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More information# Tracks

## Track title: Choosing a track allows you to tailor the programme to your personal interests

Track description:

The programme offers 6 specialized tracks you can choose from:

__Algebraic Geometry and Number Theory____Differential Geometry, Topology and Lie Theory____Logic____Differential Equations and Dynamical Systems____Applied Mathematics, Complex Systems and Scientific Computing____Probability and Statistics__

You can also choose to do a research project on History of Mathematics.

## Track title: Algebraic Geometry and Number Theory

Track description:

These areas have their roots in classical Greek mathematics, and thus belong to the oldest branches of mathematics. As you pursue this track, you’ll examine how these ancient fields have evolved into exciting areas in modern mathematics.

Imagine a curve in the plane given by a polynomial equation in two variables. We know these as the familiar curves of straight lines, conics, and higher degree classical curves. What happens if you need to plot a curve with coordinates that involve complex numbers? The curve becomes a real two-dimensional object, and you can demonstrate its topology. What happens if you ask for points with integer coordinates? You are now solving Diophantine equations. You might even ask for points with coordinates that are integers modulo a prime.

All these viewpoints open up completely different directions in the field, referred to as arithmetic-algebraic geometry.

You’ll use tools and methods ranging from topology to number theory and algebra as you pursue studies in this track.

The geometric part, or algebraic geometry, is an essential tool in modern mathematical physics. The art of point counting modulo a prime is used extensively in modern coding theory and cryptography. The study of Diophantine equations involves number theory and requires tools from algebra, analysis and geometry. Recent spectacular developments in Diophantine equations owe their existence to the discovery of parallels with the world of geometry.

Your studies in algebraic geometry and number theory will provide you with diverse research tools that have practical applications. The subject of your Master’s thesis will depend on your interests in the field. You’ll have the option to choose from topics ranging from analytic number theory to the theory of modular forms, or from solving Diophantine equations that lie at the intersection of logic and computability to the interface of algebraic geometry and theoretical physics.

### Prerequisites

You need a background in groups and rings. For number theory you also need Galois theory and an introductory course in number theory. For algebraic geometry you need complex analysis and topology and familiarity with Galois theory and differentiable manifolds is recommended. For either direction it is sometimes helpful to know representation theory of groups.

### Courses

Typical courses in this Master’s track include: Algebraic Geometry, Algebraic Number Theory, p-adic Numbers, Elliptic Curves, Diophantine Equations, Modular Forms, Analytic Number Theory, and Riemann Surfaces. Many of these courses are currently offered on a regular basis through the national Mastermath programme. More advanced courses in the field include topics on Galois theory/class field theory, transcendence theory, and scheme theory.

Your tutor will help you select courses based on your interests and desired topic of research.

*For more information about this track, please contact Prof. Dr. Gunther Cornelissen*

## Track title: Differential Geometry, Topology, and Lie Theory

Track description:

Differential Geometry, Topology, and Lie Theory is concerned with the study of spaces such as curves, surfaces (like the sphere, the Möbius band and the torus) and higher dimensional versions of them.

Topology studies the properties preserved under continuous deformations of objects like stretching (but not tearing or gluing). When studying these topics from an algebraic point of view by attaching algebraic invariants to them (such as the number of holes of a surface), you enter the field of algebraic topology. In contrast, differential topology involves a class of geometric objects called manifolds, on which we can perform additional types of analysis. Your studies in this track will allow you to evaluate a manifold in terms of differentiable functions, vector fields, and other related tools. You’ll also learn to analyze intricate manifold variations that result from imposing additional structures (differential geometry), notions of length (Riemannian geometry), notions of holomorphic function (complex and Kähler geometry), structures arising from classical mechanics (symplectic and Poisson geometry), or groups of symmetries (lie groups).

Most of the physical theories such as classical mechanics and general relativity acquire their most natural and insightful formulation in such terms. This may not be surprising because many of these concepts relate to the field of physics. However, in many cases, a physicist’s intuition was decisive in solving some of the most fundamental problems in geometry and topology. Your studies in this track will allow you to follow in their footsteps and to broaden understanding or break new ground in the field.

Lie groups are equipped with the structure of a differentiable manifold for which the group operation is smooth. They appear in many situations in mathematics and physics, where continuous symmetries play a role. In such situations one is often interested in Fourier (or harmonic) analysis.The non-ommutative nature of Lie groups requires the description of harmonic analysis in terms of (often infinite dimensional) representation theory. The rich geometric structure of Lie groups allows one to develop a theory which at the same time is amazingly general and surprisingly concrete. Subjects of current research are: Plancherel en Paley-Wiener theorems for symmetric spaces, parameter dependence of representations, asymptotic behaviour of matrix cofficients, Radon transformation, cusp forms for symmetric spaces, symplectic geometry and convexity theorems.

### Prerequisites

The usual prerequisites for entering the Master's programme plus basic knowledge of:

- topology, for example as taught in our bachelor courses ”Inleiding Topologie” (level 2);
- group theory, for example as taught in the first half of our bachelor course ”Groepentheorie” (level 2);
- differentiable manifolds, as taught for instance in the first part of our level 3 course ”Differentieerbare varieteiten” (level 3);
- basic theory of Banach and Hilbert spaces
- analysis of several variables: implicit function theorem, submanifolds of R^n; substitution theorem for integration, a version of Stokes theorem;
- some knowledge on fundamental groups (as taught for instance in the first part of our level 3 course ”Topologie en Meetkunde”) is recommended.

### Courses

Your tutor will assist you with choosing courses that reflect your interests. Along with your research and Master’s thesis, you’ll select from amongst the following options:

- at least two courses in
**algebraic topology,**such as Homotopy Theory, Homological Algebra, Sheaf Theory, Knot Theory, Quantum Groups and Knot theory, Category Theory, Simplicial Sets, or K-Theory and Vector Bundles; - at least two courses in
**differential geometry**, such as Analysis on Manifolds, Symplectic Geometry, Foliation Theory, Riemannian Geometry, Lie Groups, Semisimple Lie Algebras, or Differential Topology; - at least one course in
**algebraic geometry**, such as Algebraic Geometry, Riemann Surfaces, or Elliptic Curves; and - at least one course in
**pure analysis**, such as Functional Analysis, Distribution Theory, Fourier Analysis and Distributions, or Dynamical Systems.

The list above reflects recently given courses and is not exhaustive. You’ll have the opportunity to discuss your course load with your tutor and potentially select other options that suit your interests. You’ll also choose 3-4 optional courses to further tailor your coursework. Your tutor will help you develop a course load that will assist you with writing your Master’s thesis in geometry and topology.

*For more information about this track, please contact Prof. Dr Marius Crainic.*

## Track title: Differential Equations and Dynamical Systems

Track description:

We use mathematical models to understand how phenomena and mechanisms studied in various scientific disciplines are related to one another. To analyze such models, mathematical methods and computer tools are applied. In this track you will learn how to use, justify, and develop such methods and tools.

Often models take the form of differential equations (ordinary or partial or delay/functional). When you study how the state of a system changes in time, it's useful to consider the dynamical system (which is generated by the differential equation), and to study how its behavior depends on internal and external parameters. So methods to study the qualitative as well as the quantitative behavior of finite- and infinite-dimensional dynamical systems form the core of the specialisation.

These methods include:

- asymptotic analysis (perturbation theory and averaging);
- bifurcation analysis (topological equivalence, normal forms, and invariant manifolds);
- functional analysis (semigroups of operators, dual spaces, and fixed point theorems); and
- numerical analysis (continuation techniques and computation of normal forms).

The applications range over all natural sciences (as well as economics), but in this track we emphasize physics, engineering, and biology (in particular neuroscience, population dynamics, and epidemiology).

### Prerequisites

To specialise in Differential Equations and Dynamical Systems, one needs a background in differential equations, as well as in complex and functional analysis.

### Courses

For this track, you must take the Mastermath course on *Dynamical Systems* and one or more courses from the following list (additional courses may be possible in agreement with your tutor):

- Functional Analysis
- Partial Differential Equations
- Introduction to Numerical Bifurcation Analysis of ODEs and Maps
- Numerical Bifurcation Analysis of Large-Scale Systems
- Mathematical Neuroscience

Depending on your interests, you can also opt to take additional coursework in the areas of analysis and stochastics

*For more information about this track, please contact Prof. dr. Yuri Kuznetsov*

## Track title: Logic

Track description:

Studying logic allows you to ask questions about the heart of mathematical activity, such as: what is a proof? What is an algorithm? What are the limitations of provability? What is truth? Mathematicians such as Hilbert, Gödel, Gentzen, Herbrand, Turing, and Tarski posed and answered many of these and similar questions in the 1930’s. Modern logic goes beyond these fundamental issues, allowing you to study formal systems and their interpretations in the mathematical world. The field of logic has strong connections to almost every area of pure mathematics, such as number theory, algebraic geometry, and topology, and it also has great significance in the field of computer science. Most Utrecht University research in logic relates to topos theory and proof theory. However, you’ll be able to perform a research project in whatever area of logic best suits your mathematical interests.

### Prerequisites

The usual prerequisites for entering the master program. Some basic knowledge of mathematical logic is recommended, for example as taught in our bachelor (level 3) course ”Grondslagen” (Foundations). This can be incorporated in a Master Programme (there is an English-language reader).

### Courses

For your specialisation in this programme, you’ll select 9-10 courses as well as undertake a thesis on your chosen subject. You can choose 4-5 courses from the list below; you may be able to study some or all of these subjects on an individual basis in the form of a guided reading course.

In addition to these courses, you’ll select 2-3 courses from a neighbouring track, such as Differential Geometry and Topology or Algebraic Geometry and Number Theory. Along with the topic for your Master’s thesis you’ll select your remaining 2-3 courses based on your interests in the field.

You can select 4-5 courses from the following list. These courses are taught with some regularity either at Utrecht University or through the Mastermath programme:

- Model Theory
- Proof Theory
- Computability Theory
- Intuitionism
- Category Theory
- Topos Theory
- Peano Arithmetic and Gödel Incompleteness
- Set Theory
- Type Theory and §-Calculus
- Seminar Logic

* **For more information about this track, please contact Dr Jaap van Oosten.*

## Track title: Applied Mathematics, Complex Systems, and Scientific Computing

Track description:

This track focuses on applications of mathematics in modern society ranging from Complex systems and Bioinformatics, to Statistics and Scientific Computing.

The course programme provides you with a broad set of skills, such as analytical thinking, mathematical modelling, programming in Python or C++, and using high-level toolboxes such as Matlab and R. The master thesis project (45 EC) in this specialisation may be carried out as an internship in industry, a government research institution, or a research group from another department at Utrecht University where mathematics is applied. It is also possible to choose a topic within the Mathematics department. A special characteristic of the programme is the freedom to choose up to 30 EC of courses in other disciplines, provided mathematics is applicable there.

**Complex Systems:**

Complex systems demonstrate the popular principle that “the whole is greater than the sum of its parts”. More concretely, a complex system is one whose collective behavior cannot readily be deduced by a reductive study of its individual components: Stock markets cannot be predicted by studying individual investors, complex thought cannot be easily understood through the electrochemical processes of neurons, and fluid turbulence is not an obvious consequence of the molecular structure of water.

The science of complex systems is a multidisciplinary effort that draws on mathematically formulated models from a variety of fields. A university-wide focus area “Foundations of Complex Systems” strives to coordinate research efforts at the Utrecht University on this front.

The mathematical foundations of complex systems are far from mature. Inspired by applications from outside the traditional realm of applied mathematics, the study of complex systems may well lead to truly new forms of mathematics. Additionally, there is a growing demand for mathematical scientists trained to build and analyze models of complex systems in economics, social sciences, biology and medicine, as well as natural sciences, geosciences and ecology. In the MSc specialisation “Complex Systems” you will combine mathematical theory in dynamical systems, networks, stochastics and computation, with applications in one of the above disciplines. In particular, your Master's research will be jointly supervised by scientists from at least two disciplines.

**Scientific Computing**

Scientific Computing is a rapidly growing field, providing mathematical methods and software for computer simulations in a wide variety of application areas, from particle simulations for the study of protein folding to mesh calculations in climate change prediction. The area is highly interdisciplinary, bringing together methods from numerical analysis, high-performance computing, and application fields. The scientific computing specialisation focuses on analysing the large-scale systems that are central in various fields of science and in many real-world applications. Students willl learn the mathematical tools necessary to tackle these problems in an efficient manner and they will be able to provide generic solutions and apply these to different application areas. They will learn to develop mathematical software and to use modern high-performance computers, such as massively parallel supercomputers, PC clusters, multicore PCs, or machines based on Graphics Processing Units (GPUs). Expertise in scientific computing is in high demand, and graduates will be able to pursue careers in research institutions or in industry or management.

Other topics in the specialisation Applied Mathematics, Complex Systems, and Scientific Computing:

- Bioinformatics
- Data science
- Imaging
- Machine learning
- Parallel algorithms
- Statistics

### Courses

Mathematics courses that fit this track include

- Continous Optimization
- Discrete Optimization
- Introduction to Complex Systems
- Numerical Linear Algebra
- Parallel Algorithms
- Seminar Mathematical Epidemiology
- Advanced Linear Programming
- Inverse problems in Imaging
- Numerical Bifurcation Analysis of Large-scale Systems
- Numerical Methods for Time Dependent PDEs
- Seminar Applications of Mathematics in Radiation Research
- Seminar Machine Learning

Courses that fit the 30 EC in other disciplines include:

- Computing Science courses: like Data Mining, Crowd Simulation or Algorithms and Networks
- Physics courses: like Current Themes in Climate Change or Modelling & Simulation
- Biology courses: like Bioinformatics and Evolutionary Genomics or Introduction to R for Life Sciences

*For more information about this track, please contact Dr. Ivan Kryven.*

## Track title: Probability and Statistics

Track description:

**This track includes the national programme Stochastics and Financial Mathematics (SFM)**

Studying random phenomena is an exciting component of modern scientific research. A stochastic framework is often the only mathematical structure that allow us to perform an efficient analysis of the complex phenomena under scrutiny. This explains the pervasive use of probabilistic descriptions in almost all fields of knowledge: physics, biology, economics, medicine, social sciences...

Stochastic techniques on the other hand, have emerged as surprisingly effective tools in technological applications involving delicate calculations. Examples are important algorithms for image and sound processing, information compression and a growing number of simulation techniques. Beyond their applied side, however, probability and statistics are also fully developed areas of mainstream mathematics, subject to rapid and exciting development, with a strong presence in most universities and research institutions throughout the world.

The specialisation in probability and statistics provides you with a balanced programme combining core foundational knowledge with a wide selection of optional courses—many offered within the multi-university SFM program. This structure allow you to create a personalized course load tailored to your interests in pure and applied stochastics.

### Prerequisites

Students are expected to have completed course work introducing them to intermediate-level notions on probability and statistics.

These notions include:

- For probability: Borel Cantelli lemmas, conditional expectation for discrete and continuous random variables, law of large numbers and central limit theorems.
- For statistics: empirical distributions, estimation, sampling, hypothesis testing. Bachelor courses at our department can act as remedial courses.

### Courses

The course Measure Theoretic Probability is a prerequisite for a lot of other courses in the track. Besides Measure Theoretic Probability you take at least two of the following five courses:

- Random Walks
- Bayesian Statistics
- Stochastic Processes
- Time Series
- Nonparametric Statistics

You’ll select additional courses from an extensive list of courses (available through Mastermath and the SFM program) in consultation with your tutor, along with choosing the topic of your thesis. The offered courses will allow you to study both applied subjects and topics on foundational aspects. You’ll also have the opportunity to take courses in other departments (physics, medicine, sociology, economics, etc.) that relate to your interests.

*For more information about this track, please contact dr. Cristian Spitoni.*

## Track title: History of Mathematics

Track description:

Students may choose to do their Master's project in the History of Mathematics. This may be interesting for students who want to pursue a career in mathematics education or history of science. The theme of the M.Sc. thesis should have a substantial mathematical content in the area of the specialisation. In practice, the subject of a M.Sc. thesis in History of Mathematics is often one in the 17th century or later, although exceptions are possible. M.Sc. theses in history of mathematics may be supervised by two staff members, including one member who is specialised in the field of mathematics related to the thesis.

The student should have basic knowledge of history of mathematics, for example as taught in our bachelor course History of Mathematics. Courses in History of Science and in Concrete Geometry are an advantage.

The student should take the Master's course History of Classroom Mathematics or an equivalent Master's course, subject to availability. Depending on the subject of the Master's thesis, Mathematics courses may be necessary. For example, a student who writes a Master's thesis in the history of elliptic curves will also take Mathematics courses in this field. The programme is subject to the approval of the tutor.

Students pursuing a Master's degree in History of Science or Science Teacher Education may also choose to do their Master's project in the History of Mathematics.

*For more information, please contact dr. Steven Wepster.*