# Search result: Catalogue data in Autumn Semester 2018

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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401-3225-00L | Introduction to Lie Groups | W | 8 credits | 4G | M. Burger | |

Abstract | Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups. | |||||

Objective | The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it. | |||||

Literature | A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser) A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73) F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer) H. Samelson: "Notes on Lie algebras" (Springer, '90) S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78) A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press) | |||||

Prerequisites / Notice | Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester. Course webpage: https://metaphor.ethz.ch/x/2018/hs/401-3225-00L/ | |||||

401-3001-61L | Algebraic Topology I | W | 8 credits | 4G | P. Biran | |

Abstract | This is an introductory course in algebraic topology. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms, cohomology. Along the way we will introduce the basics of homological algebra and category theory. | |||||

Objective | ||||||

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. Book can be downloaded for free at: http://www.math.cornell.edu/~hatcher/AT/ATpage.html See also: http://www.math.cornell.edu/~hatcher/#anchor1772800 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||

Prerequisites / Notice | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level usually covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary. Some (elementary) group theory and algebra will also be needed. | |||||

401-3132-00L | Commutative Algebra | W | 10 credits | 4V + 1U | P. D. Nelson | |

Abstract | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. | |||||

Objective | We shall cover approximately the material from --- most of the textbook by Atiyah-MacDonald, or --- the first half of the textbook by Bosch. Topics include: * Basics about rings, ideals and modules * Localization * Primary decomposition * Integral dependence and valuations * Noetherian rings * Completions * Basic dimension theory | |||||

Literature | Primary Reference: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) Secondary Reference: 2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013) Tertiary References: 3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer) | |||||

Prerequisites / Notice | Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory). | |||||

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

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

401-3651-00L | Numerical Methods for Elliptic and Parabolic Partial Differential Equations (University of Zurich)Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students. Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester. No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT802 Mind the enrolment deadlines at UZH: https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html | W | 9 credits | 4V + 2U | S. Sauter | |

Abstract | This course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods. | |||||

Objective | Participants of the course should become familiar with * concepts underlying the discretization of elliptic and parabolic boundary value problems * analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems * methods for the efficient solution of discrete boundary value problems * implementational aspects of the finite element method | |||||

Content | A selection of the following topics will be covered: * Elliptic boundary value problems * Galerkin discretization of linear variational problems * The primal finite element method * Mixed finite element methods * Discontinuous Galerkin Methods * Boundary element methods * Spectral methods * Adaptive finite element schemes * Singularly perturbed problems * Sparse grids * Galerkin discretization of elliptic eigenproblems * Non-linear elliptic boundary value problems * Discretization of parabolic initial boundary value problems | |||||

Lecture notes | Course slides will be made available to the audience. | |||||

Literature | S. C. Brenner and L. Ridgway Scott: The mathematical theory of Finite Element Methods. New York, Berlin [etc]: Springer-Verl, cop.1994. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). | |||||

Prerequisites / Notice | Practical exercises based on MATLAB | |||||

401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 credits | 4V + 1U | S. van de Geer | |

Abstract | The course covers the basics of inferential statistics. | |||||

Objective | ||||||

401-4889-00L | Mathematical Finance | W | 11 credits | 4V + 2U | M. Schweizer | |

Abstract | Advanced course on mathematical finance: - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - additional topics | |||||

Objective | Advanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes) | |||||

Content | This is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models. Topics include - semimartingales and general stochastic integration - absence of arbitrage and martingale measures - fundamental theorem of asset pricing - option pricing and hedging - hedging duality - optimal investment problems - and probably others | |||||

Lecture notes | The course is based on different parts from different books as well as on original research literature. Lecture notes will not be available. | |||||

Literature | (will be updated later) | |||||

Prerequisites / Notice | Prerequisites are the standard courses - Probability Theory (for which lecture notes are available) - Brownian Motion and Stochastic Calculus (for which lecture notes are available) Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts. This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II. For an overview of courses offered in the area of mathematical finance, see Link. | |||||

401-3901-00L | Mathematical Optimization | W | 11 credits | 4V + 2U | R. Weismantel | |

Abstract | Mathematical treatment of diverse optimization techniques. | |||||

Objective | Advanced optimization theory and algorithms. | |||||

Content | 1) Linear optimization: The geometry of linear programming, the simplex method for solving linear programming problems, Farkas' Lemma and infeasibility certificates, duality theory of linear programming. 2) Nonlinear optimization: Lagrange relaxation techniques, Newton method and gradient schemes for convex optimization. 3) Integer optimization: Ties between linear and integer optimization, total unimodularity, complexity theory, cutting plane theory. 4) Combinatorial optimization: Network flow problems, structural results and algorithms for matroids, matchings, and, more generally, independence systems. | |||||

Literature | 1) D. Bertsimas & R. Weismantel, "Optimization over Integers". Dynamic Ideas, 2005. 2) A. Schrijver, "Theory of Linear and Integer Programming". John Wiley, 1986. 3) D. Bertsimas & J.N. Tsitsiklis, "Introduction to Linear Optimization". Athena Scientific, 1997. 4) Y. Nesterov, "Introductory Lectures on Convex Optimization: a Basic Course". Kluwer Academic Publishers, 2003. 5) C.H. Papadimitriou, "Combinatorial Optimization". Prentice-Hall Inc., 1982. | |||||

Prerequisites / Notice | Linear algebra. | |||||

Bachelor Core Courses: Pure Mathematics Further restrictions apply, but in particular: 401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme. Analogously for: 401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II 401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II 401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3461-00L | Functional Analysis I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | M. Einsiedler | |

Abstract | Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces; Fourier transform and applications. | |||||

Objective | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||

Literature | We will be using the book Functional Analysis, Spectral Theory, and Applications by Manfred Einsiedler and Thomas Ward and available by SpringerLink. Other useful, and recommended references include the following: Lecture Notes on "Funktionalanalysis I" by Michael Struwe Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. | |||||

Prerequisites / Notice | Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces). | |||||

401-3531-00L | Differential Geometry I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | W. Merry | |

Abstract | This will be an introductory course in differential geometry. Topics covered include: - Smooth manifolds, submanifolds, vector fields, - Lie groups, homogeneous spaces, - Vector bundles, tensor fields, differential forms, - Integration on manifolds and the de Rham theorem, - Principal bundles. | |||||

Objective | ||||||

Lecture notes | I will produce full lecture notes, available on my website at www.merry.io/differential-geometry | |||||

Literature | There are many excellent textbooks on differential geometry. A friendly and readable book that covers everything in Differential Geometry I is: John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag. A more advanced (and far less friendly) series of books that covers everything in both Differential Geometry I and II is: S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volumes I and II (1963, 1969) Wiley. | |||||

Bachelor Core Courses: Applied Mathematics ... Further restrictions apply, but in particular: 401-3601-00L Probability Theory can only be recognised for the Master Programme if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme. 402-0205-00L Quantum Mechanics I is eligible as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. | ||||||

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

401-3601-00L | Probability Theory At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | E- | 10 credits | 4V + 1U | A.‑S. Sznitman | |

Abstract | Basics of probability theory and the theory of stochastic processes in discrete time | |||||

Objective | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||

Content | This course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned: Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains. | |||||

Lecture notes | available, will be sold in the course | |||||

Literature | R. Durrett, Probability: Theory and examples, Duxbury Press 1996 H. Bauer, Probability Theory, de Gruyter 1996 J. Jacod and P. Protter, Probability essentials, Springer 2004 A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006 D. Williams, Probability with martingales, Cambridge University Press 1991 | |||||

402-0205-00L | Quantum Mechanics I | W | 10 credits | 3V + 2U | M. Gaberdiel | |

Abstract | Introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, wave functions, the description of observables as operators on a Hilbert space, as well as the formulation of symmetries, will be discussed. Basic phenomena will be analysed and illustrated by generic examples. | |||||

Objective | Introduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems. | |||||

Content | Keywords: Schrödinger equation, basic formalism of quantum mechanics (states, operators, commutators, measuring process), symmetries (translations, rotations, discrete symmetries), quantum mechanics in one dimension, spherically symmetric problems in three dimensions, hydrogen atom, harmonic oscillator, angular momentum, spin, addition of angular momenta, relation between QM and classical physics. | |||||

Literature | J.J. Sakurai: Modern Quantum Mechanics A. Messiah: Quantum Mechanics I S. Weinberg: Lectures on Quantum Mechanics | |||||

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 | |

401-3113-68L | Exponential Sums over Finite Fields | W | 8 credits | 4G | E. Kowalski | |

Abstract | Exponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory. | |||||

Objective | The goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields. | |||||

Content | Examples of elementary exponential sums The Riemann Hypothesis for curves and its applications Definition of trace functions over finite fields The formalism of the Riemann Hypothesis of Deligne Selected applications | |||||

Lecture notes | Lectures notes from various sources will be provided | |||||

Literature | Kowalski, "Exponential sums over finite fields, I: elementary methods: Iwaniec-Kowalski, "Analytic number theory", chapter 11 Fouvry, Kowalski and Michel, "Trace functions over finite fields and their applications" | |||||

401-3100-68L | Introduction to Analytic Number Theory | W | 8 credits | 4G | I. N. Petrow | |

Abstract | This course is an introduction to classical multiplicative analytic number theory. The main object of study is the distribution of the prime numbers in the integers. We will study arithmetic functions and learn the basic tools for manipulating and calculating their averages. We will make use of generating series and tools from complex analysis. | |||||

Objective | The main goal for the course is to prove the prime number theorem in arithmetic progressions: If gcd(a,q)=1, then the number of primes p = a mod q with p<x is approximately (1/phi(q))*(x/log x), as x tends to infinity, where phi(q) is the Euler totient function. | |||||

Content | Developing the necessary techniques and theory to prove the prime number theorem in arithmetic progressions will lead us to the study of prime numbers by Chebyshev's method, to study techniques for summing arithmetic functions by Dirichlet series, multiplicative functions, L-series, characters of a finite abelian group, theory of integral functions, and a detailed study of the Riemann zeta function and Dirichlet's L-functions. | |||||

Lecture notes | Lecture notes will be provided for the course. | |||||

Literature | Multiplicative Number Theory by Harold Davenport Multiplicative Number Theory I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan Analytic Number Theory by Henryk Iwaniec and Emmanuel Kowalski | |||||

Prerequisites / Notice | Complex analysis Group theory Linear algebra Familiarity with the Fourier transform and Fourier series preferable but not required. | |||||

401-3059-00L | Combinatorics IIDoes not take place this semester. | 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. | |||||

Selection: Geometry | ||||||

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

401-3057-00L | Finite Geometries II | 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. | |||||

401-3111-68L | Elliptic Curves and Cryptography | W | 8 credits | 3V + 1U | L. Halbeisen | |

Abstract | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt. Insbesondere wird der Satz von Mordell bewiesen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt, wie z.B. der Diffie-Hellman-Schluesselaustausch. | |||||

Objective | Rationale Punkte auf elliptischen Kurven, insbesondere Arithmetik auf elliptischen Kurven, Satz von Mordell, Kongruente Zahlen Anwendungen der elliptischen Kurven in der Kryptographie, wie zum Beispiel Diffie-Hellman-Schluesselaustausch, Pollard-Rho-Methode | |||||

Content | Im ersten Teil der Vorlesung wird die algebraische Struktur von elliptischen Kurven behandelt und die Menge der rationalen Punkte auf elliptischen Kurven untersucht. Insbesondere wird mit Hilfe von Saetzen aus der Algebra wie auch aus der projektiven Geometrie gezeigt, dass die Menge der rationalen Punkte auf einer elliptischen Kurven unter einer bestimmten Operation eine endlich erzeugte abelsche Gruppe bildet. Zudem werden elliptische Kurven untersucht, welche mit rationalen, rechtwinkligen Dreiecken mit ganzzahligem Flaecheninhalt zusammenhaengen. Im zweiten Teil der Vorlesung werden dann Anwendungen elliptischer Kurven in der Kryptographie gezeigt. Solche Anwendungen sind zum Beispiel ein auf elliptischen Kurven basierendes Kryptosystem oder ein Algorithmus zur Faktorisierung grosser Zahlen. | |||||

Literature | Joseph Silverman, John Tate: "Rational Points on Elliptic Curves", Undergraduate Texts in Mathematics, Springer-Verlag (1992) Ian Blake, Gadiel Seroussi, Nigel Smart: "Elliptic Curves in Cryptography", Lecture Notes Series 265, Cambridge University Press (2004) | |||||

Prerequisites / Notice | Voraussgesetzt werden Algebra I und Grundbegriffe der projektiven Geometrie. | |||||

Selection: Analysis | ||||||

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

401-4115-00L | Introduction to Geometric Measure Theory | W | 6 credits | 3V | U. Lang | |

Abstract | Introduction to Geometric Measure Theory from a metric viewpoint. Contents: Lipschitz maps, differentiability, area and coarea formula, rectifiable sets, introduction to the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, normal currents, relation to BV functions, slicing, compactness theorem for integral currents and applications. | |||||

Objective | ||||||

Content | Extendability and differentiability of Lipschitz maps, metric differentiability, rectifiable sets, approximate tangent spaces, area and coarea formula, brief survey of the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, currents with finite mass and normal currents, relation to BV functions, rectifiable and integral currents, slicing, compactness theorem for integral currents and applications. | |||||

Literature | - Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1995 - Herbert Federer, Geometric Measure Theory, 1969 - Leon Simon, Introduction to Geometric Measure Theory, 2014, web.stanford.edu/class/math285/ts-gmt.pdf - Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta math. 185 (2000), 1-80 - Urs Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742 | |||||

401-4463-62L | Fourier Analysis in Function Space Theory | W | 4 credits | 2V | T. Rivière | |

Abstract | In the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. | |||||

Objective | ||||||

Content | During the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces. In the second part of the course we will study fundamental properties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occasion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings. In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course. If time permits we shall present the notion of Paraproduct, Paracompositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE. | |||||

Literature | 1) Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions" (PMS-30) Princeton University Press. 2) Javier Duoandikoetxea, "Fourier Analysis" AMS. 3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer. 4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer. | |||||

Prerequisites / Notice | Notions from ETH courses in Measure Theory, Functional Analysis I and II (Fundamental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions) | |||||

Selection: Further Realms | ||||||

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

401-3502-68L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 2 credits | 4A | Professors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-3503-68L | Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION. Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information: 1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register; 2) in which semester; 3) for which degree programme; 4) your name and first name; 5) your student number; 6) the name and first name of the supervisor of the Reading Course. | W | 3 credits | 6A | Professors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective |

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