Foundation Modules MSc Mathematics. Winter Term 2018/19

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1 F4A1-V3A2 Algebra II Prof. Dr. Catharina Stroppel The first part of the course will start from linear group actions and study some invariant theory questions with several applications. We will learn basic facts about semisimple algebras and their appearance in mathematics. The Artin-Wedderburn theorem and Centralizer Theorems will be proved. The second part of the course will give an introduction into affine algebraic groups, their tangent spaces and mention some classification results. We will study in particular natural actions of algebraic groups and several explicit examples in detail. A solid background in groups rings and modules is required. The main theorem of Galois-theory (the Galois correspondence) and a basic understanding of the involved techniques is assumed. A basic understanding of tensor products and universal properties will be assumed. Knowledge about affine varieties and Zariski topology is helpful. Any book on groups rings and modules and on Galois theory. For instance Thomas W. Hungerford Algebra, Graduate Text in Mathematics, Springer covers all the necessary background and much more.

2 F4A1-V3A3 Foundations in Representation Theory Dr. Hans Franzen In this course we are going to treat basic concepts of homological algebra and their application to representation theory. We will cover the following material in the lecture: categories and functors abelian categories homology of chain complexes derived functors homological dimension indecomposable modules and simple functors. Required for the course: Basic knowledge of rings and modules (rings, ideals, tensor products, localization, Nakayama's lemma). These prerequisites can be found in: Jacobson - Basic Algebra I (Ch. 2) Atiyah/Macdonald - Commutative Algebra (Ch. 1,2,3) Literature for the course (on homological algebra and representation theory): Weibel - An Introduction to Homological Algebra Hilton/Stammbach - A Course in Homological Algebra Assem/Simson/Skowronski - Elements of the Representation Theory of Associative Algebras

3 F4B1-V3B1 PDE & Functional Analysis Prof. Dr. Barbara Niethammer What is functional analysis? Functional analysis is, roughly speaking, analysis on infinite dimensional spaces. The functional analytic point of view has meanwhile become essential in many areas of mathematics, e.g. in geometry and topology, probability, numerical analysis, mathematical physics and partial differential equations. This course will give an introduction to functional analysis and covers in particular the central theorems of linear functional analysis. The course is also suitable for students who have not followed the course Einführung in die partiellen Differentialgleichungen (Introduction to partial differential equations). Partial differential equations will be used as examples of applications of methods from functional analysis, but no previous knowledge of partial differential equations is assumed. A typical problem is the following: Let X and Y be infinite vector spaces and let A : X Y be a linear and continuous map. Under which conditions does A have a continuous inverse? An important theorem of linear algebra states that for X = Y = R d we have A injective A surjective A 1 exists and is continuous. This is no longer true for infinite dimensional spaces. The course will cover four big themes: 1. spaces and topologies / notions of convergence 2. linear operators 3. Hilbert spaces 4. eigenvalues and spectral theory Analysis III (i.e. measure and integration, in particular the Lebesgue integral) The course will be mostly based on the following book: H.W. Alt: Linear functional analysis, Springer

4 F4B1-V3B4 Global Analysis II Dr. Batu Güneysu The aim of this course is to give an introduction to microlocal analysis (that is, the analysis of pseudodifferential operators) on manifolds. The main topics of the course are: Distribution theory and Fourier transforms, oscillatory integrals and Fourier integral operators, pseudodifferential operators and Sobolev spaces on manifolds, regularity theory and spectral theory for elliptic operators on closed manifolds, applications of microlocal analysis such as e.g. Hodge-Theory on closed manifolds. Basics notions of differential geometry such as manifolds and vector bundles, basics of partial differential operators on manifolds. These topics can be found e.g. in the following books: Güneysu, Batu: Covariant Schrödinger semigroups on Riemannian manifolds. Operator Theory: Advances and Applications, 264. Birkhäuser/Springer, Cham, Nicolaescu, Liviu: Lectures on the geometry of manifolds. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, xviii+589 pp.

5 F4D1-V3D1 Topology I Prof. Dr. Wolfgang Lück This course is part of a series of four courses about topology. In this course we introduce one of the fundamental invariants in algebraic topology, namely, homology and cohomology. We introduce singular and cellular (co)homology, discuss invariants such as the Euler characteristic and the Lefschetz number, and introduce some basic tools for computations such as the universal coefficient theorem and the Kuenneth formula. We will also deal with product structures. We will include some interesting applications to questions outside of topology. The long term goal, which may be not be achieved this semester, but in the next semester, is Poincare duality which relates the homology and the cohomology of a closed manifold. We will need all the material of the course V2D1 - Einführung in die Geometrie und Topologie, held by myself in the summer semester There are many textbooks about this material, we mention the following ones: Hatcher, Allen: Algebraic topology, Cambridge University Press, Laures, Gerd and Szymik, Markus: Grundkurs Topologie, Springer-Lehrbuch, Springer, Lück, Wolfgang: Algebraic topology: Homology and manifolds, Vieweg Studium: Aufbaukurs Mathematik, Vieweg, 2005.

6 F4D1-V3D3 Geometry I Dr. Christian Blohmann Description: The course will equip the student with the knowledge of the basic concepts, methods, and results of differential geometry. These are the basis for other fields such as PDEs, geometric analysis, Lie theory, mathematical physics, etc. and enable the student to pursue more advanced directions in geometry such as symplectic or complex geometry. Smooth manifolds; tangent vectors; the differential of a smooth map, submersions, immersions, étale maps; the tangent bundle; vector fields, Lie bracket, flows, integration of vector fields; Lie groups, matrix groups, equivariant maps, standard series, exponential map; foliations, distributions, integrability, involutivity, Frobenius theorem, integration of Lie algebras to Lie groups; differential forms, inner derivative, de Rham differential, Lie derivative, Cartan calculus, Poincaré lemma; Riemannian metrics, geodesics, variation of energy and length, geodesic distance, Riemann normal coordinates, Hopf-Rinow theorem; connections, Ehresmann connection, linear connections, covariant derivative, parallel transport, curvature, torsion, Levi-Civita connection, Ricci curvature, scalar curvature, sectional curvature; Riemannian submanifolds, first and second fundamental form, hypermanifolds, Gauß' theorema egregium Point set topology, multilinear algebra (e.g. tensor products, exterior algebra), analysis of several variables (e.g. implicit function theorem), ordinary differential equations (e.g. Picard-Lindelöf), basic group theory. In addition, basic knowledge of homological algebra will be useful. John M. Lee: Introduction to Smooth Manifolds, Springer 2002 Jürgen Jost: Riemannian Geometry and Geometric Analysis, 6th edition, Springer 2011 Yvonne Choquet-Bruhat, Cecile DeWitt-Morette: Analysis, Manifolds and Physics, Part 1: Basics, revised edition, North Holland 1982.

7 F4E1-V3E1 Scientific Computing I Prof. Dr. Carsten Burstedde We will develop the theory of finite element methods for elliptic partial differential equations. We will apply and extend these methods and techniques in theoretical and programming exercises. Requirements are multivariate calculus / vector analysis, basic knowledge of polynomial interpolation and approximation, numerical integration, and the numerical solution of systems of equations. The programming exercises will require programming in C. The course loosely follows the book "Finite elements" by D. Braess, Cambridge University Press 2010, which is a translation of the classic German textbook.

8 F4F1-V3F2 Introduction to Stochastic Analysis Prof. Dr. Andreas Eberle Description: In stochastic analysis, more general stochastic processes in continuous time are constructed from Brownian motion. Stochastic differential calculus (Itō calculus) enables us to compute corresponding expectation values in an elegant way, and to set up connections to differential equations. Stochastic Analysis is important for many application areas (including mathematical finance, but also natural sciences and engineering). It also has fundamental connections to many other mathematical disciplines, and it is the basis for most probability courses in the Master programme. Contents and prerequisites: I plan to cover Chapters 3-9 of the lecture notes on my webpage Chapter 1 on Brownian motion will not be discussed during this course, since this material is part of the Stochastic Processes course. Knowledge of the basics about Brownian motion will be assumed from week 5 onwards. During the first 4 weeks of the course, an introduction to martingales in continuous time will be given. Measure theoretic probability and conditional expectations w.r.t. σ-algebras are required as a prerequisite, see for instance Appendix A in the lecture notes or Chapter 9 in Probability with martingales by D. Williams. In the beginning, we will briefly review the material on martingales in discrete time in Sections of the lecture notes which has already been discussed in the Stochastic Processes course. The figure shows a typical sample path of geometric Brownian motion. The process is an exponential martingale. In particular, the expectations are constant in time, but nevertheless the sample paths converge to 0 with probability one.

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