Search result: Catalogue data in Autumn Semester 2017
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 | 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 | W. Merry | |
Abstract | This is an introductory course in algebraic topology. Topics covered include: the fundamental group, covering spaces, singular homology, cell complexes and cellular homology and the Eilenberg-Steenrod axioms. Along the way we will introduce the basics of homological algebra and category theory. | |||||
Objective | ||||||
Lecture notes | I will produce full lecture notes, available on my website at Link | |||||
Literature | "Algebraic Topology" (CUP, 2002) by Hatcher is excellent and covers all the material from both Algebraic Topology I and Algebraic Topology II. You can also download it (legally!) for free from Hatcher's webpage: Link Another classic book is Spanier's "Algebraic Topology" (Springer, 1963). This book is very dense and somewhat old-fashioned, but again covers everything you could possibly want to know on the subject. | |||||
Prerequisites / Notice | You should know the basics of point-set topology (topological spaces, and what it means for a topological space to be compact or connected, etc). Some (very elementary) group theory and algebra will also be needed. | |||||
401-4147-67L | Algebraic Geometry II | W | 10 credits | 4V + 1U | R. Pink | |
Abstract | Quasicoherent sheaves, cohomology, Serre duality, Riemann-Roch theorem, algebraic curves, moduli schemes | |||||
Objective | ||||||
Literature | Primary reference: * Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer. * 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. * 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 | Algebraic Geometry I Spring 2017 | |||||
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). | |||||
401-3581-67L | Symplectic Geometry | W | 8 credits | 4V + 1U | A. Cannas da Silva | |
Abstract | This course is an introduction to symplectic geometry -- the geometry of manifolds equipped with a closed non-degenerate 2-form. We will discuss symplectic manifolds and transformations, the relation of symplectic to other geometries and some of the interplay with dynamics, eventually in the presence of symmetry groups. Guided homework assignments will complement the exposition. | |||||
Objective | Introduction to symplectic geometry |
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