Syllabus

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

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 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.

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, 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

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

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.

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

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.

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.

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

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: 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.

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.

• 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

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.

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.

• 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.

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.