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The course of study comprises 20 modules. Each semester, you will earn 20 ECTS points (credits), divided between 2-3 modules. The proseminar courses are an important and integral part of the curriculum. Thus, the B Sc in Mathematics degree encompasses 180 ECTS points.

The course takes an average of 4½ years.

In our experience, studying mathematics takes time and patience, as new mathematical concepts have to be learnt, practised and mastered. The time and commitment required to follow a mathematics course should not be underestimated and is usually around 30 hours per week.

Our bridging course and the way we design our modules are intended to (re)introduce you to the subject matter and make the best use of your weekly study time.

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 1

DescriptionReading recommendations

Students will acquire basic competences in algorithmically oriented mathematics.

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

• Induction, Relations and Matlab

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

• 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

• Symmetry and Cholesky

• Linear Independence and Orthogonality

• Cubic Splines

• Approximation and Normal equations

Mandatory literature:

Class script

Additional literature/special activities:

The most fitting literature is the following: www.schulthess.com/buchshop/detail/ISBN-9783642419515/Harbrecht-Helmut-Multerer-Michael/Algorithmische-Mathematik, but it is currently only available

in German. A translation into English is planned in the future. The class script, provided in English, will cover the necessary content for this class.

• Otto Forster, Analysis I

• Gilbert Strang, Linear Algebra

• Herbert Amann and Joachim Escher, Analysis I

• Stephen Abbott, Understanding Analysis (Second Edition)

• Kenneth A. Ross, Elementary Analysis (Second Edition)

• H.-D. Ebbinghaus, Numbers

• Walter Rudin, Principles of Mathematical Analysis (Third Edition)

• Allen B. Downey, Think Python (Second Edition)

• Timothy Sauer, Numerical Analysis (Second Edition)

• Matlab (officially supported) or Octave

• Tablet for writing (e.g., HP Envy, Lenovo Yoga , Microsoft Surface, Wacom Intuos, Samsung Galaxy Tab, iPad,

etc.)

(SS24)

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 1

DescriptionReading recommendations

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

Mandatory literature

Class script

Additional literature/special activities

• H. Harbrecht and M. Multerer, Algorithmische Mathematik. Graphen, Numerik und Probabilistik, Springer

• B. Korte and C. Vygen, Combinatorial Optimization, Springer

• U. Krengel, Einführung in die Wahrscheinlichkeitstheorie und Statistik, Vieweg

(SS24)

Prof. Dr Helmut Harbrecht

BSc Ilja Kalmykov

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 2

DescriptionReading recommendations

The student learns the basics of one-dimensional real analysis.

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,

• unction sequences, uniform convergence.

Mandatory literature:

Lecture notes provided by the lecturer.

Additional literature/special activities:

Books mentioned in the lecture notes.

(SS24)

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 2

DescriptionReading recommendations

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

Mandatory literature:

• Course Lecture Notes (available on Moodle)

Additional literature/special activities:

• M. Artin, Algebra

• G. Strang, Introduction to Linear Algebra

(SS24)

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 3

DescriptionReading recommendations

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

Content:

• Normed, metric, and topological spaces,

• convergence in metric spaces,

• closedness, compactness, connectedness,

• differentiability and differentiation of functions in several variables,

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

• inverse and implicit function theorem,

• integration on curves, surfaces, and other geometric bodies,

• integral theorems

Mandatory literature:

Lecture notes provided by the lecturer.

Additional literature/special activities:

Books mentioned in the lecture notes.

(SS24)

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semester 3

DescriptionReading recommendations

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

Mandatory literature:

• Course Lecture Notes (available on Moodle)

Additional literature/special activities:

• M. Artin, Algebra

• G. Strang, Introduction to Linear Algebra

(SS24)

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

Folgt

Folgt

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 6 and 7

DescriptionReading recommendations

This is a first course on probability theory. Topics to be covered:

• Probability spaces, random variables, expectation

• The classical laws of probability

• Independence and the Borel-Cantelli lemmas

• The Borel-Cantelli lemmas

• Sums of independent random variables

• Convergence of random variables

• The law of large numbers

• Convergence in law

• Characteristics functions

• The central limit theorem

Lecture notes (available on Moodle).

(SA23)

Prof. Dr Antti Knowles

Dr Volodymyr Fomichov

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

This "Numerics" course focuses on computer-based methods for solving mathematical problems that arise in various scientific and engineering applications. The primary objective is to develop a fundamental understanding of the construction of numerical methods. A significant portion of the course is dedicated to analyzing these methods to determine their usability, limitations, and applicability, with particular emphasis on three key aspects: (1) assessing the accuracy provided by different methods, (2) evaluating their efficiency, and (3) addressing issues of stability. The course covers a range of standard methods commonly used in numerical computation, including:

A) direct solution methods for linear systems

B) Iterative solution methods for linear systems

C) Root finding techniques for nonlinear equations

D) Interpolation and approximation methods for functions

E) Numerical integration and quadrature.

An essential component of numerical analysis involves the practical implementation of the methods discussed in the course. This hands-on approach allows students to gain firsthand experience with the challenges related to accuracy, computational effort, and stability. The coursework includes computational experiments, which students will conduct using the MATLAB programming language or another language of their choice.

Mandatory literature:

Quarteroni, Sacco, Saleri, Numerical Mathematics, Second edition, 2007.

Additional literature/special activities:

There are several other books available that cover similar topics to the course:

"Numerical Analysis in Modern Scientific Computing: An Introduction" by Peter Deuflhard and Andreas Hohmann.

"Concise Numerical Mathematics" by Robert Plato.

"Numerical Mathematics and Computing" (7th edition) by Ward Cheney and David Kinclad.

For the implementation aspect, although it is not mandatory, the following text can be useful:

"Scientific Computing with MATLAB and Octave" by Alfio Quarteroni, Fausto Saleri, and Paola Gervasio.

This book provides valuable insights into implementing numerical methods using MATLAB and Octave, which can enhance students' practical understanding.

Additionally, the course may provide additional papers as supplementary materials for further in-depth study and exploration of the topics covered. These papers will serve as additional resources to broaden students' knowledge and understanding of the subject matter.

(SA23)

Dr Marco Favino

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

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.

Standard literature:

Lecture notes made available on moodle.

Additional literature/special activities:

To be determined and announced during the course

(SS24)

Dr Emanuele Delucchi

Dr Andrea Adriani

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 6 and 7

DescriptionReading recommendations

This lecture covers the classical theory of Ordinary Differential Equations (ODEs), including explicit solution techniques, existence and uniqueness theory and numerical solutions. Content:

•Basics

o definition of an ODE

o definition of a solution of an ODE

o Initial Value Problems

o graphical methods/tangent fields

• Solving ODEs explicitly

o linear ODEs

▪ homogenous equations

▪ superposition principle

▪ particular solutions

o linear ODEs with constant coefficients

o first order separable equations

o integrating factors

o Laplace transform

o variation of constants

o series solutions

• Existence and uniqueness theory

o Picard existence and uniqueness theorem

o example of non-uniqueness

o example of non-existence

• Numerical solutions of ODEs

o Euler one step method

o backward Euler method

o Runge-Kutta methods

o accuracy, consistency

o stability

o stiffness

Mandatory resources:

The lecture notes for the course.

A tablet for writing (for solving exercises and for the oral exam)

Additional literature (if helpful – the lecture notes will be self-contained):

• H S Bear - Differential Equations: A Concise Course

• Arnold - Ordinary Differential Equations

• Trefethen - Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations

• Hairer, Norsett, Wanner - Solving Ordinary Differential Equations I

(SS24)

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 6 and 7

DescriptionReading recommendations

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.

Folgt

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 8 and 9

Description

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.

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 8 and 9

Description

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.

ECTS10

Semesterspring and autumn semester

Target AudienceStudents from semesters 8 and 9

Description

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.

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

• 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

Mandatory literature:

Course Lecture Notes (available on Moodle)

Additional literature/special activities:

• M. W. Hirsch, S. Smale : Differential equations, dynamical systems and linear algebra, Chapters 1-3

• M. Chipot : Element of nonlinear analysis, Chapter 1 + mentioned in the lecture notes

(SS24)

Prof. Dr Michel Chipot

Dr Jianfang Lin

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 4 and 5

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 6 and 7

Description

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.

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 6 and 7

Description

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

ECTS10

Semesterspring semester

Target AudienceStudents from semesters 8 and 9

Description

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.