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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

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

(AS24)

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

(AS24)

Prof. Dr Helmut Harbrecht

Marc Salvadó Benasco

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,

• function 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.

(AS24)

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

(AS24)

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

The student learns advanced topics of analysis such as measure and integration theory and complex analysis.

Content:

• Rings, sigma-algebras, contents, and measures,

• measurable functions and Lebesgue integration, L^p-spaces,

• convergence theorems (Beppo-Levi theorem, Fatou’s lemma, Lebesgue’s dominated convergence theorem),

• analysis in the complex plane (holomorphic functions, power series expansions, integration along curves),

• Cauchy’s integral theorem and formula,

• maximum principle,

• isolated singularities and meromorphic functions,

• Laurent series expansions,

• residue calculus.

Folgt

ECTS6

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).

(AS24)

Prof. Dr Antti Knowles

Dr Volodymyr Fomichov

ECTS6

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 vari-ous 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 ac-curacy, 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.

(AS24)

Dr Marco Favino

PhD. Hardik Kothari

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

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

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 4 and 5

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 6 and 7

DescriptionReading recommendations

Elementary number theory: prime factorisation of integers; congruences and modular arithmetic; applications to cryptography (RSA algorithm). Quadratic residues, Legendre symbol, quadratic reciprocity (via Gauss and Eisenstein’s lemmas).

Algebraic number theory: Gaussian integers, applications to sums of two squares. Number fields, algebraic integers; unique factorisation of ideals. Finiteness of the ideal class group (statement).

P-adic number theory: p-adic metric, completion; Hensel’s lemma. Local-global principles.

• Lecture script available from Moodle

• Textbooks:

o Davenport, The Higher Arithmetic (available free as e-book)

o Coutinho, The Mathematics of Ciphers

o Stewart & Tall, Algebraic Number Theory and Fermat’s Last Theorem

o Gouvea, P-adic Numbers: An Introduction

ECTS8

Semesterautumn semester

Target AudienceStudents from semesters 4 and 5

DescriptionReading recommendations

Holomorphic functions

Cauchy’s integral Theorem

Properties of analytic functions

Singularities

Residues

Complements

Banach space techniques

Hilbert spaces

Fourier series

Sobolev spaces

Operators in Hilbert spaces

Applicationss

- W. Rudin : Real and complex analysis

- J. Marsden, M. Hoffman : Basic complex analysis

- H. Cartan : Elementary theory of analytic functions of one or several complex variables

- M. Chipot : Elliptic equations : an introductory course

(SA23)

ECTS10

Semesterautumn semester

Target AudienceStudents from semesters 6 and 7

DescriptionReading recommendations

• OPTIMIZATION

o Optimization by hand: Finding maxima/minima of multivariate optimization problems, including

▪ unconstrained problems

▪ constrained problems with Lagrange multipliers

▪ KKT conditions

o Numerical optimization algorithms

▪ Gradient Descent

▪ Stochastic Gradient Descent

▪ Stochastic Gradient Descent with momentum

▪ Implementation of the above

▪ Mathematical analysis of the above

▪ (If time permits) Newton’s method

o Significance of convexity/concavity

o (If time permits) Linear programming

• MACHINE LEARNING

o Basic framework of supervised learning

▪ Classification vs regression tasks

▪ Hypotheses classes

▪ Generalization

▪ Train error, validation error, test error, generalization error

▪ Overfitting and underfitting, regularization

o Linear regression and logistic regression

▪ (If time permits) Mathematical analysis of generalization error for these

o Dimensionality reduction, Principal Component Analysis (PCA)

o (If time permits) Bayesian machine learning and graphical models

o Kernel regression

o Neural networks

▪ Different architectures: feedforward, convolutional (CNN)

▪ Training algorithms

▪ (If time permits) Scaling limits and mathematical analysis thereof

o (If time permits) Generative models, (Large) Language models, transformers.

Mandatory resources:

The lecture notes for the course.

A tablet for writing (for solving exercises, collaborating live during interactive sessions, and for the oral exam).

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

PDFs of all the books listed below can be downloaded for free by following the corresponding links.

Hardt, Recht - Patterns, predictions, and actions - mlstory.org

Bishop - Pattern Recognition and Machine Learning - www.microsoft.com/en-us/research/publication/patternrecognition-machine-learning/

Deiseinroth, Faisal, Ong - Mathematics for Machine Learning - mml-book.github.io

Mohri, Rostamizadeh, Talwalkar - Foundations of Machine Learning - cs.nyu.edu/~mohri/mlbook/

Shwartz, Ben-David - Understanding Machine Learning: From Theory to Algorithms - www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/

James, Witten, Hastie, Tibshirani - An Introduction to Statistical Learning - www.statlearning.com

Hasie, Tibshirani, Friedman - The Elements of Statistical Learning - hastie.su.domains/ElemStatLearn/

The last book is a comprehensive reference work, while the others are textbooks.

(AS24)

ECTS14

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.

ECTS6

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 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.

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.