### Specialisations

The system of specialisations is intended to optimise both freedom and structure in the Master's programme. It is flexible enough to leave considerable freedom to a student with wide interests, while at the same time providing guidance to students with a more focused interest. Mathematical Sciences has eight specialisations. Changes of specialisation during your Master are also allowed.

Fundamental Mathematics tracks

## Algebraic geometry and number theory

These are the areas which have their roots in classical Greek mathematics and thus belong to the oldest branches of Mathematics. They are more alive than ever and have evolved in a spectacular way into an exciting area in modern Mathematics.
As an example, imagine a curve in the plane given by a polynomial equation in two variables. These are the familiar curves like straight lines, conics and other higher degree classical curves. But what if we ask for points on such a curve with coordinates in the complex numbers? Then the curve is in fact a real two-dimensional object and we can ask for its topology. Or what happens if we ask for points with integer coordinates? We are then in the business of solving Diophantine equations. As a third alternative we can ask for points which have coordinates which are integers modulo a prime.
All these viewpoints open up completely different directions in the field which is now known as arithmetic-algebraic geometry. The methods used range from topology to number theory and algebra in a fertile mix.
The geometric part, algebraic geometry, has become 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 is a part of number theory and requires tools from algebra, analysis and, last but not least, geometry. Several recent spectacular developments in Diophantine equations now owe their existence to the discovery of parallels with the world of geometry.
The tools of the trade in this specialization can be very diverse, and as a result the choice of the courses taken, and a Master’s thesis subject depends very much upon the individual student. For instance, the subject could belong to analytic number theory, the theory of modular forms, be about solving Diophantine equations, lie in an area that makes contact with logic and computability or be on 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.

### Requirements

Some typical basic courses in the Master curriculum: Algebraic Geometry, Algebraic Number Theory, p-adic Numbers, Elliptic Curves, Diophantine Equations, Modular Forms, Analytic Number Theory, Riemann Surfaces, many of which are currently offered on a regular basis by the national MasterMath programme. More advanced courses would be in the areas of Galois theory/class field theory, transcendence theory, and scheme theory.
Clearly, some research directions require taking courses in (or familiarity with) adjacent areas.

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

## Differential geometry and topology

Differential Geometry and Topology is concerned with the study of spaces such as curves, surfaces (think of the sphere, the Möbius band and the torus) and higher dimensional versions of them.
Topology studies those properties that are preserved under continuous deformations of objects like stretching (but no tearing or gluing). But when these are studied from an algebraic point of view by attaching algebraic invariants to them (such as the ”number of holes” of a surface) we have entered the field of Algebraic Topology. Differential Topology on the other hand is involved with a class of geometric objects, called manifolds, on which we can do the rudiments of analysis; on such a manifold it makes sense to talk about differentiable functions, vector fields, and the like. The situation becomes much more intricate and interesting if one imposes certain additional structures on a manifold (we then speak of Differential Geometry): that could be a notion of length (Riemannian Geometry), a notion of holomorphic function (Complex and Kähler Geometry), structures arising from Classical Mechanics (Symplectic Geometry, Poisson Geometry), or groups of symmetries (Lie Groups). Most of the major physical theories, such as classical mechanics and general relativity acquire their most natural and insightful formulation in such terms. This is perhaps not so surprising, as many of these notions have a physics ancestry, but what still is surprising that in many a case physicists intuition was decisive in solving some of the most fundamental problems in Geometry and Topology.

### 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 variteiten” (level 3);
• some knowledge on fundamental groups (as taught for instance in the first part of our level 3 course ”Topologie en Meetkunde”) is recommended.

### Requirements

From the 9-10 courses (which, together with the thesis, constitute a student’s program), the student should take:

• at least two from the list of courses in the direction of Algebraic Topology, such as: Homotopy Theory, Homological Algebra, Sheaf Theory, Knot Theory, Quantum groups and Knot theory, Category Theory, Simplicial sets, K-theory and vector bundles;
• at least two from the list of courses in the direction of Differential Geometry, such as: Analysis on Manifolds, Symplectic Geometry, Foliation Theory, Riemannian Geometry, Lie groups, Semisimple Lie Algebras, Differential Topology;
• at least one from the courses in the direction of Algebraic Geometry, such as Algebraic Geometry, Riemann surfaces, Elliptic Curves;
• at least one from the courses in the direction of Pure Analysis, such as: Functional Analysis, Distribution Theory, Fourier Analysis and Distributions, Dynamical Systems.

(The lists are based on courses that were recently given and are not meant to be exhaustive.) In practice, this will leave 3 or 4 courses for the student to choose freely. Finally, the student should also write a Master's thesis that belongs to ”Geometry and Topology”.

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

Prof. Dr. Marius Crainic

## Logic

Logic starts out asking basic questions at the heart of mathematical activity, such as: what is a proof? What is an algorithm? What are the limitations of provability? What is truth? Many of these and similar questions were posed and answered in the 1930’s by Hilbert, Gödel, Gentzen, Herbrand, Turing and Tarski. Modern logic goes beyond these fundamental issues and studies formal systems and their interpretations in the mathematical world. It has strong connections to almost any area of pure mathematics; in particular number theory, algebraic geometry and topology. But logic is also of great importance in computer science. In Utrecht, research in logic is mostly in topos theory and proof theory. However, you can do a research project in whatever area of logic that suits your mathematical taste.

### 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).

### Requirements

From the 9 or 10 courses together with the thesis that constitute a student’s program, the student should take 4 or 5 courses from the list below; these are courses taught with some regularity, either locally or in the Mastermath programme. These courses can also be taken on an individual basis in the form of a (guided) reading course.
Furthermore, the student should follows 2 or 3 courses in a “neighbouring” (or relevant) track, such as Geometry, Topology, or Algebra and Number Theory.
Finally, the student should also write a master thesis in Logic.
In practice, this will leave 2 or 3 courses for the student to choose freely.
List of Logic courses taught with some regularity:

• Model Theory
• Proof Theory
• Computability Theory
• Intuitionism
• Category Theory
• Topos Theory
• Peano Arithmetic and Gödel Incompleteness
• Set Theory
• Type theory and §-calculus
• Master student seminar in a specialized topic in mathematical logic.

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

Dr. Jaap van Oosten

## Pure analysis

The pure analysis programme focuses on analysis on manifolds, with a strong geometric flavor. On the one hand this encompasses analysis on Lie groups and on the other hand the study of dynamical systems.

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.

Dynamical systems describe the evolution of deterministic behavior. This can be conservative (think of Hamiltonian mechanics) or dissipative. Typical examples come from physics (celestial mechanics, resonant circuit), chemistry (reaction-diffusion equations) biology (predator-prey systems) or economy (market models) --- this short list is far from exhaustive. In mathematical terms they often can be described by a vector field on phase space, preserving the inherent geometric structure. Given a concrete example (model), the aim is to describe as much of the dynamics as possible. This is helped by theoretical considerations --- one can only find what one is looking for. The emphasis in this specialisation is given to bifurcations, explaining how to pass from one robust regime to another.

### Prerequisites

Minimal requirements for a successful start in this programme are a good basic knowledge of the following subjects at the bachelor level:

• theory of series, in particular power series and fourier series;
• ordinary differential equations;
• basic theory of Banach and Hilbert spaces;
• analysis of several variables: implicit function theorem, submanifolds of Rn; substitution theorem for integration, a version of Stokes theorem;
• topology, in particular the notion of compactness;
• groups.

For successful participation it is desirable to also have basic knowledge of differentiable manifolds, complex functions and the Lebesgue integral.

### Requirements

Courses in the Master programme are selected in close contact with a tutor, in order to get tailor made preparation for writing a Master's thesis.

• For specialization towards manifolds and Lie groups, a master course in either Lie groups, or both Lie algebras and differentiable manifolds is required.
• For specialization towards dynamical systems, a master course in dynamical systems is required.

In view of the geometric nature of the programme it is advisable to follow master courses in differential geometry, Riemannian or symplectic. For obtaining a good general background in analysis it is advisable to also follow master level courses in functional analysis, distribution theory, and/or partial differential equations.
The tutor will assist in finding an appropriate specialisation and thesis supervisor. In some cases it may be quite natural to specialise in subjects on the overlap with the programmes in applied analysis, geometry and topology, or mathematical physics.

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

Prof. dr. Erik van den Ban

Mathematical Modeling tracks

## Applied analysis

To understand how phenomena and mechanisms studied in various scientific disciplines are related to one another, one needs mathematical models. To analyze such models, one applies mathematical methods and computer tools.The aim of the ”Applied Analysis” track is to 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 one studies how the state of a system changes in time, it is 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 track.
These methods include:

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

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

### Prerequisites

To specialise in Applied Analysis, one needs a background in differential equations, as well as in complex and functional analysis.

### Requirements

The student should take the MasterMath master course on:

• Dynamical Systems

and at least one and preferably more master courses from the
following list (that can be extended in agreement with the tutor):

• Functional Analysis
• Fourier Analysis and Distribution Theory
• Partial Differential Equations
• Introduction to Numerical Bifurcation Analysis
• Mathematical Biology

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

Additional courses may also be chosen in the areas analysis and of stochastics.

Dr. Yuri Kuznetsov

## 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”  (see website) 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 master specialization “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 masters research will be jointly supervised by scientists from at least two disciplines.

### Requirements

The student must take:

• Introduction to Complex Systems (7.5 EC)

Additionally, at least one of the following mathematics courses (or an equivalent) should be chosen:

• Complex Networks (8 EC)
• Introduction to Numerical Bifurcation Analysis (8 EC)
• Mathematical Biology (8 EC)

Also required: you will follow at least 15 EC from other disciplines (other courses may be acceptable, discuss with your advisor). Some examples:

• Sociological Theory Construction and Model Building (Social Sciences)
• Economic Geography (Geosciences)
• Modelling and Simulation (Physics)
• Simulation of Ocean, Atmosphere and Climate (Physics)
• Computational Biology (Biology)
• Courses at the Utrecht School of Economics

Additional mathematics courses (43 EC) should be selected, in consultation with your tutor, from the local course list or that of Mastermath.

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project. For example:  Seminar stochastics, Seminar differential delay equations.

## Probability, Statistics and Mathematical Modelling

(this includes the national programme Stochastics and Financial Mathematics)

The study of random phenomena is an unavoidable component of modern scientific research. A stochastic framework is often the only mathematical structure allowing an efficient treatment 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 specialization in Probability, Statistics and Mathematical Modelling in the master offered by Utrecht Mathematics Department offers a balanced program combining a basic core of foundational knowledge with a wide selection of optional courses—many offered within the multi-university SFM program— allowing personalized student profiles both 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:

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

### Requirements

The student must take:

• Measure theoretical probability

and (at least) two of the following three courses:

• Stochastic processes
• Time series
• Applied statistics

The students will choose further courses from an extensive list of courses available through Mastermath and the SFM program. They include both applied courses and courses on foundational aspects. Students are also encouraged to take related courses in other departments (physics, medicine, sociology, economics, etc.)

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

## 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.

None

### Requirements

Choose at least 4 courses from:

• Laboratory class Scientific Computing
• Numerical Linear Algebra
• Numerical PDEs: Time-dependent
• Numerical PDEs: Stationary
• Parallel algorithms
• Modelling and Simulation (Physics)
• Simulation of Ocean, Atmosphere, and Climate
• Introduction to Numerical Bifurcation Analysis
• Wavelets and Fourier Transforms (not part of the curriculum anymore)

Students must follow an advanced seminar (at least 7.5 EC) in which they themselves have to give oral presentations. This seminar can also be followed while the student is working on the research project.

Prof. dr. Rob Bisseling

## Applied Mathematics

The track Applied Mathematics focuses on applications of mathematics in modern society ranging from Bioinformatics, Complex systems, to Statistics.
The course programme provides the students with a broad set of skills, such as analystical thinking, mathematical modelling, programming in Python or C++, and using high-level toolboxes such as Matlab and R.

The master thesis project (45 ECTS) in this track 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 30 ECTS of courses in other disciplines, provided mathematics is applicable there.

Topics:
- Bioinformatics
- Complex systems
- Data science
- Imaging
- Machine learning
- Parallel algorithms
- Scientific computing
- Statistics

## History of mathematics

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 advisor.

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.