Courses

The course part of this programme consists of;

  • 15 EC compulsory courses
  • 45 EC primary electives
  • 15 EC secondary electives

This Master's is part of the national Mastermath Programme, which offers you an exceptional list of mathematical courses to choose from. You can choose both local courses (at Utrecht University), as well as courses offered by another Dutch University. The local courses are listed below. For an overview of all courses offered in the Mastermath programme: please visit the Mastermath website.

Local courses

Mathematics Colloquium (compulsory)

There is no content available for this course.

Mathematics for Industry (compulsory)

There is no content available for this course.

Orientation on Mathematical Research (compulsory)

There is no content available for this course.

Categories and Topology

More information on: https://sites.google.com/site/gijsheuts/teaching/categories-and-topology

Day, time and place. Will be organized online Mondays 15.15-17.00.

Constructions of topological spaces from categories play a central role in many parts of mathematics, notably in algebraic K-theory and in algebraic topology itself. These constructions make use of the theory of simplicial sets, which form "combinatorial models" for topological spaces.

Aim. The purpose of this course is to provide an introduction to the theory of simplicial sets together with its relation to the homotopy theory of topological spaces, and to study some of the constructions used in higher algebraic K-theory.

Contents. The topics we intend to cover include the following:

  • Basics of simplicial sets
  • Geometric realization of simplicial sets
  • Classifying spaces of groups and of categories
  • Kan complexes and Kan fibrations
  • Homotopy (co)limits
  • Simplicial sets and algebraic K-theory (after Quillen and Waldhausen)

Format and prerequisites. The course will be mostly self-contained, requiring not much more than basic knowledge of topological spaces. Some experience with algebraic topology, for example as in the course Algebraic Topology I, will be useful, although not strictly necessary.

Introduction to Complex Systems

There is no content available for this course.

Seminar Mathematical Epidemiology

Day, time and place. Mondays 09.00-10.45. On campus, with an online alternative.

Contents. Infectious diseases are world wide a major problem in the human and the veterinarian sector. To acquire the disease, one should have ‘contact’ with an individual or animal who is infectious. The risk for acquisition, therefore, depends on the infection status of other individuals. This dependence leads to non-linear mathematical models. In this seminar we will study several aspects on these models to understand the spread, to predict the effect of interventions and to estimate model parameters.

Material. Mathematical Tools for Understanding Infectious Disease Dynamics (http://press.princeton.edu/titles/9916.html) by Odo Diekmann, Hans Heesterbeek and Tom Britton.

Format. The participants, in turns, will study a part of the book and give a (constructive critical) presentation for the other participants who should have read that part as well. After each presentation there will be a discussion. All students are expected to contribute to this discussion by sending ideas/questions/suggestions about the presented material the day before the presentation to the teacher.

Aim. During this course students learn how to construct, analyze, interpret and present mathematical models on the spread of infectious diseases.

Prerequisites. This course can be taken by students with a background in mathematics, as well as by students with a background in infectious disease epidemiology (medicine/veterinary sciences). For the mathematics students we require a basic knowledge of analysis, differential equations and probability theory. The non-mathematics students should not fear mathematical formulas, but we try to let them present the less technical parts of the course material.

Evaluation. Grades will be based on, at least two, presentations (60%), an oral exam (20%) and participation in the discussions (20%).

Learning goals and evaluation matrix. presentations 60%oral exam 20%in class participation 20%has in-depth knowledge of several important models on the spread of infectious diseasesxxxunderstands the biological implications of mathematical assumptions and how biological assumptions translate into mathematicsxxxis able to analyze both basic deterministic and stochastic models on the spread of infectious diseases xxhas a basic understanding of how data on incidence of infectious disease cases can be used to estimate epidemiological parameters in various settings xxis able to summarize literature on a specific topic and present results to fellow studentsx

Seminar Machine Learning

There is no content available for this course.

Seminar Logic

There is no content available for this course.

Seminar Algebraic Topology: Spectral Sequences

There is no content available for this course.

Seminar Number Theory: Arithmetic Dynamics

There is no content available for this course.

Seminar Delay Differential Equations

Day, time and place. Mondays 15.15-17.00.

In this seminar, we will explore the mathematical theory of infinite dimensional dynamical systems by focusing on a relatively simple yet rich class of dynamical systems described by functional differential equations (FDE). An important class of FDE is given by delay differential equations in which the derivatives of the variable at a given time do not only depend on the variables at that time but also on the variables at earlier times. During the last 50 years, the theory of functional differential equations has been developed extensively and has become a mature field of research. It has become part of the vocabulary of researchers dealing with specific applications such as control theory, distributed networks, epidemiology, physiology, viscoelasticity, and most recently gene regulatory networks. Delay equations are infinite dimensional dynamical systems and this implies that, in contrast to the study of ordinary differential equations, linear algebra has to be replaced by functional analysis and the mathematical analysis cannot use arguments based on local compactness of the state space.
Material.

  • Infinite dimensional dynamical systems: state space approach, semigroups, perturbation theory, linearized stability, invariant manifolds and attractors, small solutions and completeness
  • Differential Delay Equations: basic linear theory, local nonlinear theory, normal forms, bifurcation theory, global theory
  • Numerical bifurcation analysis of delay equations and applications: Computation of eigenvalues, detection of bifurcations, computation of normal form coefficients with the extended DDE-BIFTOOL software, neural networks with delays in the connections

The material will be available in pdf. We will mostly follow chapters from material in progress, research papers and also chapters of the earlier books: Hale and Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York.

Prerequisites. Knowledge of complex analysis, functional analysis and ordinary differential equations is useful, but not necessary, and depending on the background of the participants, the basic notions in these fields can be recalled. The final talks can be surveys of active research areas.
Format and evaluation. The first meetings will be prepared by the organizer. Participants are expected to give at least one seminar talk using a blackboard presentation. They will pose one hand-in exercise to the other seminar participants. The other students have to hand this in at the next lecture. The hand-in exercise should be approved beforehand by the seminar organizers. The speaker is responsible for grading this hand-in exercise. In case of discussions, the seminar organizers decide. It is obligatory to be present at all talks in this seminar (unless force majeure). The final grade for the seminar is based on a grade that the seminar organizers give to your talk (40%), and on your homework grades (60%).
Learning goals. After completion of the course, the student is able to:

  • convert a part of a graduate-level textbook on infinite dimensional dynamical systems into a coherent and comprehensible presentation for fellow students and mathematicians in general
  • choose appropriate means of communicating applied mathematics to fellow students and mathematicians, in written and oral form
  • formulate and correct exercises and projects that have a balance between relevance, interest, and feasibility
  • explain specific topics from the geometric theory of functional differential equations to fellow students, and put them in perspective as far as their relevance to wider mathematics and applications is concerned
  • perform basic numerical analysis of DDEs with DDE-BIFTOOL

Evaluation matrix.
presentations = presentations and following discussionhomework = combined output for homework assignments presentationshomeworkunderstanding the material200effective communication of the material200formulating and correcting homework010homework grades050

Seminar Mathematical Statistical Mechanics

There is no content available for this course.

Seminar Differential Geometry: Characteristic Classes

Day, time and place. Mondays 11.00-12.45.

Cohomology is one of the tools used to tackle the general question regarding the shape of a topological space: if two spaces are homotopy equivalent, they have the same cohomology. Cohomology is an algebra and on an orientable manifold it comes equipped with a perfect pairing. Understanding of this structure leads to better understanding of the underlying space. Besides the algebra structure and perfect pairing of cohomology, every manifold has distinguished cohomology classes which are better understood using the framework of characteristic classes. These are cohomology classes associated to vector bundles over manifolds. These classes often have geometrical interpretation with corresponding geometrical/topological implications. In particular, the characteristic classes of the tangent bundle of a manifold give information about the structure of the manifold itself, for example the Euler class of an orientable manifold counts the number of zeros of vector fields on the manifold. In this course we will develop the basics of cohomology needed to introduce characteristic classes, such as compact support cohomology, Poincare lemma and Mayer–Vietoris sequences, and then introduce these classes from a differentiable view point and see some of their applications.

Material. The main reference for this course and the book we will follow in the seminar is Bott and Tu’s Differential Forms in Algebraic Topology. In some occasions we may refer to other literature. Particularly relevant second source will be Milnor and Stasheff’s Characteristic classes.

Prerequisites. The bachelor course on differentiable manifolds is a pre-requisite and the bachelor course on algebraic topology is highly desirable.

Format. ​The course will be run in seminar style. Each group preparing a lecture will be responsible for proposing exercises, assigning one to be handed in and marking.

Evaluation. The final mark will be determined by the oral presentations (70%), hand-in exercises (15%) and overall participation (15%).

Learning goals. After completion of the course, the student is able to:

  • learn conceptual and computational aspects of cohomology and characteristic classes (Euler, Chern and Pontrjagin classes) and their implications to manifold topology
  • practice presenting mathematics in lecture style to a group of peers

Evaluation matrix. oral presentations 70%hand-in exercises 15%overall participation 15%learn conceptual and computational aspects of cohomology and characteristic classes (Euler, Chern and Pontrjagin classes) and their implications to manifold topologyxxxpractice presenting mathematics in lecture style to a group of peersx

Laboratory Class Scientific Computing

There is no content available for this course.

Higher Category Theory

There is no content available for this course.

High-Dimensional Probability Theory with its Applications in Data Analysis

There is no content available for this course.

Seminar Random Graphs

There is no content available for this course.

Seminar Bifurcation Theory

The course description will be published later.

Seminar Ergodic Theory

The roots of ergodic theory go back to Boltzmann's ergodic hypothesis concerning the equality of the time mean and the space mean of molecules in a gas, i.e., the long term time average along a single trajectory should equal the average over all trajectories. The hypothesis was quickly shown to be incorrect, and the concept of ergodicity (`weak average independence') was introduced to give necessary and sufficient conditions for the equality of these averages. Nowadays, ergodic theory is known as the probabilistic (or measurable) study of the average behavior of ergodic systems, i.e., systems evolving in time that are in equilibrium and ergodic. The evolution is represented by the repeated application of a single map (in case of discrete time), and by repeated applications of two (or more) commuting maps in case of `higher dimensional discrete time'. The first major contribution in ergodic theory is the generalization of the strong law of large numbers to stationary and ergodic processes (seen as sequences of measurements on your system). This is known as the Birkhoff ergodic theorem. The second contribution is the introduction of entropy to ergodic theory by Kolmogorov. This notion was borrowed from the notion of entropy in information theory defined by Shannon. Roughly speaking, entropy is a measure of randomness of the system, or the average information acquired under a single application of the underlying map. Entropy can be used to decide whether two ergodic systems are not `the same' (not isomorphic).

Contents. In this seminar the following concepts will be represented:

  • The notion of measure preserving (stationarity), several interpretations, examples and the Poincare Recurrence Theorem
  • The notion of ergodicity (which is a weak notion of independence), and its characterization
  • Ergodic Theorems (generalizations of the Strong Law of Large Numbers) such as Birkhoff and Von Neumann’s Ergodic Theorems
  • Some consequences of the Ergodic Theorems and the notions of weakly and strongly mixing, and isomorphism between measure preserving systems
  • Notion of entropy, the Shannon-Mcmillan Breiman Theorem, and Lochs Theorem
  • Perron-Frobenius Operator and the existence of finite and infinite invariant
  • Construction of invariant and ergodic measures for continuous transformations, unique ergodicity, uniform distribution and Benford’s Law
  • Topological dynamics, topological entropy and the variational principle
  • Introduction to infinite ergodic theory

Material. The presentations will be based on the book A First Course in Ergodic Theory by Karma Dajani and Charlene Kalle, ISBN 9780367226206, published 5 July 2021 by Chapman and Hall/CRC.

Recommended literature.

Prerequisites. Basic knowledge of measure theory and an exposure to functional analysis.

Format. This seminar is aimed at master students with a background in measure theory, and some elementary knowledge of undergraduate functional analysis is useful in some parts. The maximum number of (active) participants of the seminar is 10.

Schedule. Wednesdays 13.15-15.00.

Learning goals.

  • convert material from part of graduate-level textbook or a scientific paper into a coherent and comprehensible presentation for fellow students and mathematicians in general
  • choose appropriate means of communicating theoretical mathematics to fellow students and mathematicians, in written and oral form
  • formulate and correct exercises that keep a balance between relevance, interest, and feasibility
  • explain specific topics from the content list of the seminar to fellow students, and put them in perspective as far as their relevance to wider mathematics is concerned

Evaluation. Participants are expected to give two seminar talks (i.e., two times 2 x 45 minutes presentation), possibly more depending on the number of participants. They will study the material beforehand, hold a blackboard presentation about it. They will pose a hand-in exercise to the other seminar participants (to be handed in at the next lecture), that should be approved beforehand by the seminar organizers. The speaker is responsible for grading this hand-in exercise. In case of discussion about the solutions, the seminar organizers decide. It is obligatory to be present at all talks in this seminar (unless force majeure). The final grade for the seminar is based on the average grade that the seminar organizers give to your talks and handouts (60%), and on your homework grades (40%).
PresentationsHomeworkUnderstanding the material300Effective communication300Formulating and correcting homework010Homework grades030

Seminar History of Mathematics

Seminar on Neural Networks and Finance

The course description will be published later.

Computational Finance

The course description will be published later.

Invariant Theory

The course description will be published later.

Complex Manifolds

The course description will be published later.