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Course Descriptions

300 Level Courses

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.

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.

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.

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.

400 Level Courses

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.

 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.

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.

100591: Mathematics Colloquium

 Short Name: MathColloquium Type: Seminar Credit Points: None Prerequisites: None Corequisites: None Tutorial: No

Course contents

The weekly mathematics colloquium features talks by international scientists for the entire mathematical community, broadening horizons and encouraging formal or informal interactions.

 Short Name: GradResearchSem Type: Seminar Credit Points: 5 Prerequisites: None Corequisites: None Tutorial: No

Course contents

This course is intended for beginning graduate students to help them identify interesting areas of research and possible thesis subjects and advisors. It consists of lectures mainly by professors, but also by other faculty, about current areas of research in mathemaical sciences, with particular emphasis on research areas of Jacobs faculty. Students get involved in discussions of all the areas of research; during the course of the semester, they choose at least three topics which they investigate further and which they elaborate into a research report. At the end of the semester, every student presents at least one of these reports. Participation is also open for advanced undergraduates looking for topics for their undergraduate theses, the results of which are presented as well.