# Specialisations

### 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 six specialisations. Changes of specialisation during your master's are allowed.

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 specialisation 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 courses in the master's curriculum:

- Algebraic Geometry 1
- Algebraic Geometry 2
- Algebraic Number Theory
- Commutative Algebra
- p-adic Numbers
- Elliptic Curves
- Diophantine Approximation
- Modular Forms
- Analytic Number Theory
- Riemann Surfaces
- Advanced Algebraic Geometry
- Algebraic Methods in Combinatorics
- Invariant Theory
- Algebraic Curves
- Seminar Number Theory

Many of these courses are currently offered on a regular basis by the national Mastermath programme. Clearly, some research directions require taking courses in (or familiarity with) adjacent areas.

### For more information about this specialisation, please contact Prof. Dr. Gunther Cornelissen

Differential Geometry, Topology, and Lie Theory 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.

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-commutative 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 coefficients, 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 our bachelor course “Inleiding groepen en ringen” (level 1);
- differentiable manifolds, as taught for instance in the first part of our level 3 course “Differentieerbare variëteiten” (level 3);
- 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;
- 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 around 8 elective 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: Algebraic Topology 1 and 2, Category Theory, Topology of Geometric Structures, Seminar Algebraic Topology;
- at least two from the list of courses in the direction of Differential Geometry, such as: Differential Geometry, Symplectic Geometry, Lie Groups, Lie Algebras, Poisson Geometry, Complex Manifolds, Seminar Differential Geometry;
- at least one from the courses in the direction of Algebraic Geometry, such as Algebraic Geometry 1 and 2, Riemann Surfaces, Elliptic Curves;
- at least one from the courses in the direction of Pure Analysis, such as: Functional Analysis, Partial Differential Equations, Dynamical Systems, Operator Algebras.

The lists are based on courses that were recently given and are not meant to be exhaustive. In practice, this will leave around 2 courses for the student to choose freely. Finally, the student should also write a master's thesis that belongs to “Differential Geometry, Topology, and Lie Theory”.

### For more information about this specialisation, please contact Prof. Dr. Marius Crainic

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's programme (we work from the booklet “Sets, Models and Proofs”).

### Study programme

In the master Mathematical Sciences, you select 9-10 courses and you write a master thesis on your chosen subject. When you choose the logic track, you naturally select some courses in logic; but we recommend that apart from logic, you choose a second focus area within mathematics, for example number theory, topology or algebraic geometry. Suppose you select 4 logic courses (see below) and 3 courses in mathematical topic X (leaving some courses for side interests you may have); then you have a sound basis for doing a research project on the interaction of logic with topic X.

### Courses

Since logic is a very small part of mathematics, there generally is not enough staff to give the courses needed; apart from the bachelor-level introductory course Foundations, there is every year a seminar in logic for master's students. Then there are courses in the Mastermath programme: at the moment these are

- Set theory
- Category Theory
- Topos Theory

Every other year there is also a course in Complexity theory. Then students can enroll as “bijvakstudenten” and follow courses at other universities. You should be aware that these courses need approval of the Board of Examiners; the courses Model Theory and Proof Theory, given yearly at Amsterdam University, are usually approved and recommended.

In the logic seminar, we have had the following topics in recent times:

- Boolean-valued models
- Intuitionism
- Constructible sets
- Hilbert's 10th Problem
- Ultracategories
- Algebraic set theory
- Stability Theory

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

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 ordinary and partial differential equations (including existence and classification of solutions, and methods to compute them), as well as in complex and functional analysis (including the implicit function theorem and contraction mapping principle).

### Requirements

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
- Calculus of Variations
- Introduction to Numerical Bifurcation Analysis of ODEs and Maps
- Numerical Bifurcation Analysis of Large-Scale Systems
- Mathematical Neuroscience
- Mathematical Biology
- Group Theory and Dynamical Systems
- Ergodic Theory

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

(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 specialisation in Probability and Statistics 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:

- 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.

### 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.

### For more information about this specialisation, please contact Dr. Cristian Spitoni

Applied mathematics is the intersection point where all branches of fundamental mathematics meet together to bloom. Here you will find complex problems that can only be addressed by combining several seemingly distant mathematical tools. Think number theory combined with analysis to find numerical approximations for the solutions of partial differential equations, or graph theory combined with measure theory to analyze complex dynamical systems, or even functional analysis and discrete mathematics to solve ill-posed problems. Consider diving into this field not only because some of the hardest mathematics problems require the applied mathematics lens, but also because some of the largest scientific challenges in the real world can be tamed only when viewed from the applied mathematics perspective. Whether it is climate physics, neuroscience, bioinformatics, space engineering, data science, machine learning or prediction of financial markets – it is the applied mathematicians who create the analytical infrastructure for the new science of tomorrow. In short, if you would like to change the world to be a better place, applied mathematics is a good place to start.

While perpetually keeping track of new developments of this field, we have experience of teaching this programme for almost 20 years. The programme consists of local courses taught at Utrecht and a wide selection of electives from the national programme. The programme provides you with a broad set of skills, such as analytical thinking, advanced mathematical modeling, 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 at the Mathematical Institute or as an internship at industry, a government research institution, or a research group from another department at Utrecht University where mathematics is applied. A special characteristic of the programme is the freedom to choose up to 30 EC of courses in adjacent disciplines, such as physics, computer science, biology and others, provided mathematics is applicable there. The programme gives a perfect preparation for entering the job market as a research mathematician in industry or academia.

### Complex Systems

Complex systems demonstrate the popular principle that “the whole is greater than the sum of its parts”. A complex system is a mathematical model composed of multiple components whose collective behaviour cannot readily be deduced by studying the individual components: Stock markets cannot be predicted by studying individual investors, brain activity cannot be easily understood through the electrochemical processes of neurons taken apart, and fluid turbulence is not an obvious consequence of the molecular structure of water. Mathematicians tackle such problems using probabilistic modelling, measure theoretic principles, numerical methods and other tools. This is an actively developing area of research, and there is a growing demand for mathematical scientists trained to build and analyse models of complex systems in economics, social sciences, biology and medicine, as well as natural sciences, clime and earth sciences and ecology. Inspired by such new applications, the study of complex systems may well lead to truly new forms of mathematics.

In “Complex Systems” you will combine mathematical theory in dynamical systems, networks, stochastics and computation, with applications in one of the above disciplines. Your Master's research will be jointly supervised by scientists from at least two disciplines. The department of mathematics coordinates the research in this area with the interdisciplinary Complex Systems Studies Centre at Utrecht, and you will be invited to attend the research talks and workshops organized at the centre.

### 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 will 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).

### Prerequisites

Apart from standard mandatory courses in a mathematics bachelor's programme, we expect you are familiar with analysis and mathematical modelling at large. For example, at least with one of the following:

- analytical solutions and analysis of ODEs and PDEs
- theory of linear operators
- numerical schemes for integration and the notion for their stability
- basic methods of numerical linear algebra

For complex systems, you need to be familiar with probability theory and stochastic processes (or measure theory).

For scientific computing, we expect you have background in numerical analysis and be comfortable with at least one programming language (e.g. Python, C++).

### Mathematics courses that fit the specialisation:

- Introduction to Complex Systems
- Partial Differential Equations
- Functional Analysis
- Calculus of Variations
- Continuous Optimization
- Inverse Problems in Imaging
- Mathematical Theory for Tomography
- Machine Learning Theory
- Numerical Linear Algebra
- Numerical Methods for Time-Dependent PDEs
- Stochastic Gradient Techniques in Optimization and Learning
- Measure Theoretic Probability
- Stochastic Processes
- Ergodic Theory
- Time Series
- Nonparametric Statistics
- Asymptotic Statistics
- Forensic Probability and Statistics
- Scheduling
- Queueing Theory
- Discrete Optimization
- Parallel Algorithms
- Advanced Linear Programming
- Coding Theory
- Algorithms Beyond the Worst Case
- Quantum Information Theory
- Dynamical Systems
- Group Theory and Dynamical Systems
- Numerical Bifurcation Analysis of Large-Scale Systems
- Mathematical Neuroscience
- Mathematical Biology
- Systems and Control
- Laboratory Class Scientific Computing
- Computational Finance
- High-Dimensional Probability Theory with its Applications in Data Analysis
- Seminar on Neural Networks and Finance
- Seminar Machine Learning
- Seminar Interacting Random Systems
- Seminar Mathematical Epidemiology
- Seminar Random Graphs

Note that the above courses may not be offered every academic year.

### Courses outside of mathematics that fit the 30 EC courses in other disciplines:

Computing science courses:

- INFOMDM - Data mining
- INFOGA - Geometric algorithms
- INFOMCRWS - Crowd simulation
- INFOAN - Algorithms and networks
- INFOEA - Evolutionary computing
- INFOMGP - Game physics
- INFOMNWSC - Network science

Physics courses:

- NS-MO402M - Dynamical meteorology
- NS-MO434M - Current themes in climate change
- NS-TP432M - Modelling and simulation

Biology courses:

- B-MQBIO - Introductory Course Quantitative Biology
- B-MBIEG06 - Bioinformatics and Evolutionary Genomics
- BMB502114 - Advanced Bioinformatics: Data Mining and Data Integration for Life Sciences
- BMB502219 - Introduction to R for Life Sciences

Chemistry courses:

- SK-MCBIM21 - Structural Bioinformatics and Modelling

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

### For more information about this specialisation, please contact Dr. Ivan Kryven

## 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.

For more information, please contact: Dr. Steven Wepster