In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

During the past decade interest in the formation of complex disorderly patterns far from equilibrium has grown rapidly. This interest has been stim­ ulated by the development of new approaches (based primarily on fractal geometry) to the quantitative description of complex structures, increased understanding of non-linear phenomena and the introduction of a variety of models (such as the diffusion-limited aggregation model) that provide paradigms for non-equilibrium growth phenomena. Advances in computer technology have played a crucial role in both the experimental and theoret­ ical aspects of this enterprise. Substantial progress has been made towards the development of comprehensive understanding of non-equilibrium growth phenomena but most of our current understanding is based on simple com­ puter models. Pattern formation processes are important in almost all areas of science and technology, and, clearly, pattern growth pervades biology. Very often remarkably similar patterns are found in quite diverse systems. In some case (dielectric breakdown, electrodeposition, fluid-fluid displacement in porous media, dissolution patterns and random dendritic growth for example) the underlying causes of this similarity is quite well understood. In other cases (vascular trees, nerve cells and river networks for example) we do not yet know if a fundamental relationship exists between the mechanisms leading the formation of these structures.

During the past decade interest in the formation of complex disorderly patterns far from equilibrium has grown rapidly. This interest has been stim­ ulated by the development of new approaches (based primarily on fractal geometry) to the quantitative description of complex structures, increased understanding of non-linear phenomena and the introduction of a variety of models (such as the diffusion-limited aggregation model) that provide paradigms for non-equilibrium growth phenomena. Advances in computer technology have played a crucial role in both the experimental and theoret­ ical aspects of this enterprise. Substantial progress has been made towards the development of comprehensive understanding of non-equilibrium growth phenomena but most of our current understanding is based on simple com­ puter models. Pattern formation processes are important in almost all areas of science and technology, and, clearly, pattern growth pervades biology. Very often remarkably similar patterns are found in quite diverse systems. In some case (dielectric breakdown, electrodeposition, fluid-fluid displacement in porous media, dissolution patterns and random dendritic growth for example) the underlying causes of this similarity is quite well understood. In other cases (vascular trees, nerve cells and river networks for example) we do not yet know if a fundamental relationship exists between the mechanisms leading the formation of these structures.

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka''s Theorem).

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka's Theorem).

During the past decade interest in the formation of complex disorderly patterns far from equilibrium has grown rapidly. This interest has been stim­ ulated by the development of new approaches (based primarily on fractal geometry) to the quantitative description of complex structures, increased understanding of non-linear phenomena and the introduction of a variety of models (such as the diffusion-limited aggregation model) that provide paradigms for non-equilibrium growth phenomena. Advances in computer technology have played a crucial role in both the experimental and theoret­ ical aspects of this enterprise. Substantial progress has been made towards the development of comprehensive understanding of non-equilibrium growth phenomena but most of our current understanding is based on simple com­ puter models. Pattern formation processes are important in almost all areas of science and technology, and, clearly, pattern growth pervades biology. Very often remarkably similar patterns are found in quite diverse systems. In some case (dielectric breakdown, electrodeposition, fluid-fluid displacement in porous media, dissolution patterns and random dendritic growth for example) the underlying causes of this similarity is quite well understood. In other cases (vascular trees, nerve cells and river networks for example) we do not yet know if a fundamental relationship exists between the mechanisms leading the formation of these structures.

During the past decade interest in the formation of complex disorderly patterns far from equilibrium has grown rapidly. This interest has been stim­ ulated by the development of new approaches (based primarily on fractal geometry) to the quantitative description of complex structures, increased understanding of non-linear phenomena and the introduction of a variety of models (such as the diffusion-limited aggregation model) that provide paradigms for non-equilibrium growth phenomena. Advances in computer technology have played a crucial role in both the experimental and theoret­ ical aspects of this enterprise. Substantial progress has been made towards the development of comprehensive understanding of non-equilibrium growth phenomena but most of our current understanding is based on simple com­ puter models. Pattern formation processes are important in almost all areas of science and technology, and, clearly, pattern growth pervades biology. Very often remarkably similar patterns are found in quite diverse systems. In some case (dielectric breakdown, electrodeposition, fluid-fluid displacement in porous media, dissolution patterns and random dendritic growth for example) the underlying causes of this similarity is quite well understood. In other cases (vascular trees, nerve cells and river networks for example) we do not yet know if a fundamental relationship exists between the mechanisms leading the formation of these structures.

In the slightly more than thirty years since its formulation, the Hubbard model has become a central component of modern many-body physics. It provides a paradigm for strongly correlated, interacting electronic systems and offers insights not only into the general underlying mathematical structure of many-body systems but also into the experimental behavior of many novel electronic materials. In condensed matter physics, the Hubbard model represents the simplest theoret­ ical framework for describing interacting electrons in a crystal lattice. Containing only two explicit parameters - the ratio ("Ujt") between the Coulomb repulsion and the kinetic energy of the electrons, and the filling (p) of the available electronic band - and one implicit parameter - the structure of the underlying lattice - it appears nonetheless capable of capturing behavior ranging from metallic to insulating and from magnetism to superconductivity. Introduced originally as a model of magnetism of transition met­ als, the Hubbard model has seen a spectacular recent renaissance in connection with possible applications to high-Tc superconductivity, for which particular emphasis has been placed on the phase diagram of the two-dimensional variant of the model. In mathematical physics, the Hubbard model has also had an essential role. The solution by Lieb and Wu of the one-dimensional Hubbard model by Bethe Ansatz provided the stimulus for a broad and continuing effort to study "solvable" many-body models. In higher dimensions, there have been important but isolated exact results (e. g. , N agoaka''s Theorem).