Syllabus

Students will acquire basic competences in algorithmically oriented mathematics.

• Basics – Sets, Functions, Numbers, Absolute Value and Matlab I

• Induction, Relations and Matlab II

• Groups, Construction of Z, Minimum and Maximum and Matlab III

• Floating-Point Representation, Cancellation of Leading Digits and b-adic Representation

• Practical Floating-Point system

• Condition Number and Landau Symbol

• Polynomial interpolation

• Divided differences

• Linear systems and Gaußian elimination

• LU-decomposition and Matrix norms

• Stability of Gaußian elimination

• Symmetry and Cholesky

• Linear independence and Orthogonality

• Cubic Splines

• Approximation and Normal equations

Students acquire basic competences in discrete structures, probability theory and basic statistics. The topics are

• Logic and Sets

• Sorting algorithms and complexity

• Graphs and trees

• Graph colouring and algorithms on graphs

• Probability spaces and probability

• Conditional probability and independency

• Random variables and discrete distributions

• Expectation and variance

• Elementary statistics

• Random number generators and simulation

The student learns the basics of one-dimensional calculus.

Content:

• construction of the real numbers,

• sequences (convergence and convergence criteria),

• series (convergence and convergence criteria),

• functions of one real variable,

• continuity, uniform and Lipschitz continuity,

• complex numbers,

• function types: exponential, logarithm, sine, cosine, …

• differentiability and differentiation, differentiation rules,

• introductory examples for an ordinary differential equation (exp(x)),

• Taylor’s theorem,

• Riemann-Integral, fundamental theorem of calculus,

• function sequences, uniform convergence.

Students acquire basic competences in linear algebra:

• Fields and complex numbers

• Matrices and vectors

• Matrix products

• Subspaces

• Linear independence and bases

• Dimension

• Matrices and linear maps

• Change of basis

• Determinants (characterization, existence, properties, permutations)

• Endomorphisms (trace, eigenvalues and eigenvectors)

• Affine spaces and quotient vector spaces

The student learns how to differentiate and integrate functions in several variables.

Content:

• Normed, metric, and topological spaces,

• convergence in metric spaces,

• closedness, compactness, connectness,

• differentiability and differentiation of functions in several variables,

• classical differential operators (nabla, rot, curl),

• inverse and implicit function theorem,

• measure theory,

• Lebesgue integration and L^p spaces,

• Theorems of Tonelli and Fubini

Students acquire basic competences in linear algebra:

• Symmetry and groups

• Bilinear forms

• Euclidean spaces

• orthonormal bases

• Self adjoint endomorphism

• Quadratic forms

• Unitary spaces

• Jordan normal form

• Duality

Description: This lecture gives an introduction into function theory and functional analysis

Content: Function theory: analytical functions, Cauchy’s integral theorem, properties of analytical functions, singu-larities, residue theorem Functional analysis: normed pscaes, dual, bounded and linear mappings (functionals), fixed point theorems, Hahn-Banach, Reflexivity, Hilbertspaces, Orthonormalbases and Fourier series, Theorem of Riesz, Lax-Milgram, adjoint operators, compact operators.

Description: This lecture gives an introduction to the mathematical theorey of stochastics

Content: Probability measures, probability spaces, random variables, distributions (normal, hypergeometric, Poissson, ...), density, Chebycheff, expectation, variance, moments, correlation, central limit theorem, Monte-Carlo methods, Markov-chains.

Description: This lecture introduces fundamental numerical methods

Content: Condition number, Interpolation, approximation, direct solution methods, matrix factorization, singular value decomposition, adaptivity.

• Ordinary differential equations

• First order systems, existence, uniqueness

• Applications : prey-predators, population dynamics, finance…

• Resolution of typical cases, linear systems

• Celestial mechanics

• Diffusion and heat equation, weak formulations

• Wave equation

• Elasticity

• Fluid mechanics

• Numerical approximations

This course will cover the foundations of abstract algebra, focusing on the notions of group, ring and field. For each object we will present basic concept and some applications. In the case of groups: quotients, isomorphism theorems, the symmetric group, Sylow theorems. For rings: Ideals, polynomial rings, Principal Ideal Domains and notions of divisibility. For fields: field extensions, splitting fields. The narrative arc of the lecture will be organized around five “applicative goals”, each abutting one of the five blocks of the lecture. A tentative list includes: classification of regular polyhedra, applications in combinatorial game theory, the Chinese Reminder Theorem, Pick’s theorem, and the impossibility of angle trisection with ruler and compass.

Description: This lectures gives an overview on theory and numerics for dynamical systems governed by ordinary differential equations.

Content: Picard-Lindelöff, Initial Value Problems, Existence, uniqueness of solutions, smoothness of solutions,lineare ODEs, stability, attractors, differential algebraic equations (DAEs), one-step methods, stability of numerical methods, stiff/non-stiff OFEs, Runge-Kutta methods, step-size control and global error, multi-step methods, collocation, multiple shooting, symplectic integrators, numerical methods for DAEs, parallel-in-tome methods (e.g. parareal).

M13a Geometry

Projective geometry and the study of curves in the plane or surfaces in the space is a classical field in mathematics and an active area of research with a lot of beautiful results. The classification of quadrics and elliptic curves is a first step in the theory, followed by the study of intersections and singularities.

M13b Number theory

This is a very classical subject, starting with the field of rational numbers and its extensions, called number fields. A first highlight is Galois Theory, setting up a beautiful interaction between number fields and groups. Modern number theory is still one of the main research areas in mathematics. On the other hand, elementary number theory plays a fundamental role in cryptology and communication.

Description: This lecture gives an introduction to Differential Geometry

Content: Curves, manifolds, tangential space, vector fields, tangent bundles, symplectic forms. Total differential, Theorem of Stokes, Lie-groups.

M15a Topology

Description: This lecture gives an introduction to general topology or in Algebraic Topology. Topology is concerned with properties of geometrical objects that are preserved under continuous deformations. The abstract setting is a topological space and its behaviour under continuous maps. In the algebraic topology one can attach to a topological spaces algebraic objects, such as the homology groups or the fundamental group, which allows to distinguish topological spaces

Content: Topological Spaces, topologies, continuous mappings, separation axioms, filter, compactness, compactification, metrization

M15b Theory of Groups

The theory of groups and their representations appears in many fields in mathematics and also plays an important role in theoretical physics, e.g. in quantum mechanics and recently in quantum computing. Symmetries can be described by groups. Typical examples are the platonic solids which correspond to finite symmetry groups in space, and the crystallographic groups in space used for the classification of crystals.

M15c Combinatorics

Combinatorics is mainly concerned with enumerating and counting, and with structures on finite sets. It has many applications, in particular in statistics and computer science. Combinatorial problems arise in many areas of mathe-matics, e.g. in algebra, topology and in group theory. A classical subject is graph theory, going back to the 18th century.

Description: The course gives an introduction in Optimization and Machine Learning.

Content: Minima, stationary points, convex minimization, unconstrained minimization, KKT conditions, constraints (equality/inequality), Trust-region methods, SQP, BFGS, CG, linear Programing, Simplex, interior point, learning, supervised, unsupervised, re-inforcement learning, neuronal nets, convolutional networks, autoencoder, u-net, stochastic gradient method, backpropagation, accelerated gradient methods

Description: This lecture gives an introduction to the theory and numerics of partial differential equations (PDEs) and includes an integrated Lab.

Content: Definition of PDEs, clasification, boundary value problems, Sobolev spaces, elliptic problems, Poisson equation, heat equation, Green’s function, fundamental solution, weak formulation, solution, Lax-Milagram, Finite differences, finite elements, parabolic problems, stability, eigenfunctions, Fouriersolutions, method of Lines, Rothe’s method, advection, characteristics, hyperbolic equations, stability of finite diffrenece schemes, CFL-condition, conservation laws, finite Volume, error estimators, adaptivity, space-time formulations.

Integrated Lab

Content: Finite Differences, Finite Elements, mesh generation, non-linear finite elements, boundary conditions, a simple geometric multigrid, visualization.

The Module may content special topics from the fields of Algebraic Geometry, Geometry, Theory of Groups, Representation Theory of Groups, Number Theory, Topology or Combinatorics.

Students can choose from a catalogue of modules from the study courses of the other Faculties offered by FernUni Schweiz. The available modules from the Faculties of Economics, Psychology, Law and History are published every semester. The modules are given in the language of the study course.

This two-semester module consists of a seminar followed by a bachelor thesis. In the guided seminar, students will read, summarize, and present a mathematically advanced paper with the aim to consolidate the essential competence of mathematicians: “Comprehend – Transmit – Write up”. This is the basis for the bachelor thesis.