Lectures in Mathematics. ETH Zürich – serie

These lecture notes describe the Aubry-Mather-Theory within the calculus of variations. The text consists of the translated original lectures of Jurgen Moser and a bibliographic appendix with comments on the current state of the art in this field of interest. Students will find a rapid introduction to the calculus of variations, leading to modern dynamical systems theory. Differential geometric applications are discussed, in particular billiards and minimal geodesics on the two-dimensional torus. Many exercises and open questions make this book a valuable resource for both teaching and research.

Disclosing a fascinating connection between optimal stopping problems in probability and free-boundary problems this comprehensive book covers classic methods of solution and more recent ones. Using minimal tools and key examples the book exposes optimal stopping problems at its basic principles.

Der Begriff der Splinefunktionen wurde von I. J. Schoenberg 1946 eingefUhrt. "Spline" ist der Name eines Zeichengerates, welches auf mechanischem Weg Interpolatio- aufgaben lost. Dieses Gerat besteht aus einer flexiblen, oft mehrere Meter langen Latte, die auf dem Zeichenbrett aufliegt und dort an bestimmten Stellen durch Gewichte festgehalten wird. Die Form, die die Latte annimmt, hangt von den Elastizitatseigenschaften der Latte abo -, " , , , , , , \ , \ \ I , , , ," -"', , , , , J::>----" , I I , I I I , , , , , , , , , , , " ) Fig. 1: Latteninterpo1ation Po1ynominterpo1ation _ - - - - - - - --0 Wir konnen natUrlich versuchen, ein mathematisches Modell fUr dieses mechanische Zeichengerat zu machen, d. h. die Gestalt solcher Kurven mathematisch zu erfassen. - 2 - Die Theorie der Balkenbiegung verlangt, dass die mittlere 2 K quadratische KrUmmung, ("strain energy", Spannungs- J energie) minimiert wird.Lasst sich die Kurve als Graph einer Funktion f auf dem Intervall [a,b] schreiben, so erhalt man mit dem bekannten Ausdruck fUr die Krlimmung K [f" (t) P --------------dt ~ min (1) t [1 +f' (t)2J5/2 a Statt dieses schwierige Extremalproblem zu losen, begnUgt man sich damit, (2) zu minimieren. Die Extremalfunktion fUr das Funktional (2) ist stUckweise ein kubisches Polynom; die Polyn- stUcke gehen an den Bruchstellen so glatt ineinander Uber, dass die Funktion zweimal stetig differenzierbar ist.

These notes were developed for a graduate-level course on the theory and numerical solution of nonlinear hyperbolic systems of conservation laws. Pa rt 1 deals with the basic mathematical theory of the equations: the notion of weak solutions, entropy conditions, a detailed description of the wave structure of solutions to the Riemann problem. The emphasis is on tools and techniques that are indispensible in developing good numerical methods for discontinuous solutions. Part 2 is devoted to the development of high resolution shock-capturing methods, including the theory of total variation diminishing (TDV) methods and the use of limiter functions. This book is intended for a wide audience and should be of use both to numerical analysts and to computational researchers in a variety of applications.

These lecture notes are based on the joint work of the author and Arthur Fischer on Teichmiiller theory undertaken in the years 1980-1986. Since then many of our colleagues have encouraged us to publish our approach to the subject in a concise format, easily accessible to a broad mathematical audience. However, it was the invitation by the faculty of the ETH Ziirich to deliver the ETH N achdiplom-Vorlesungen on this material which provided the opportunity for the author to develop our research papers into a format suitable for mathematicians with a modest background in differential geometry. We also hoped it would provide the basis for a graduate course stressing the application of fundamental ideas in geometry. For this opportunity the author wishes to thank Eduard Zehnder and Jiirgen Moser, acting director and director of the Forschungsinstitut fiir Mathematik at the ETH, Gisbert Wiistholz, responsible for the Nachdiplom Vorlesungen and the entire ETH faculty for their support and warm hospitality. This new approach to Teichmiiller theory presented here was undertaken for two reasons. First, it was clear that the classical approach, using the theory of extremal quasi-conformal mappings (in this approach we completely avoid the use of quasi-conformal maps) was not easily applicable to the theory of minimal surfaces, a field of interest of the author over many years. Second, many other active mathematicians, who at various times needed some Teichmiiller theory, have found the classical approach inaccessible to them.

These notes form the contents of a Nachdiplomvorlesung given at the Forschungs- institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. K. Chandrasekharan and Prof. Jurgen Moser have encouraged me to write them up for inclusion in the series, published by Birkhiiuser, of notes of these courses at the ETH. Dr. Albert Stadler produced detailed notes of the first part of this course, and very intelligible class-room notes of the rest. Without this work of Dr. Stadler, these notes would not have been written. While I have changed some things (such as the proof of the Serre duality theorem, here done entirely in the spirit of Serre's original paper), the present notes follow Dr. Stadler's fairly closely. My original aim in giving the course was twofold. I wanted to present the basic theorems about the Jacobian from Riemann's own point of view. Given the Riemann-Roch theorem, if Riemann's methods are expressed in modern language, they differ very little (if at all) from the work of modern authors.

As an interesting object of arithmetic, algebraic and analytic geometry the complex ball was born in a paper of the French Mathematician E. PICARD in 1883. In recent developments the ball finds great interest again in the framework of SHIMURA varieties but also in the theory of diophantine equations (asymptotic FERMAT Problem, see ch. VI). At first glance the original ideas and the advanced theories seem to be rather disconnected. With these lectures I try to build a bridge from the analytic origins to the actual research on effective problems of arithmetic algebraic geometry. The best motivation is HILBERT'S far-reaching program consisting of 23 prob­ lems (Paris 1900) " . . . one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field". This message can be found in the 12-th problem "Extension of KRONECKER'S Theorem on Abelian Fields to Any Algebraic Realm of Rationality" standing in the middle of HILBERTS'S pro­ gram. It is dedicated to the construction of number fields by means of special value of transcendental functions of several variables. The close connection with three other HILBERT problems will be explained together with corresponding advanced theories, which are necessary to find special effective solutions, namely: 7. Irrationality and Transcendence of Certain Numbers; 21.

Discussing the convergence of Krylov subspace methods for solving fixed point problems, this work focuses on the dynamical aspects of the iteration processes. For example, there are many similarities between the evolution of a Krylov subspace process and that of linear operator semigroups, in particular in the beginning of the iteration. A lifespan of an interation might typically start with a fast but slowing phase. Such a behaviour is sublinear in nature, and is essentially independent of whether the problem is singular or not. Then, for nonsingular problems, the iteration might run with a linear speed before a possible superlinear phase. All these phases are based on differential mathematical mechanisms which the book outlines. Its aim is to understand how to precondition effectively, both in the case of "numerical linear algebra" (where one usually thinks of first fixing a finite dimensional problem to be solved) and in function spaces where the "preconditioning" corresponds to software which approximately solves the original problem.

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level.

The notes in this volume were written as a part of a Nachdiplom course that I gave at the ETH in the summer semester of 1995. The aim of my lectures was the development of some of the basics of the interaction of homological algebra, or more specifically the cohomology of groups, and modular representation theory. Every time that I had given such a course in the past fifteen years, the choice of the material and the order of presentation of the results have followed more or less the same basic pattern. Such a course began with the fundamentals of group cohomology, and then investigated the structure of cohomology rings, and their maximal ideal spectra. Then the variety of a module was defined and related to actual module structure through the rank variety. Applications followed. The standard approach was used in my University of Essen Lecture Notes [e1] in 1984. Evens [E] and Benson [B2] have written it up in much clearer detail and included it as part of their books on the subject.

Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.

The aim of these lecture notes is to give an essentially self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments concerning the structure of the singular set and asymptotics on approach to the singular set. Specialized knowledge in partial differential equations or the geometric calculus of variations is not required; a good general background in mathematical analysis would be adequate preparation.

This book is organized into nine chapters, the first six of which are on expansion of filtration formulae, Burkholder-Gundy inequalities up to any random time, martingales which vanish on the zero set of Brownian motion, the Azema-Emery martingales and chaos representation, the filtration of truncated Brownian motion, and attempts to characterize the Brownian filtration. The three remaining chapters discuss principle value diffusion times, probabilistic representations of the Riemann zeta function, and progress made on some topics covered in part one. Most of the contents of this text are the subjects of active research, centred on real-value martingales and Brownian mothion, and this volume may be of interest to researchers in probability theory or in more applied fields, such as mathematical finance.

The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpos­ itive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of generalized nonpositive curvature. The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi­ tive curvature. Such constructions have been put forward by Wald, Alexandrov, Busemann, and others, and they will be systematically explored in Chapter 2. Our focus and treatment will often be different from the existing literature. In the first Chapter, we consider several classes of examples of Riemannian manifolds of nonpositive curvature, and we explain how conditions about nonpos­ itivity or negativity of curvature can be exploited in various geometric contexts.

Disordered systems are statistical mechanics models in random environments. This text considers the equilibrium properties of a few examples of disordered Ising models from a probability theory and mathematical physics approach, but the material should be of interest to condensed matter physicists, material scientists, applied mathematicians and theoretical computer scientists. The two main types of systems considered are disordered ferromagnets and spin glasses. The emphasis is on questions concerning the nunmber of ground states (at zero temperature) or the nunber of pure Gibbs states (at nonzero temperatures), and these questions are often connected to issues concerning percolation and related models of geometric or combinatorial probability. The conceptual issue of how to approach the thermodynamic (i.e. infinite volume) in systems which may have many complex competing states is considered, along with the more specific problem of whether and in what sense Parisi's analysis of the meanfield (or "infinite-range") model is relevant with regard to the low temperature behaviour of short-range spin glasses.

In these notes we study time-dependent partial differential equations and their numerical solution. The analytic and the numerical theory are developed in parallel. For example, we discuss well-posed linear and nonlinear problems, linear and nonlinear stability of difference approximations and error estimates. Special emphasis is given to boundary conditions and their discretization. We develop a rather general theory of admissible boundary conditions on energy estimates or Laplace transform techniques. These results are fundamental for the mathematical and numerical treatment of large classes of applications like Newtonian and non-Newtonian flows, two-phase flows and geophysical problems.

In these lectures, we give an account of certain recent developments of the theory of spatial branching processes. These developments lead to several fas­ cinating probabilistic objects, which combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial dif­ ferential equations. Our first objective is to give a short self-contained presentation of the measure­ valued branching processes called superprocesses, which have been studied extensively in the last twelve years. We then want to specialize to the important class of superprocesses with quadratic branching mechanism and to explain how a concrete and powerful representation of these processes can be given in terms of the path-valued process called the Brownian snake. To understand this representation as well as to apply it, one needs to derive some remarkable properties of branching trees embedded in linear Brownian motion, which are of independent interest. A nice application of these developments is a simple construction of the random measure called ISE, which was proposed by Aldous as a tree-based model for random distribution of mass and seems to play an important role in asymptotics of certain models of statistical mechanics. We use the Brownian snake approach to investigate connections between super­ processes and partial differential equations. These connections are remarkable in the sense that almost every important probabilistic question corresponds to a significant analytic problem.

There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical Kahler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kahler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.

This book is an extended version of the notes of my lecture course given at ETH in spring 1999. The course was intended as an introduction to combinatorial torsions and their relations to the famous Seiberg-Witten invariants. Torsions were introduced originally in the 3-dimensional setting by K. Rei- demeister (1935) who used them to give a homeomorphism classification of 3-dimensional lens spaces. The Reidemeister torsions are defined using simple linear algebra and standard notions of combinatorial topology: triangulations (or, more generally, CW-decompositions), coverings, cellular chain complexes, etc. The Reidemeister torsions were generalized to arbitrary dimensions by W. Franz (1935) and later studied by many authors. In 1962, J. Milnor observed 3 that the classical Alexander polynomial of a link in the 3-sphere 8 can be interpreted as a torsion of the link exterior. Milnor's arguments work for an arbitrary compact 3-manifold M whose boundary is non-void and consists of tori: The Alexander polynomial of M and the Milnor torsion of M essentially coincide.

The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the niminal amount of energy required in order to generate a given mechanical motion? This variational problem admits an interpretation in terms of a remarkable geometry on the group discovered by Hofer in 1990. Hofer's geometry serves as a source of interesting problems and gives rise to new methods and notions which extend significantly our vision of the symplectic world. In the past decade this new geometry has been extensively studied in the framework of symplectic topology with the use of modern techniques such as Gromov's theory of pseudo-holomorphic curves, Floer homology and Guillemin-Sternberg-Lerman theory of symplectic connections. Furthermore, it opens up the intriguing prospect of using an alternative geometric intuition in dynamics. The book provides an essentially self-contained introduction into these developments and includes recent results on diameter, geodesics and growth of one-parameter subgroups in Hofer's geometry, as well as applications to dynamics and ergodic theory.It is addressed to researchers and students from the graduate level onwards.

One of the most striking development of the last decades in the study of minimal surfaces, constant mean surfaces and harmonic maps is the discovery that many classical problems in differential geometry - including these examples - are actually integrable systems. This theory grew up mainly after the important discovery of the properties of the Korteweg-de Vries equation in the sixties. After C. Gardner, J. Greene, M. Kruskal et R. Miura [44] showed that this equation could be solved using the inverse scattering method and P. Lax [62] reinterpreted this method by his famous equation, many other deep observations have been made during the seventies, mainly by the Russian and the Japanese schools. In particular this theory was shown to be strongly connected with methods from algebraic geom- etry (S. Novikov, V. B. Matveev, LM. Krichever...), loop techniques (M. Adler, B. Kostant, W. W. Symes, M. J. Ablowitz ...) and Grassmannian manifolds in Hilbert spaces (M. Sato ...). Approximatively during the same period, the twist or theory of R. Penrose, built independentely, was applied successfully by R. Penrose and R. S.Ward for constructing self-dual Yang-Mills connections and four-dimensional self-dual manifolds using complex geometry methods. Then in the eighties it became clear that all these methods share the same roots and that other instances of integrable systems should exist, in particular in differential ge- ometry. This led K.

This is a self-contained exposition of the well-posedness theory for nonlinear hyperbolic systems of conservation laws, completed by the author together with his collaborators. The systems of partial differential equations under consideration arise in many areas of continuum physics. No familiarity with the subject is assumed, so the book can be used by graduate students and researchers interested in developments concerning nonlinear partial differential equations and the mathematical aspects of shock waves and propagating phase boundaries. The text covers the existence, uniqueness and continuous dependence of classical (compressive) entropy solutions. It also introduces the reader to the developing theory of nonclassical (undercompressive) entropy solutions. The study of nonclassical shock waves is based on the concept of a kinetic relation introduced by the author for general hyperbolic systems and derived from singular limits of hyperbolic conservation laws with balanced diffusion and dispersion terms.

A quick description of the conjecture The Baum-Connes conjecture is part of Alain Connes'tantalizing "noncommuta- tive geometry" programme [18]. It is in some sense the most "commutative" part of this programme, since it bridges with classical geometry and topology. Let r be a countable group. The Baum-Connes conjecture identifies two objects associated with r, one analytical and one geometrical/topological. The right-hand side of the conjecture, or analytical side, involves the K- theory of the reduced C*-algebra c;r, which is the C*-algebra generated by r in 2 its left regular representation on the Hilbert space C(r). The K-theory used here, Ki(C;r) for i = 0, 1, is the usual topological K-theory for Banach algebras, as described e.g. in [85]. The left-hand side of the conjecture, or geometrical/topological side RKf(Er) (i=O,I), is the r-equivariant K-homology with r-compact supports of the classifying space Er for proper actions of r. If r is torsion-free, this is the same as the K-homology (with compact supports) of the classifying space Br (or K(r,l) Eilenberg-Mac Lane space). This can be defined purely homotopically.

These notes had their origin in a postgraduate lecture series I gave at the Eid­ genossiche Technische Hochschule (ETH) in Zurich in the Spring of 2000. I am very grateful to my hosts, the Forschungsinstitut fUr Mathematik at ETH, for providing the ideal opportunity to develop and present this material in what I hope is a reasonably coherent manner, and also for encouraging and assisting me to record the proceedings in these lecture notes. The subject of the lecture series was counting (of combinatorial structures) and related topics, viewed from a computational perspective. As we shall see, "related topics" include sampling combinatorial structures (being computationally equivalent to approximate counting via efficient reductions), evaluating partition functions (being weighted counting) and calculating the volume of bodies (being counting in the limit). We shall be inhabiting a different world to the one conjured up by books with titles like Combinatorial Enumeration or Graphical Enumeration. There, the prob­ lems are usually parameterised on a single integer parameter n, and the required solutions are closed form or asymptotic estimates obtained using very refined and precise analytical tools.

The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost.At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.