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Program Contact:

Dierk Schleicher

Professor of Mathematics

dierk@jacobs-university.de

Research I, 115

+49 421 200 3213

 
 
 

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Courses

First year Mathematics and ACM


110101: General Mathematics and Computational Science I

  Short Name: GenMathCPS I
  Type: Lecture
  Credit Points: 5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

General Mathematics and Computational Science I and II are the introductory first year courses for students in Mathematics and Applied and Computational Mathematics. In addition, these courses address anyone with an interest in mathematics and mathematical modeling. Each semester includes a selection of ``pure'' and ``applied'' topics which provide a solid foundation for further study, convey the pleasure of doing mathematics, and relate mathematical concepts to real-world applications.

Topics covered in the first semester are:

  • Fundamental concepts: sets, relations, functions, equivalence classes.
  • Numbers: Peano axioms, proof by induction, construction of integers and rational numbers.
  • Discrete Mathematics: combinatorics, binomial coefficients, generating functions, applications to elementary discrete probability.
  • Inequalities: Geometric-arithmetic mean inequalities, Cauchy inequality; Laplace's method and Stirling's approximation.
  • Difference equations: linear first order difference equations, nonlinear first order difference equations, equilibrium points and their stability, linear second order difference equations; modeling with difference equations.


110102: General Mathematics and Computational Science II

  Short Name: GenMathCPS II
  Type: Lecture
  Credit Points: 5
  Prerequisites:  110101
  Corequisites: None
  Tutorial: Yes

Course contents

This course continues General Mathematics and Computational Science I with the following selection of topics:
  • Groups: Basic properties and simple examples, Euclidean symmetries of the plane, symmetry groups of subsets of the plane, symmetry groups of polyhedra.
  • Graph Theory: Graphs and parity, trees, Euler's formula and Euler characteristic, pairings, Eulerian graphs.
  • Stochastic Modeling: Simple discrete stochastic systems, continuum limits, introduction to entropy.
  • Linear Programming: graphical method, simplex method, duality.
  • Fourier Transform: Discrete Fourier transform, fast Fourier transform, Fourier transform on groups.


110111: Natural Science Lab Unit - Symbolic Software

  Short Name: NatSciLab SymbSoft
  Type: Lab
  Credit Points: 2.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

The Natural Science Lab Units in Mathematics and ACM will introduce the computer as a tool for the working mathematician, as well as for scientists in many other fields.

The Lab Unit Symbolic Software introduces Mathematica, a software package that can perform complex symbolic manipulations such as solving algebraic equations, finding integrals in closed form, or factoring mathematical expressions. Mathematica also has powerful and flexible graphing capabilities that are useful for illustrating concepts as well as numerical data. The computer will be used as a tool in this course so that you will also learn some mathematics alongside learning to use the computer program.


110112: Natural Science Lab Unit - Numerical Software

  Short Name: NatSciLab NumSoft
  Type: Lab
  Credit Points: 2.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

The Natural Science Lab Units in Mathematics and ACM will introduce the computer as a tool for the working mathematician, as well as for scientists in many other fields.

The Lab Unit Numerical Software introduces Matlab and its free cousin Octave, software packages that allow easy and in many cases efficient implementations of matrix-based ``number crunching''. The software is ideal for numerical work such as solving differential equations or analyzing large amounts of laboratory data. The computer will be used as a tool in this course so that you will also learn some mathematics alongside learning to use the computer program.

This Lab Unit is particularly suited for students from both schools interested in experiments, as Matlab is used as a standard tool for analyzing and visualizing data in many fields of research.

Engineering and Science Mathematics


120101: ESM 1A - Single Variable Calculus

  Short Name: ESM 1A
  Type: Lecture
  Credit Points: 5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

The courses from the Engineering and Science Mathematics 1 series provide the foundation for all other Engineering and Science Mathematics courses. Taking at least one of them is mandatory for all Engineering and Science majors. Emphasis is on the use of basic mathematical concepts and methods in the sciences, rather than on detailed proofs of the underlying mathematical theory.

The course ESM 1A covers basic differential and integral calculus of functions of one variable. It starts with a brief review of number systems and elementary functions, then introduces limits (for both sequences and functions) and continuity, and finally derivatives and differentiation with applications (tangent problem, error propagation, minima/maxima, zero-finding, curve sketching). A short chapter introduces complex numbers.

The second half of the semester is devoted to integration (anti-derivatives and Riemann integral) with applications, and concluded by brief introductions to scalar separable and linear first-order differential equations, and the convergence of sequences and power series.

Compared to ESM 1C which covers similar material, this course assumes a rigorous high school preparation in Mathematics and leaves more room for explaining mathematical concepts (as needed for the majority of SES majors).


120111: ESM 1B - Multivariable Calculus, ODE

  Short Name: ESM 1B
  Type: Lecture
  Credit Points: 5
  Prerequisites:  None
  Corequisites: 120101
  Tutorial: Yes

Course contents

Engineering and Science Mathematics 1B introduces multivariable calculus and ordinary differential equations, topics of particular importance to the physical sciences. Students of ACM, Physics, and Electrical Engineering are strongly encouraged to take this course in their first semester. The curriculum is designed so that ESM 1A and ESM 1B can be taken at the same time.

The course covers vector algebra (three-dimensional vectors, dot product, cross product), equations of lines, planes, and spheres, Euclidean distance, vector-valued functions, space curves, functions of several variables, partial derivatives, chain rule, gradient, directional derivative, extrema, Lagrange multipliers, double and triple integrals with applications, change of variables, vector fields, divergence, curl, cylindrical and spherical coordinates, line integrals, Green's theorem in the plane, surface and volume integrals, divergence theorem, Stokes' theorem, introduction to ordinary differential equations (direction field, the question of existence and uniqueness of solutions), separable and exact equations, integrating factors, and linear higher order ODEs with constant coefficients.


120102: ESM 2A - Linear Algebra, Probability, Statistics

  Short Name: ESM 2A
  Type: Lecture
  Credit Points: 5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

Second semester Engineering and Science Mathematics is offered in two parallel classes that cover a common set of core topics at approximately the same level of difficulty. However, style of exposition and selection of additional material will vary slightly to meet the needs of different groups of majors.

ESM 2A is recommended for students majoring in Life Sciences or Chemistry. It covers the following topics: Linear Algebra (equations of lines and planes, matrix algebra, system of linear equations, matrix inverse, vector spaces, linear independence, basis, dimension, linear transformations, change of basis, eigenvalues and eigenvectors, diagonalization). Probability (basic notions of set theory, outcomes, events, sample space, probability, conditional probability, Bayes' rule, permutations and combinations, random variables, expected value, variance, binomial, Poisson, and normal distributions, central limit theorem). Statistics (one-sample hypothesis testing, two sample hypothesis testing, chi-square hypothesis testing, analysis of variance, bivariate association, simple linear regression, multiple regression and correlation).


120112: ESM 2B - Linear Algebra, Fourier, Probability

  Short Name: ESM 2B
  Type: Lecture
  Credit Points: 5
  Prerequisites:  120101 or 120111 or 120121
  Corequisites: None
  Tutorial: Yes

Course contents

Second semester Engineering and Science Mathematics is offered in two parallel classes that cover a common set of core topics at approximately the same level of difficulty. However, style of exposition and selection of additional material will vary slightly to meet the needs of different groups of majors.

ESM 2B is recommended for students who do not intend to major in the Life Sciences or Chemistry. It covers the following topics:

  • Linear Algebra (equations of lines and planes, matrix algebra, system of linear equations, matrix inverse, vector spaces, linear independence, basis, dimension, linear transformations, change of basis, eigenvalues and eigenvectors, diagonalization, inner products, orthonormalization)
  • Fourier methods (expanding functions in terms of orthonormal function systems, Fourier series, Fourier transform, Dirac delta-function)
  • Probability (basic notions of set theory, outcomes, events, sample space, probability, conditional probability, Bayes' rule, permutations and combinations, random variables, expected value, variance, binomial, Poisson, and normal distributions, central limit theorem).


120202: ESM 4A - Numerical Methods

  Short Name: ESM 4A
  Type: Lecture
  Credit Points: 5
  Prerequisites:  120112 or 100221
  Corequisites: None
  Tutorial: No

Course contents

Engineering and Science Mathematics 4A is mandatory for students of Electrical Engineering, Computer Science, and Applied and Computational Mathematics. It is also recommended as a home school elective for students who would like to get a short, one-semester introduction to Numerical Methods.

This course is a hands-on introduction to numerical methods. It covers root finding methods, solving systems of linear equations, interpolation, numerical quadrature, solving ordinary differential equations, the fast Fourier transform, and optimization. These methods are crucial for anyone who wishes to apply mathematics to the real world, i.e. computer scientists, electrical engineers, physicists and, of course, mathematicians themselves.

Second year Mathematics


100211: Analysis I

  Short Name: Analysis I
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

Analysis I/II is one of the fundamental courses in the mathematical education (together with Linear Algebra I/II). Its goal is to develop calculus in a rigorous manner and in sufficient generality to prepare the student for advanced work in mathematics. At the same time, the content is chosen so that students arrive quickly at central concepts which are used in essentially all mathematics courses, and which are needed in the exact sciences.

The Analysis sequence begins with a quick review of natural, rational and real numbers (which are assumed as known), and introduces the field of complex numbers. The axiom of completeness distinguishes the real numbers from the rationals and marks the beginning of Analysis. The complex exponential and trigonometric functions are defined.

Metric spaces are introduced and used to define continuity and convergence in a general framework. The Bolzano-Weierstraß and the Heine-Borel theorems are proved. The intermediate and maximal value theorems for functions on the real line are discussed as consequences of connectedness and compactness on metric spaces. Sequences of functions are discussed, in particular uniform convergence, as well as the continuity, differentiability, integrability of the limit function.

Differentiability of functions on the real line is introduced. The mean value theorem and Taylor's theorem is discussed.

The Riemann integral in one variable is introduced. The relation between the derivative and the integral, i.e., the fundamental theorem of calculus is proved.

This course has no formal prerequisites; incoming students with a strong mathematics background are encouraged to take this class in their first semester. However, a familiarity with mathematical reasoning and proof (e.g. proof by induction or by contradiction), such as introduced in General Mathematics and Computational Science I, is required.


100212: Analysis II

  Short Name: Analysis II
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100211
  Corequisites: 100221 or 120102 or 120112 (if not already taken)
  Tutorial: Yes

Course contents

This course is a continuation of Analysis I. Its main theme is to extend the concepts from Analysis I, in particular differentiation and integration, to functions of several variables. Taylor's theorem in several variables, the implicit function theorem and the inverse function theorem are proved. (Riemann) integration in several real variables is introduced, including the transformation formula for integrals in several variables.


100221: Linear Algebra I

  Short Name: LinAlg I
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

Together with Analysis I, this is one of the basic mathematics courses. It introduces vector spaces and linear maps, which play an important role throughout mathematics and its applications.

The course begins by introducing the concept of a vector space over an arbitrary field (for example, the real or complex numbers) and the concept of linear independence, leading to the notion of ``dimension''. We proceed to define linear maps between vector spaces and discuss properties such as nullity and rank. Linear maps can be represented by matrices and we show how matrices can be used to compute ranks and kernels of linear maps or to solve linear systems of equations.

In order to study some geometric problems and talk about lengths and angles, we introduce an additional structure called the inner or scalar product on real vector spaces. Properties of Euclidean vector spaces and orthogonal maps are treated, including the Cauchy-Schwarz inequality, Gram-Schmidt orthonormalization and orthogonal and unitary groups.

An endomorphism is a linear map from a vector space to itself and is represented by a square matrix. We study the trace and determinant of endomorphisms and matrices and discuss eigenvalues and eigenvectors. We discuss the question whether a matrix is diagonalizable and state the theorem on Jordan Normal Form which provides a classification of endomorphisms.

This course has no formal prerequisites; incoming students with a strong mathematics background are encouraged to take this class in their first semester. However, a familiarity with mathematical reasoning and proof (e.g. proof by induction or by contradiction), such as introduced in General Mathematics and Computational Science I, is required.


100222: Linear Algebra II

  Short Name: LinAlg II
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100221
  Corequisites: None
  Tutorial: Yes

Course contents

This course continues Analysis I and complements Analysis II. It continues with the classification of matrices and introduces elements of tensor algebra.

In the first part, we continue the discussion of endomorphisms, discussing the Cayley-Hamilton Theorem and minimal polynomials, and giving some versions of Jordan normal form over non-algebraically closed fields.

The second part of the course deals with dual spaces and quadratic, symmetric and skew-symmetric forms. We introduce the dual vector space and dual linear maps and their relation with bilinear forms. Classifications are given of symmetric and skew-symmetric real bilinear forms and of Hermitian and skew-Hermitian forms over the complex numbers.

The last part is concerned with tensors -- objects that play an important role in many branches of physics and mathematics. The tensor product of two (or more) vector spaces is introduced and properties are discussed including the universal property, contraction of tensors, `outer' and `inner' products, and the relationship between linear maps and tensors.


100291: Perspectives of Mathematics I

  Short Name: Perspectives I
  Type: Lecture
  Credit Points: 5
  Prerequisites:  None
  Corequisites: 100211 or 100221
  Tutorial: No

Course contents

This course is an overview and an introduction to selected topics from different areas of mathematics which are not usually covered in introductory classes. The goal is to develop an understanding for interesting mathematical questions beyond the standard first and second year curriculum. Rather than proving all results in full detail, we visit several different areas of mathematics, thus illustrating the breadth and beauty of mathematics.

Topics vary; past instances of this course have included combinatorial game theory, hyperbolic and spherical geometries, manifolds, Fourier analysis and wavelets, the Banach-Tarski-paradox, dynamical systems and chaos, and others.

For students in Mathematics and ACM, this course is designed to be taken at the same time as Analysis I and/or Analysis I. It is open to anyone with interest and some experience in mathematics (for others, General Mathematics and Computational Science I is recommended). Instead of a final exam, students write a term paper on a mathematical subject of their choice and present it to the class.


100292: Perspectives of Mathematics II

  Short Name: Perspectives II
  Type: Lecture
  Credit Points: 5
  Prerequisites:  100211 or 100221
  Corequisites: None
  Tutorial: No

Course contents

The spring semester instance of Perspectives of Mathematics is usually, but not always, independent of the fall semester. Format and goals of this course are as for Perspectives of Mathematics I.

Third year Mathematics


110341: Numerical Analysis

  Short Name: NumAnal
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

This course an advanced introduction to Numerical Analysis. It complements ESM 4A - Numerical Methods, placing emphasis, on the one hand, on the analysis of numerical schemes, on the other hand, focusing on problems faced in large-scale computations. Topics include sparse matrix linear algebra, large scale and/or stiff systems of ordinary differential equations, and a first introduction to methods for partial differential equations.


110361: Mathematical Modeling in Biomedical Applications

  Short Name: MathMod BioMed
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

The course discusses the area of mathematical modeling in biomedical applications. It includes an introduction into the basic principles of mathematical modeling, and it covers a variety of models for growth and treatment of cancer with increasing complexity ranging from simple ordinary differential equations to more complicated free boundary problems and partial differential equations. Further models for the description of physiology in the human body like blood flow and breathing are briefly touched as well.


100312: Introductory Complex Analysis

  Short Name: ComplexAnal
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212 and 100221
  Corequisites: None
  Tutorial: Yes

Course contents

This course introduces the theory of functions of one complex variable. It centers around the notion of complex differentiability and its various equivalent characterizations. Unlike differentiability for real functions, complex differentiability is a very strong property; for example it implies that the function is differentiable infinitely often and that it is represented by its Taylor series in a neighborhood of every point in its domain of definition. This results in a very nice and elegant theory that is used in many areas of mathematics.

Topics include holomorphic functions, Cauchy integral theorem and formula, Liouville's theorem, fundamental theorem of algebra, isolated singularities and Laurent series, analytic continuation and monodromy theorem, residue theorem, normal families and Montel's theorem, and the Riemann mapping theorem.

Possible further topics are elliptic and modular functions, the Riemann zeta function, introduction to Riemann surfaces.


100313: Real Analysis

  Short Name: RealAna
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

Real Analysis is one of the core advanced courses in the Mathematics curriculum. It introduces measures, integration, elements from functional analysis, and the theory of function spaces. Knowledge of these topics, especially Lebesgue integration, is instrumental in many areas, in particular, for stochastic processes, partial differential equations, applied and harmonic analysis, and is a prerequisite for the graduate course in Functional Analysis.

The course is suitable for undergraduate students who have taken Analysis I/II, and Linear Algebra I; it should also be taken by incoming students of the Graduate Program in the Mathematical Sciences. Due to the central role of integration in the applied sciences, this course provides an excellent foundation for mathematically advanced students from physics and engineering.


100321: Introductory Algebra

  Short Name: IntroAlgebra
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100221
  Corequisites: None
  Tutorial: Yes

Course contents

This course gives an introduction to three basic types of algebraic structures: groups, commutative rings, and fields. (If time permits, a fourth one: modules.) Here is a more detailed list of topics to be covered.

Group Theory: Definitions and key examples. Cosets and Lagrange's theorem. Group homomorphisms and basic constructions including quotient groups, direct and semi-direct products. Some examples of (important) groups. Group actions and orbit-stabilizer theorem. Possibly: Sylow theorems.

(Commutative) Rings: Definitions and elementary properties. Ideals, ring homomorphisms and quotient rings. Domains, Euclidean domains, principal ideal domains and unique factorization. Polynomial rings.

Field extensions: Roots of polynomials. Irreducibiliy criteria. Finite and algebraic field extensions. Finite fields. Possibly: Splitting fields and algebraic closure. Constructions with straightedge and compass.

If time permits Modules: Definitions and basic constructions. Linear maps and exact sequences. Direct products and sums. Structure theory for finitely generated modules over a principal ideal domain.


100331: Introductory Number Theory

  Short Name: IntroNumTheory
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100211 and 100321
  Corequisites: None
  Tutorial: Yes

Course contents

This course gives a Þrst introduction to number theory. It starts with Elementary Number Theory, covering topics such as congruences, the Chinese Remainder Theorem, Fermat's Little Theorem and Euler's extension; these have interesting applications to cryptography (such as the famous RSA algorithm). Further topics include Gaussian integers, quadratic reciprocity, Diophantine equations, Minkowski's lattice point theorem, as well as sums of two, three, and four squares.

The course will then move on beyond elementary number theory. Depending on the interests of students and instructor, possible topics are Pell's equation and continued fractions, the Prime Number Theorem, Dirichlet's theorem about primes in arithmetic progressions, or elliptic curves.


100332: Discrete Mathematics

  Short Name: DiscMath
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: Yes

Course contents

This course is open to anyone with interest and some experience in mathematics (for others, General Mathematics and Computational Science I and/or General Mathematics and Computational Science II is recommended).

Discrete mathematics is a branch of mathematics that deals with discrete objects and has naturally many applications to computer science. This course introduces the basics of the subject, in particular (enumerative) combinatorics, graph theory, as well as mathematical logic.

Enumerative combinatorics includes the binomial and multinomial coefficients, the pigeonhole principle, the inclusion-exclusion formula, generating functions, partitions, and Young diagrams.

Fundamental topics in graph theory include trees (spanning trees, enumeration of trees), cycles (Eulerian and Hamiltonian cycles), planar graphs (Kuratowski's theorem), colorings, and matching (perfect matchings, Hall's theorem).

In mathematical logic, among the basic topics are the Zermelo-Fraenkel axioms, as well as cardinal and ordinal numbers and their properties.

Additional topics may be chosen depending on interests of instructor and students.


100341: Introductory Topology

  Short Name: IntroTopology
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100211 and 100221
  Corequisites: None
  Tutorial: Yes

Course contents

This course is an introduction to some basic concepts and techniques in topology. The first part of the course builds on material from Analysis I, in particular the topology of metric spaces. We introduce topological spaces and continuous maps and proceed to discuss properties of spaces including connectedness, compactness and the Hausdorff property. Basic constructions such as the product and quotient of spaces are also treated.

The second part of the course deals with basic concepts of algebraic topology. We introduce the notion of homotopy, construct the fundamental group of a space and introduce the Seifert-van Kampen theorem, a key tool for computing fundamental groups. We discuss covering spaces and their relation with the fundamental group, including the construction of the universal covering space.

The course concludes with a basic treatment of homology groups and their properties, which are a fundamental tool for distinguishing topological spaces and mappings between them.


100353: Manifolds and Topology

  Short Name: ManifoldsTop
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212 and 100221
  Corequisites: None
  Tutorial: Yes

Course contents

This course is an introduction to the language and some of the fundamental concepts of modern geometry. Manifolds are among the most fundamental concepts of mathematics: curves and surfaces are important special cases that have historical significance.

The course starts with introducing the notion of a manifold, followed by examples that naturally arise in various areas of mathematics. Differentiability, tangent spaces and vector fields are then defined. This will be followed by establishing the notion of integration on manifolds. We will then formulate and prove Stokes' theorem, which is the higher-dimensional generalization of the fundamental theorem of calculus. Among the further topics that are discussed in the course are: orientation, degree of a map, Lie groups and their actions. The classification of one- and two-dimensional manifolds and the Poincaré-Hopf theorem will be some of the highlights of the course.


100361: Ordinary Differential Equations and Dynamical Systems

  Short Name: DynSystems
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212 and 100221
  Corequisites: None
  Tutorial: Yes

Course contents

Dynamical systems is an topic which links pure mathematics with applications in physics, biology, electrical engineering, and others. The course will furnish a systematic introduction to ordinary differential equations in one and several variables, focusing more on qualitative aspects of solutions than on explicit solution formulas in those few cases where such exist. It will be shown how simple differential equations can lead to complicated and interesting, often ``chaotic'' dynamical behavior, and that such arise naturally in the ``real world''. We will also discuss time-discrete dynamical systems (iteration theory) with its relations and differences to differential equations.


100362: Introductory Partial Differential Equations

  Short Name: Intro PDE
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

This course is a rigorous, but elementary introduction to the theory of partial differential equations: classification of PDEs, linear prototypes (transport equation, Poisson equation, heat equation, wave equation); functional setting, function spaces, variational methods, weak and strong solutions; first order nonlinear PDEs, introduction to conservation laws; exact solution techniques, transform methods, power series solutions, asymptotics.

This course alternates with Partial Differential Equations which takes a functional analytic approach to partial differential equations.


100382: Stochastic Processes

  Short Name: StochProc
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

This course is an introduction to the theory of stochastic processes. The course will start with a brief review of probability theory including probability spaces, random variables, independence, conditional probability, and expectation.

The main part of the course is devoted to studying important classes of discrete and continuous time stochastic processes. In the discrete time case, topics include sequences of independent random variables, large deviation theory, Markov chains (in particular random walks on graphs), branching processes, and optimal stopping times. In the continuous time case, Poisson processes, Wiener processes (Brownian motion) and some related processes will be discussed.

This course alternates with Applied Stochastic Processes.


100383: Applied Stochastic Processes

  Short Name: ApplStochProc
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100212
  Corequisites: None
  Tutorial: Yes

Course contents

This course aims at an introduction to the mathematical theory of financial markets that discusses important theoretical concepts from the theory of stochastic processes developed in parallel to their application to the mathematical finance.

The applied part of this course revolves around the central question of option pricing in markets without arbitrage which will be first posed and fully solved in the case of binomial model. Interestingly enough, many of the fundamental concepts of financial mathematics such as arbitrage, martingale measure, replication and hedging will manifest themselves, even in this simple model. After discussing conditional expectation and martingales, more sophisticated models will be introduced that involve multiple assets and several trading dates. After discussing the fundamental theorem of asset pricing in the discrete case, the course will turn to continuous processes. The Wiener process, Ito integrals, basic stochastic calculus, combined with the main applied counterpart, the Black-Scholes model, will conclude the course.

This course alternates with Stochastic Processes.


100391: Guided Research Mathematics I

  Short Name: GR Math I
  Type: Self Study
  Credit Points: 7.5
  Prerequisites:  permission of instructor
  Corequisites: None
  Tutorial: No

Course contents

Guided Research allows study, typically in the form of a research project, in a particular area of specialization that is not offered by regularly scheduled courses. Each participant must find a member of the faculty as a supervisor, and arrange to work with him or her in a small study group or on a one-on-one basis.

Guided research has three major components: Literature study, research project, and seminar presentation. The relative weight of each will vary according to topic area, the level of preparedness of the participant(s), and the number of students in the study group. Possible research tasks include formulating and proving a conjecture, proving a known theorem in a novel way, investigating a mathematical problem by computer experiments, or studying a problem of practical importance using mathematical methods.

Third year students in Mathematics and ACM are advised to take 1-2 semesters of Guided Research. The Guided Research report in the spring semester will typically be the Bachelor's Thesis which is a graduation requirement for every Jacobs University undergraduate. Note that the Bachelor's Thesis may also be written as part of any other course by arrangement with the respective instructor of record.

Students are responsible for finding a member of the faculty as a supervisor and report the name of the supervisor and the project title to the instructor of record no later than the end of Week 4. A semester plan is due by the end of Week 6.


100392: Guided Research Mathematics and BSc Thesis II

  Short Name: GR Math II
  Type: Self Study
  Credit Points: 7.5
  Prerequisites:  permission of instructor
  Corequisites: None
  Tutorial: No

Course contents

As for Guided Research Mathematics I.

Advanced courses


110411: Topics in Applied Analysis

  Short Name: ApplAnalysis
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

The course Topics in Applied Analysis introduces to a variety of fundamental analytic tools and methods used in the theory, modeling, and numerical simulation of phenomena in the natural sciences. The course is offered with different contents in different years, the choice will depend on the instructor. Examples of areas currently covered are applied harmonic analysis and operator theory, perturbation theory and asymptotic analysis, approximation theory, and others. Students specializing in applied mathematics or applied sciences may participate in this course more than once.


100412: Topics in Complex Analysis

  Short Name: CompAnalysis
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

Topics in Complex Analysis builds on the material taught in the undergraduate Complex Variables course. After a quick review of the most important results and concepts, some more advanced topics are covered. Possible subjects are Riemann Surfaces, Elliptic Functions and Modular Forms, Complex Dynamics, Geometric Complex Analysis, or Several Complex Variables. Which subjects are chosen will depend on the instructor and on the students' interests. This course may also provide an introduction to a specific area of research, leading to possible PhD thesis projects.

Due to the varying content, this course can be taken multiple times for credit.


100421: Algebra

  Short Name: Algebra
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

Advanced topics from algebra, including groups, rings, ideals, fields, and modules, continuing the course Introductory Algebra.


100422: Advanced Algebra

  Short Name: AdvAlg
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

This course develops more advanced topics in algebra beyond those from Algebra, including Galois theory, commutative algebra and its relation to algebraic geometry, as well as elements of noncommutative algebra.


100423: Algebraic Geometry

  Short Name: AlgGeometry
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

Algebraic geometry is the study of geometry using algebraic tools: the geometric objects are the common roots of a set of polynomials in several variables. Many geometric properties can be studied in terms of algebraic properties of these polynomials, using the powerful machinery of algebra to study geometry.

Basic concepts from Algebra and Introductory Algebra are used in this course. Among the studied subjects are affine and projetive varieties, schemes, curves, and cohomology.


100442: Algebraic Topology

  Short Name: AlgebrTopology
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

This course is mostly concerned with the comprehensive treatment of the fundamental ideas of singular homology/cohomology theory and duality. The knowledge of fundamental concepts of algebra as well as of general topology is assumed (at a level of Introductory Topology and Introductory Algebra.

The first part studies the definition of homology and the properties that lead to the axiomatic characterization of homology theory. Then further algebraic concepts such as cohomology and the multiplicative structure in cohomology are introduced. In the last section the duality between homology and cohomology of manifolds is studied and few basic elements of obstruction theory are discussed.

The graduate algebraic topology course gives a solid introduction to fundamental ideas and results that are used nowadays in most areas of pure mathematics and theoretical physics.


100451: Differential Geometry

  Short Name: DiffGeom
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

Differential geometry is the study of differentiable manifolds. Assuming basic concepts such as manifolds, differential forms, and Stokes' theorem, the focus in this course is on Riemannian geometry: the study of curved spaces which is at the heart of much current mathematics as well as mathematical physics (for example, General Relativity).


100452: Lie Groups and Lie Algebras

  Short Name: LieGroups
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

A Lie group is a group with a differentiable structure, the tangent space at the identity element of a Lie group is its Lie algebra. Lie groups and Lie algebras are indispensable in many areas of mathematics and physics. As a mathematical subject on its own, Lie theory has led to many beautiful results, such as the famous classification of semisimple Lie algebras. In physics, Lie groups and their representations are essential to the theory of elementary particles and its current developments. Due to the close correspondence of physical phenomena and abstract mathematical structures, the theory of Lie groups has become a showcase of mathematical physics.

The course presents fundamental concepts, methods and results of Lie theory and representation theory. It covers the relation between Lie groups and Lie algebras, structure theory of Lie algebras, classification of semisimple Lie algebras, finite-dimensional representations of Lie algebras, and tensor representations and their irreducible decompositions.

A solid background in multivariable real analysis and linear algebra is presumed. Familiarity with some basic algebra and group theory will also be helpful. No prior knowledge of differential geometry is necessary.


100453: Modern Geometry

  Short Name: ModernGeometry
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

The course serves as an introduction, at the advanced level, to the basic concepts of modern geometry.

The following concepts, known from the 300-level courses, should be briefly reviewed: concept of a manifold, the simplest examples of manifolds, and the concept of homotopy.

The core of the course will consist of explaining material related to the following topics: Lie groups, homogeneous spaces, symmetric spaces, fiber bundles, vector bundles, Morse theory, differential topology of mappings and submanifolds.

This material will provide a solid background for the 400-level courses, Differential Geometry and Algebraic Topology.


100461: Dynamical Systems

  Short Name: DynSystems
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

Dynamical systems is the study of the long-term behavior of anything in motion. The classical motivating topic is the stability of the solar system or, more recently, the study of weather prediction.

One theme in the course is the study of the underlying questions and difficulties in terms of model equations that are much simpler, often 1-, 2-, or at most 3-dimensional, but yet show rich and interesting dynamical features. A fundamental tool is to describe the dynamics of flows in terms of iterated maps of lower dimension, which are of great interest in their own right. Among the topics covered are circle homeomorphisms and endomorphisms, including rotation numbers, the quadratic family, toral automorphisms, horseshoes and the solenoid, the Lorenz systems, symbolic dynamics and shifts, and Sharkovski's theorem.

A second topic are ways to describe and quantify how complicated dynamical systems are: recurrence, topological transitivity and periodic orbits, mixing dynamics, topological and metric entropy, Lyapunov exponents, ergodicity and Birkhoff's theorem, and more.

Finally, there will be a discussion of general hyperbolic dynamics, including the stable/unstable manifold theorem and the shadowing lemma (not necessarily with detailed proofs in full generality).


100471: Functional Analysis

  Short Name: FunctAnalysis
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  None
  Corequisites: None
  Tutorial: No

Course contents

This course assumes basic knowledge of measure and integration theory, and of classical Banach and Hilbert spaces of measurable functions. Functional Analysis focuses on the description, analysis, and representation of linear functionals and operators defined on general topological vector spaces, most prominently on abstract Banach and Hilbert spaces.

Even though abstract in nature, the tools of Functional Analysis play a central role in applied mathematics, e.g., in partial differential equations. To illustrate this strength of Functional Analysis is one of the goals of this course.


100472: Partial Differential Equations

  Short Name: PDE
  Type: Lecture
  Credit Points: 7.5
  Prerequisites:  100313
  Corequisites: None
  Tutorial: No

Course contents

The course is an introduction to the theory of partial differential equations in a Sobolev space setting. Topics include Sobolev spaces, second order elliptic equations, parabolic equations, semi-groups, and a selection of nonlinear problems.

This course differs from the approach taken in Introductory Partial Differential Equations which focuses on solutions in classical function spaces via Greens functions. It may therefore be taken by students who have attended Introductory Partial Differential Equations, but we will again start from basic principles so that Introductory Partial Differential Equations is not a prerequisite.



 

 
Last updated 2014-08-07, 18:04. © Jacobs University Bremen. All rights reserved.