# Search result: Catalogue data in Spring Semester 2023

Mathematics Master | |||||||||||||||||||||||||||

Core Courses For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | |||||||||||||||||||||||||||

Core Courses: Pure Mathematics | |||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||
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401-3112-23L | Number Theory II: Introduction to Modular Forms Be aware that there is a big overlap with former course units on modular forms, in particular with 401-4118-22L taught in the Spring Semester 2022 as an elective course. Only one of the two course units may be recognised for credits. More precisely, it is also not allowed to have recognised one course unit for the Bachelor's and the other course unit for the Master's degree. | W | 8 credits | 4G | Ö. Imamoglu | ||||||||||||||||||||||

Abstract | This is a introductory course on automorphic forms covering its basic properties with emphasis on connections with number theory. | ||||||||||||||||||||||||||

Objective | The aim of the course is to cover the classical theory of modular forms | ||||||||||||||||||||||||||

Content | Basic definitions and properties of SL(2,Z), its subgroups and modular forms for SL(2,Z). Eisenstein and Poincare series. L-functions of modular forms. Hecke operators. Theta functions. Possibly Maass forms. Possibly automorphic forms for more general groups. | ||||||||||||||||||||||||||

Literature | J.P. Serre, A Course in Arithmetic; N. Koblitz, Introduction to Elliptic Curves and Modular Forms; D. Zagier, The 1-2-3 of Modular Forms; H. Iwaniec, Topics in Classical Automorphic Forms. | ||||||||||||||||||||||||||

Prerequisites / Notice | Function theory, Algebra I and II | ||||||||||||||||||||||||||

401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | S. Kalisnik Hintz | ||||||||||||||||||||||

Abstract | This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality. | ||||||||||||||||||||||||||

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Literature | 1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. The book can be downloaded for free at: Link 3) E. Spanier, "Algebraic topology", Springer-Verlag | ||||||||||||||||||||||||||

Prerequisites / Notice | General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I"). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary. | ||||||||||||||||||||||||||

401-3146-12L | Algebraic GeometryDoes not take place this semester. | W | 10 credits | 4V + 1U | not available | ||||||||||||||||||||||

Abstract | This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes). | ||||||||||||||||||||||||||

Objective | Learning Algebraic Geometry. | ||||||||||||||||||||||||||

Literature | Primary References: * John Ottem, Geir Ellingsrud: Introduction to algebraic varieties, Link * James Milne: Algebraic Geometry, Link Secondary References: * Miles Reid: Undergraduate Algebraic Geometry, Cambridge University Press. * Ravi Vakil: Foundations of Algebraic Geometry, Link * David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer. Other textbooks: * Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications. * Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer. * Igor Shafarevich: Basic Algebraic geometry 1 & 2, Springer-Verlag. * Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer. | ||||||||||||||||||||||||||

Prerequisites / Notice | Requirement: Basic knowledge of Commutative Algebra. | ||||||||||||||||||||||||||

401-8142-21L | Algebraic Geometry II (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT517 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 1U | University lecturers | ||||||||||||||||||||||

Abstract | We continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc. | ||||||||||||||||||||||||||

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Prerequisites / Notice | MAT507 Algebraic Geometry = 401-8141-00L Algebraic Geometry (University of Zurich) offered in the Autumn Semester 2022. This is also a good follow-up course unit for ETH students who visited Maria Yakerson's lecture Algebraic Geometry in Spring 2022. | ||||||||||||||||||||||||||

401-3226-00L | Symmetric Spaces | W | 8 credits | 4G | A. Iozzi | ||||||||||||||||||||||

Abstract | * Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples * Symmetric spaces of non-compact type: flats and rank, roots and root spaces * Iwasawa decomposition, Weyl group, Cartan decomposition * Geometry at infinity | ||||||||||||||||||||||||||

Objective | Learn the basics of symmetric spaces | ||||||||||||||||||||||||||

401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | J. Serra | ||||||||||||||||||||||

Abstract | This is a continuation course of Differential Geometry I. Topics covered include: Introduction to Riemannian geometry: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, and isoperimetric inequalities. | ||||||||||||||||||||||||||

Objective | Providing an introductory invitation to Riemannian geometry. | ||||||||||||||||||||||||||

Literature | - M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992 - I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP, - S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004 - S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley, | ||||||||||||||||||||||||||

Prerequisites / Notice | Differential Geometry I (or basics of differentiable manifolds) | ||||||||||||||||||||||||||

401-3462-00L | Functional Analysis II | W | 10 credits | 4V + 1U | P. Hintz | ||||||||||||||||||||||

Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles. Basic results for hyperbolic PDE. | ||||||||||||||||||||||||||

Objective | Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods to the study of elliptic boundary value problems, and to initial value problems for hyperbolic PDE. | ||||||||||||||||||||||||||

Literature | Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001. Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011. Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003. Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. | ||||||||||||||||||||||||||

Prerequisites / Notice | Functional Analysis I and fluency in multivariable calculus. | ||||||||||||||||||||||||||

Core Courses: Applied Mathematics and Further Appl.-Oriented Fields ¬ | |||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||

401-3052-10L | Graph Theory | W | 10 credits | 4V + 1U | B. Sudakov | ||||||||||||||||||||||

Abstract | Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem | ||||||||||||||||||||||||||

Objective | The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems. | ||||||||||||||||||||||||||

Lecture notes | Lecture will be only at the blackboard. | ||||||||||||||||||||||||||

Literature | West, D.: "Introduction to Graph Theory" Diestel, R.: "Graph Theory" Further literature links will be provided in the lecture. | ||||||||||||||||||||||||||

Prerequisites / Notice | Students are expected to have a mathematical background and should be able to write rigorous proofs. | ||||||||||||||||||||||||||

401-3642-00L | Brownian Motion and Stochastic Calculus | W | 10 credits | 4V + 1U | D. Possamaï | ||||||||||||||||||||||

Abstract | This course gives an introduction to Brownian motion and stochastic calculus. It includes the construction and properties of Brownian motion, basics of Markov processes in continuous time and of Levy processes, and stochastic calculus for continuous semimartingales. | ||||||||||||||||||||||||||

Objective | This course gives an introduction to Brownian motion and stochastic calculus. The following topics are planned: - Definition and construction of Brownian motion - Some important properties of Brownian motion - Basics of Markov processes in continuous time - Stochastic calculus, including stochastic integration for continuous semimartingales, Ito's formula, Girsanov's theorem, stochastic differential equations and connections with partial differential equations - Basics of Levy processes | ||||||||||||||||||||||||||

Lecture notes | Lecture notes will be made available in class. | ||||||||||||||||||||||||||

Literature | - R.F. Bass, Stochastic Processes, Cambidge University Press (2001). - I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991). - J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016). - D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005). - L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000). | ||||||||||||||||||||||||||

Prerequisites / Notice | Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in - J. Jacod, P. Protter, Probability Essentials, Springer (2004). - R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010). | ||||||||||||||||||||||||||

401-3632-00L | Computational Statistics | W | 8 credits | 3V + 1U | M. Mächler | ||||||||||||||||||||||

Abstract | We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R. | ||||||||||||||||||||||||||

Objective | The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R. | ||||||||||||||||||||||||||

Content | See the class website | ||||||||||||||||||||||||||

Prerequisites / Notice | At least one semester of (basic) probability and statistics. Programming experience is helpful but not required. | ||||||||||||||||||||||||||

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401-3602-00L | Applied Stochastic Processes | W | 8 credits | 3V + 1U | V. Tassion | ||||||||||||||||||||||

Abstract | Poisson processes; renewal processes; Markov chains in discrete and in continuous time; some applications. | ||||||||||||||||||||||||||

Objective | Stochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We present several classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable". | ||||||||||||||||||||||||||

Literature | R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: Link R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: Link M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: Link S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005) | ||||||||||||||||||||||||||

Prerequisites / Notice | Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L). | ||||||||||||||||||||||||||

401-3652-00L | Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich) No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT827 Mind the enrolment deadlines at UZH: Link | W | 10 credits | 4V + 1U | University lecturers | ||||||||||||||||||||||

Abstract | This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB. | ||||||||||||||||||||||||||

Objective | The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations. | ||||||||||||||||||||||||||

Content | * Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering. * Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes. * Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes. * Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods. * Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes. * Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory. | ||||||||||||||||||||||||||

Lecture notes | Lecture slides will be made available to participants. However, additional material might be covered in the course. | ||||||||||||||||||||||||||

Literature | H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online. R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online. E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991. | ||||||||||||||||||||||||||

Prerequisites / Notice | Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite. Programming exercises in MATLAB Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations" | ||||||||||||||||||||||||||

Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | |||||||||||||||||||||||||||

Electives: Pure Mathematics | |||||||||||||||||||||||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | |||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||

401-3058-00L | Combinatorics I | W | 4 credits | 2G | N. Hungerbühler | ||||||||||||||||||||||

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | ||||||||||||||||||||||||||

Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | ||||||||||||||||||||||||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | ||||||||||||||||||||||||||

Prerequisites / Notice | Recognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008). | ||||||||||||||||||||||||||

401-3035-00L | Forcing: An Introduction to Independence Proofs | W | 8 credits | 3V + 1U | L. Halbeisen | ||||||||||||||||||||||

Abstract | Mit Hilfe der Forcing-Technik werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese von den Axiomen der Mengenlehre unabhaengig ist. | ||||||||||||||||||||||||||

Objective | Die Forcing-Technik kennenlernen und verschiedene Unabhaengigkeitsbeweise fuehren koennen. | ||||||||||||||||||||||||||

Content | Mit Hilfe der sogenannten Forcing-Technik, welche anfangs der 1960er Jahre von Paul Cohen entwickelt wurde, werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese CH von den Axiomen der Mengenlehre ZFC unabhaengig ist. Weiter wird in Modellen von ZFC, in denen CH nicht gilt, die Groesse verschiedener Kardinalzahlcharakteristiken untersucht. Zum Schluss der Vorlesung wird ein Modell von ZFC konstruiert, in dem es (bis auf Isomorphie) genau n Ramsey-Ultrafilter gibt, wobei n fuer irgend eine nicht-negative ganze Zahl steht. | ||||||||||||||||||||||||||

Lecture notes | Ich werde mich weitgehend an mein Buch "Combinatorial Set Theory" halten, aus dem einige Kapitel aus Part III & IV behandelt werden. | ||||||||||||||||||||||||||

Literature | "Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2017) Link | ||||||||||||||||||||||||||

Prerequisites / Notice | Voraussetzung ist die Vorlesung "Axiomatische Mengenlehre" (Herbstsemester 2017) bzw. die entsprechenden Kapitel aus meinem Buch. | ||||||||||||||||||||||||||

Selection: Geometry | |||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||

401-3056-00L | Finite Geometries I Does not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | ||||||||||||||||||||||

Abstract | Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares. | ||||||||||||||||||||||||||

Objective | Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design. | ||||||||||||||||||||||||||

Content | Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design | ||||||||||||||||||||||||||

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | ||||||||||||||||||||||||||

401-4146-22L | Derived Algebraic Geometry | W | 4 credits | 2V | A. Bojko | ||||||||||||||||||||||

Abstract | The main goal is to introduce this subject to a wider audience in a more intuitive way. The course should ideally end with applications of derived algebraic geometry to constructing virtual fundamental classes in enumerative geometry using perverse sheaves. | ||||||||||||||||||||||||||

Objective | A keen listener should understand by the end of the course why derived algebraic geometry is useful and have an idea of where to begin in applying it to problems in enumerative geometry. | ||||||||||||||||||||||||||

Literature | B. Toën, Derived Algebraic Geometry, arXiv:1401.1044, 2014. J. Lurie. Higher topos theory, Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009. J. Lurie, On Infinity Topoi, arXiv:math/0306109, 2003. J. Lurie, Derived Algebraic Geometry, Ph.D. thesis, Massachusetts Institute of Technology, Dept. of Mathematics, 2004. B. Toën and G. Vezzosi. Homotopical algebraic geometry I: Topos theory”, Advances in mathematics, 2005. B. Toën and G. Vezzosi, From HAG to DAG: Derived Moduli Stacks:Ax-iomatic, Enriched and Motivic Homotopy Theory, 2004. B. Toën and M. Vaquié, Moduli of objects in dg-categories, Annales scien-tifiques de l’Ecole normale supérieure, 2007. C. Brav, V. Bussi, and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, Journal of the American Mathematical Society, 2019. D. Joyce , P. Safronov, A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes, In Annales de la Faculté des sciences de Toulouse: Mathématiques, 2019. D. Borisov, and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geometry & Topology, 2017. K. Tasuki, Virtual classes via vanishing cycles, arXiv:2109.06468, 2021. | ||||||||||||||||||||||||||

Prerequisites / Notice | One should have some understanding of algebraic geometry (in particular intersection theory), algebraic topology and category theory. Familiarity with some enumerative geometry using virtual fundamental classes would be helpful for understanding the goal of the course. | ||||||||||||||||||||||||||

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401-3574-61L | Introduction to Knot Theory Does not take place this semester. | W | 6 credits | 3G | |||||||||||||||||||||||

Abstract | Introduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school. | ||||||||||||||||||||||||||

Objective | The aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school. | ||||||||||||||||||||||||||

Content | Definition of a knot and of equivalent knots. Definition of a knot invariant and some elementary examples. Various operations on knots. Knot polynomials (Jones, ev. Alexander.....) | ||||||||||||||||||||||||||

Literature | An extensive bibliography will be handed out in the course. | ||||||||||||||||||||||||||

Prerequisites / Notice | Prerequisites are some elementary knowledge of algebra and topology. | ||||||||||||||||||||||||||

Selection: Analysis | |||||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | ||||||||||||||||||||||

401-4494-23L | Variational Problems and PDEs | W | 4 credits | 2V | A. Figalli | ||||||||||||||||||||||

Abstract | In this class, we will study some classical variational problems to motivate the introduction and study of a series of important tools in analysis, such as: 1) the direct method of the calculus of variations 2) the use of geometric measure theory to deal with non-smooth objects 3) the regularity theory for elliptic PDEs with measurable and smooth coefficients | ||||||||||||||||||||||||||

Objective | Learn important tools in the calculus of variations, elliptic PDEs, and geometric measure theory. | ||||||||||||||||||||||||||

Prerequisites / Notice | Functional Analysis I | ||||||||||||||||||||||||||

401-4834-23L | Nonlinear Wave Equations with Applications to General Relativity | W | 4 credits | 2V | C. Kehle | ||||||||||||||||||||||

Abstract | Introduction to linear and nonlinear wave equations, aspects of Lorentzian geometry and the Einstein equations in general relativity. | ||||||||||||||||||||||||||

Objective | In the first part of the course, the basics of linear wave equations on Minkowski space are covered. We then go beyond to nonlinear equations as well as to curved backgrounds. To this end, the relevant background from Lorentzian geometry is introduced. The ultimate goal is to rigorously study dynamics of the Einstein equations of general relativity. | ||||||||||||||||||||||||||

Prerequisites / Notice | It may be helpful if the participants have some familiarity with the basics of differential manifolds and basic functional analysis. | ||||||||||||||||||||||||||

Competencies |
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401-3374-23L | Dynamical Systems and Ergodic Theory (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT733 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 2U | University lecturers | ||||||||||||||||||||||

Abstract | Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. | ||||||||||||||||||||||||||

Objective | By the end of the unit the student: will have developed a good background in the area of dynamical systems; will be familiar with the basic concepts, results, and techniques relevant to the area; will have detailed knowledge of a number of fundamental examples that help clarify and motivate the main concepts in the theory; will understand the proofs of the fundamental theorems in the area; will have mastered the application of dynamical systems techniques for solving a range of standard problems; will have a firm foundation for further study in the area. | ||||||||||||||||||||||||||

Content | Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it in average. We will give a strong emphasis on presenting many fundamental examples of dynamical systems. Driven by the examples, we will first introduce some of the phenomena and main concepts which one is interested in studying. We will then formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications for example to number theory, information theory and Internet search engines. Topics which will be covered include: -Basic examples of dynamical systems (e.g. rotations and doubling map; baker’s map, CAT map and hyperbolic toral automorphisms; the Gauss map and continued fractions); -Elements of topological dynamics (minimality; topological conjugacy; topological mixing; topological entropy); -Elements of symbolic dynamics (shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding); -Introduction to ergodic theory: invariant measures; Poincare' recurrence; ergodicity; mixing; the Birkhoff Ergodic Theorem and applications; Markov measures; the ergodic theorem for Markov chains and applications to Internet Search; measure theoretic entropy; -Selected topics (time permitting): Shannon-McMillan-Breiman theorem; Lyapunov exponents and multiplicative ergodic theorem; continuous time dynamical systems and some mathematical billiards. | ||||||||||||||||||||||||||

Lecture notes | Lecture notes for several of the topics covered will be provided. | ||||||||||||||||||||||||||

Literature | Textbooks which can be used as additional reference for some of the topics include: -B. Hasselblatt and A. Katok, Dynamics: A first course. (Cambdirge University Press, 2003) – Chapters 7,8,10 and 15 -M. Brin and G. Stuck, Introduction to Dynamical Systems. (Cambridge University Press, 2002) – Chapters 1-4 -Omri Sarig, Lectures Notes on Ergodic Theory (Available Online), Topics from Chapter 1 and 2 | ||||||||||||||||||||||||||

Prerequisites / Notice | Prior Knowledge Basic knowledge of measure theory and integration. |

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