# Search result: Catalogue data in Spring Semester 2019

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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401-3146-12L | Algebraic Geometry | W | 10 credits | 4V + 1U | E. Kowalski | |

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

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

Literature | Primary reference: * Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer. Secondary reference: * Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications. * Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer. * Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013). Other good textbooks and online texts are: * David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer. * Ravi Vakil, Foundations of Algebraic Geometry, Link * Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link "Classical" Algebraic Geometry over an algebraically closed field: * Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer. * J.S. Milne, Algebraic Geometry, Link Further readings: * Günter Harder: Algebraic Geometry 1 & 2 * I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag. * Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA * Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag. | |||||

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

401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | P. Biran | |

Abstract | This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations. | |||||

Objective | ||||||

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

Prerequisites / Notice | General topology, linear algebra. Basic knowledge of singular homolgoy and cohomology 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. | |||||

401-3226-01L | Representation Theory of Lie Groups | W | 8 credits | 4G | P. D. Nelson | |

Abstract | This course will contain two parts: * Introduction to unitary representations of Lie groups * Introduction to the study of discrete subgroups of Lie groups and some applications. | |||||

Objective | The goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups. | |||||

Content | * Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula * Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C) * Example: Property (T) for SL(n,R) * Discrete subgroups of Lie groups: examples and some applications | |||||

Literature | Bekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press. | |||||

Prerequisites / Notice | Differential geometry, Functional analysis, Introduction to Lie Groups (or equivalent). | |||||

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

Abstract | This is a continuation course of Differential Geometry I. Topics covered include: - Connections and curvature, - Riemannian geometry, - Gauge theory and Chern-Weil theory. | |||||

Objective | ||||||

Lecture notes | I will produce full lecture notes, available from my website at Link | |||||

Literature | There are many excellent textbooks on differential geometry. A friendly and readable book that contains everything covered in Differential Geometry I is: John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag. For Differential Geometry II, the textbooks: - S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley, - I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP, are both excellent. The monograph - A. L. Besse "Einstein Manifolds", (1987), Springer, gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.) | |||||

Prerequisites / Notice | Familiarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). lecture notes for Differential Geometry I can be found on my webpage. | |||||

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

Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity, spectral theory, and unitary representations. | |||||

Objective | Acquiring the language and methods for boundary value problems, Sobolev spaces, Banach algebras, Spectral theory of bounded and unbounded selfadjoint operators, and Unitary representations. | |||||

Lecture notes | Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017 | |||||

Literature | Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017 | |||||

Prerequisites / Notice | Functional Analysis I plus a 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). |

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