# Search result: Catalogue data in Autumn Semester 2020

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 Self-service registration for this course unit in myStudies has been closed. | W | 8 credits | 4G | A. Iozzi | |

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: Link | |||||

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

Abstract | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||

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: Link See also: Link 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 covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||

401-3145-70L | Algebraic Geometry IRegistration for this course unit has been closed. | W | 10 credits | 4V + 1U | P. Yang | |

Abstract | This course is an introduction to Algebraic Geometry (algebraic varieties). | |||||

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

Literature | Primary reference: * I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag. * M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publ., 1969. Secondary reference: * Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer. * 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. * D. Eisenbud: Commutative algebra. With a view towards algebraic geometry, GTM 150, Springer Verlag, 1995. * H. Matsumura, Commutative ring theory, Cambridge University Press 1989. * N. Bourbaki, Commutative Algebra. 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 * Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA * Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag. | |||||

Prerequisites / Notice | Linear Algebra |

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