As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. I will present a geometric interpretation for D-branes in the c=1 string theory in terms of complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve comes from the profile of the Fermi sea of free fermions. The ZZ branes, which describe Dirichlet boundary conditions on the Liouville field, are associated with the singularities of the complex curves. Since in the linear dilaton background the singularities degenerate, we study the c=1 matrix model perturbed by a tachyon potential where the degeneracy disappears. The matrix model formulation allows to find the non-perturbative effects and the complex curve to all orders in the perturbation coupling. Thus, we give a prediction how D-branes flow with the perturbation and derive various multi-point disk correlation functions. .

ENS

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. I will present a geometric interpretation for D-branes in the c=1 string theory in terms of complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve comes from the profile of the Fermi sea of free fermions. The ZZ branes, which describe Dirichlet boundary conditions on the Liouville field, are associated with the singularities of the complex curves. Since in the linear dilaton background the singularities degenerate, we study the c=1 matrix model perturbed by a tachyon potential where the degeneracy disappears. The matrix model formulation allows to find the non-perturbative effects and the complex curve to all orders in the perturbation coupling. Thus, we give a prediction how D-branes flow with the perturbation and derive various multi-point disk correlation functions. .

ENS

A theory of nonequilibrium statistical mechanics that is based on stochastic dynamics can deal with this through study of the spectral properties of the underlying stochastic matrix. We have shown the relation between near-degeneracy (of the associated eigenvalues) of the slowest modes of this matrix and the occurrence of one or more stable or metastable modes. This can be applied to ordinary metastable states, e.g., supercooled water, and to more subtle situations like spin glasses (although the latter application is far from complete). An extension of these ideas is used to quantify the coarse graining process that lies at the foundations of statistical mechanics. We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. I will present a geometric interpretation for D-branes in the c=1 string theory in terms of complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve comes from the profile of the Fermi sea of free fermions. The ZZ branes, which describe Dirichlet boundary conditions on the Liouville field, are associated with the singularities of the complex curves. Since in the linear dilaton background the singularities degenerate, we study the c=1 matrix model perturbed by a tachyon potential where the degeneracy disappears. The matrix model formulation allows to find the non-perturbative effects and the complex curve to all orders in the perturbation coupling. Thus, we give a prediction how D-branes flow with the perturbation and derive various multi-point disk correlation functions. .

ENS

The non-perturbative corrections to the free energy of the two-matrix model are expressed in terms of its spectral curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. Compared with the world-sheet theory, the results for the (p,q) critical points lead to intriguing identifications between different Liouville and matter boundary conditions in non-critical string theories. De nombreux travaux récents étudient les limites continues dárbres aléatoires discrets. Ces arbres discrets peuvent être définis soit de manière combinatoire (arbre choisi au hasard parmi tous les arbres à n sommets dún certain type), soit de manière probabiliste (en donnant la loi du “nombre dénfants” de chaque sommet de lárbre). Un passage à la limite où le nombre de sommets de lárbre tend vers línfini et où simultanément la longueur de chaque arête tend vers 0, conduit à des arbres aléatoires continus, dont le prototype est le Continuum Random Tree (CRT) introduit par Aldous. Léxposé décrira la manière dont ces arbres sont codés et en quel sens ils sont limites des arbres discrets. On donnera aussi diverses propriétés géométriques et fractales de ces arbres continus. Si le temps le permet, on introduira le modèle appelé ISE (Integrated Super-Brownian Excursion) qui combine la structure de branchement du CRT avec un déplacement brownien (les ïndividus” dont le CRT décrit la généalogie se déplacent dans léspace de manière brownienne). Les travaux récents de Slade et de ses co-auteurs ont montré que lÍSE intervient dans lásymptotique de différents modèles de mécanique statistique (arbres sur réseaux, percolation, etc.). There are many situations in physics where all the hallmarks of a phase transition are present but for which the idealized notions of analyticity are difficult to apply.

A theory of nonequilibrium statistical mechanics that is based on stochastic dynamics can deal with this through study of the spectral properties of the underlying stochastic matrix. We have shown the relation between near-degeneracy (of the associated eigenvalues) of the slowest modes of this matrix and the occurrence of one or more stable or metastable modes. This can be applied to ordinary metastable states, e.g., supercooled water, and to more subtle situations like spin glasses (although the latter application is far from complete). An extension of these ideas is used to quantify the coarse graining process that lies at the foundations of statistical mechanics. We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. I will present a geometric interpretation for D-branes in the c=1 string theory in terms of complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve comes from the profile of the Fermi sea of free fermions. The ZZ branes, which describe Dirichlet boundary conditions on the Liouville field, are associated with the singularities of the complex curves. Since in the linear dilaton background the singularities degenerate, we study the c=1 matrix model perturbed by a tachyon potential where the degeneracy disappears. The matrix model formulation allows to find the non-perturbative effects and the complex curve to all orders in the perturbation coupling. Thus, we give a prediction how D-branes flow with the perturbation and derive various multi-point disk correlation functions. .

ENS

The non-perturbative corrections to the free energy of the two-matrix model are expressed in terms of its spectral curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. Compared with the world-sheet theory, the results for the (p,q) critical points lead to intriguing identifications between different Liouville and matter boundary conditions in non-critical string theories. De nombreux travaux récents étudient les limites continues dárbres aléatoires discrets. Ces arbres discrets peuvent être définis soit de manière combinatoire (arbre choisi au hasard parmi tous les arbres à n sommets dún certain type), soit de manière probabiliste (en donnant la loi du “nombre dénfants” de chaque sommet de lárbre). Un passage à la limite où le nombre de sommets de lárbre tend vers línfini et où simultanément la longueur de chaque arête tend vers 0, conduit à des arbres aléatoires continus, dont le prototype est le Continuum Random Tree (CRT) introduit par Aldous. Léxposé décrira la manière dont ces arbres sont codés et en quel sens ils sont limites des arbres discrets. On donnera aussi diverses propriétés géométriques et fractales de ces arbres continus. Si le temps le permet, on introduira le modèle appelé ISE (Integrated Super-Brownian Excursion) qui combine la structure de branchement du CRT avec un déplacement brownien (les ïndividus” dont le CRT décrit la généalogie se déplacent dans léspace de manière brownienne). Les travaux récents de Slade et de ses co-auteurs ont montré que lÍSE intervient dans lásymptotique de différents modèles de mécanique statistique (arbres sur réseaux, percolation, etc.). There are many situations in physics where all the hallmarks of a phase transition are present but for which the idealized notions of analyticity are difficult to apply.

A theory of nonequilibrium statistical mechanics that is based on stochastic dynamics can deal with this through study of the spectral properties of the underlying stochastic matrix. We have shown the relation between near-degeneracy (of the associated eigenvalues) of the slowest modes of this matrix and the occurrence of one or more stable or metastable modes. This can be applied to ordinary metastable states, e.g., supercooled water, and to more subtle situations like spin glasses (although the latter application is far from complete). An extension of these ideas is used to quantify the coarse graining process that lies at the foundations of statistical mechanics. We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. I will present a geometric interpretation for D-branes in the c=1 string theory in terms of complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve comes from the profile of the Fermi sea of free fermions. The ZZ branes, which describe Dirichlet boundary conditions on the Liouville field, are associated with the singularities of the complex curves. Since in the linear dilaton background the singularities degenerate, we study the c=1 matrix model perturbed by a tachyon potential where the degeneracy disappears. The matrix model formulation allows to find the non-perturbative effects and the complex curve to all orders in the perturbation coupling. Thus, we give a prediction how D-branes flow with the perturbation and derive various multi-point disk correlation functions. .

ENS