
Course contentsReal 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.

Short Name:  ComplexAnal  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212 and 100221  
Corequisites:  None  
Tutorial:  Yes 
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.
Short Name:  IntroAlgebra  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100221  
Corequisites:  None  
Tutorial:  Yes 
Group Theory: Definitions and key examples. Cosets and Lagrange's theorem. Group homomorphisms and basic constructions including quotient groups, direct and semidirect products. Some examples of (important) groups. Group actions and orbitstabilizer 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.
Short Name:  IntroNumTheory  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100211 and 100321  
Corequisites:  None  
Tutorial:  Yes 
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.
Short Name:  DiscMath  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  Yes 
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 inclusionexclusion 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 ZermeloFraenkel axioms, as well as cardinal and ordinal numbers and their properties.
Additional topics may be chosen depending on interests of instructor and students.
Short Name:  IntroTopology  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100211 and 100221  
Corequisites:  None  
Tutorial:  Yes 
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 Seifertvan 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.
Short Name:  ManifoldsTop  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212 and 100221  
Corequisites:  None  
Tutorial:  Yes 
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 higherdimensional 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 twodimensional manifolds and the PoincaréHopf theorem will be some of the highlights of the course.
Short Name:  DynSystems  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212 and 100221  
Corequisites:  None  
Tutorial:  Yes 
Short Name:  Intro PDE  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212  
Corequisites:  None  
Tutorial:  Yes 
This course alternates with Partial Differential Equations which takes a functional analytic approach to partial differential equations.
Short Name:  StochProc  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212  
Corequisites:  None  
Tutorial:  Yes 
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.
Short Name:  ApplStochProc  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212  
Corequisites:  None  
Tutorial:  Yes 
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 BlackScholes model, will conclude the course.
This course alternates with Stochastic Processes.
Short Name:  GR Math I  
Type:  Self Study  
Credit Points:  7.5  
Prerequisites:  permission of instructor  
Corequisites:  None  
Tutorial:  No 
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 12 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.
Short Name:  GR Math II  
Type:  Self Study  
Credit Points:  7.5  
Prerequisites:  permission of instructor  
Corequisites:  None  
Tutorial:  No 
Short Name:  NumAnal  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212  
Corequisites:  None  
Tutorial:  Yes 
Short Name:  MathMod BioMed  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100212  
Corequisites:  None  
Tutorial:  Yes 
Short Name:  CompAnalysis  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Due to the varying content, this course can be taken multiple times for credit.
Short Name:  Algebra  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Short Name:  AdvAlg  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Short Name:  AlgGeometry  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
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.
Short Name:  AlgebrTopology  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
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.
Short Name:  DiffGeom  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Short Name:  LieGroups  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
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, finitedimensional 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.
Short Name:  ModernGeometry  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
The following concepts, known from the 300level 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 400level courses, Differential Geometry and Algebraic Topology.
Short Name:  DynSystems  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
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 3dimensional, 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).
Short Name:  FunctAnalysis  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
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.
Short Name:  PDE  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  100313  
Corequisites:  None  
Tutorial:  No 
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.
Short Name:  ApplAnalysis  
Type:  Lecture  
Credit Points:  7.5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Short Name:  MathColloquium  
Type:  Seminar  
Credit Points:  None  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Short Name:  GradResearchSem  
Type:  Seminar  
Credit Points:  5  
Prerequisites:  None  
Corequisites:  None  
Tutorial:  No 
Graduate research seminar participants receive a grade on the scale 1.0 through 5.0.
Last updated
20140807, 18:04.
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