of populations. These are concerned with non homogeneous media. The first one deals with persistence of a species and biological invasions in a periodic environment. I will also describe a model addressing the question of how a species keeps pace with a warming climate. Looking at a more general class of problems, I will then mention some nonlinear Liouville type results for elliptic problems.This lecture, meant for a general audience, will describe some mathematical properties of reaction-diffusion equations as an approach to spatial propagation and diffusion. After illustrating the mechanism of reaction and diffusion on examples and reviewing some classical properties, I will present two recent works in the context of ecology of populations. These are concerned with non homogeneous media. The first one deals with persistence of a species and biological invasions in a periodic environment. I will also describe a model addressing the question of how a species keeps pace with a warming climate. Looking at a more general class of problems, I will then mention some nonlinear Liouville type results for elliptic problems. Thursday, February 8 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium EFFECTIVE CLASSES AND LAGRANGIAN TORI IN SYMPLECTIC FOUR-MANIFOLDS JEAN-YVES WELSCHINGER ENS Lyon Abstract A two-dimensional homology class in a closed symplectic manifold is called effective when it is realized by a pseudo-holomorphic curve for every almost-complex structure tamed by the symplectic form. My aim is to explain why in dimension four, effective classes and (classes realized by) Lagrangian tori are always orthogonal to each other with respect to the intersection form. I will recall what pseudo-holomorphic curves and Lagrangian manifolds are, discuss this result together with related ones and finally show how techniques from symplectic field theory make possible to prove them.A two-dimensional homology class in a closed symplectic manifold is called effective when it is realized by a pseudo-holomorphic curve for every almost-complex structure tamed by the symplectic form. My aim is to explain why in dimension four, effective classes and (classes realized by) Lagrangian tori are always orthogonal to each other with respect to the intersection form. I will recall what pseudo-holomorphic curves and Lagrangian manifolds are, discuss this result together with related ones and finally show how techniques from symplectic field theory make possible to prove them. Thursday, February 15 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/ Stanford Department of Mathematics Colloquium PASSIVE SCALAR EQUATION FOR KRAICHNAN TYPE VELOCITY FIELD BORIS ROZOVSKY BROWN Abstract Propagation of a passive scalar (e.g. temperature) by a fluid flow with velocity field v(t, x) is described by the transport equation ∂θ(t, x) ∂t = −v(t, x) · ∇θ(t, x), t > 0; θ(0, x) = θ0(x). If the velocity field is reasonably regular (e.g. Lipschitz continuous in x) the solution of the transport equation is given by θ(t, x) = θ0(X t,x 0 ), (1) where X is the backward flow related to the velocity field v: dX t,x s ds = v(s, X t,x s ), t > s, X t,x t = x. In the main statistical models of turbulence such as the Kolmogorov and Kraichnan models, the velocity field is not Lipschitz continuous. In this case formula (1) does not hold anymore. In Kraichnan’s model which will be considered in this talk, the velocity v is a statistically homogeneous, isotropic, and stationary Gaussian vector field with zero mean and covariance E(v(t, x)v(s, y)) = δ(t − s)C (x − y), where C (x) ∼ C (0)(1 − |x|), 0 < γ < 2 for |x| 1 and decays fast for x.The velocity field for this covariance is Hölder continuous (in x) with the exponent γ/2. A stochastic flow corresponding to this velocity will be constructed and a Lagrangian representation of the solution of the transport equation extending formula (1) to KraichnanPropagation of a passive scalar (e.g. temperature) by a fluid flow with velocity field v(t, x) is described by the transport equation ∂θ(t, x) ∂t = −v(t, x) · ∇θ(t, x), t > 0; θ(0, x) = θ0(x). If the velocity field is reasonably regular (e.g. Lipschitz continuous in x) the solution of the transport equation is given by θ(t, x) = θ0(X t,x 0 ), (1) where X is the backward flow related to the velocity field v: dX t,x s ds = v(s, X t,x s ), t > s, X t,x t = x. In the main statistical models of turbulence such as the Kolmogorov and Kraichnan models, the velocity field is not Lipschitz continuous. In this case formula (1) does not hold anymore. In Kraichnan’s model which will be considered in this talk, the velocity v is a statistically homogeneous, isotropic, and stationary Gaussian vector field with zero mean and covariance E(v(t, x)v(s, y)) = δ(t − s)C (x − y), where C (x) ∼ C (0)(1 − |x|), 0 < γ < 2 for |x| 1 and decays fast for x.The velocity field for this covariance is Hölder continuous (in x) with the exponent γ/2. A stochastic flow corresponding to this velocity will be constructed and a Lagrangian representation of the solution of the transport equation extending formula (1) to Kraichnan velocity field will be presented. It will be shown that the flow corresponding to Kraichnan velocity is super-unstable which leads to the dissipation of energy of the passive scalar. Moreover, it will be shown that the dissipation of energy takes place if and only if the flow is super unstable. Surprisingly, the solution of the transport equation driven by the Kraichnan velocity field is pathwise unique and square integrable as in the classical setting. It will be shown that the solution of this equations remains to be well defined and unique even for the velocity fields much less regular then the Kraichnan velocity. Thursday, February 22 4:15 p.m. Room 380-W http://math.stanford.edu/coll/0607/