# 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.

**Attention**

Starting from September 2024, the master's programme will be restructured into the new specialisations below. All study and research directions that could be pursued under the old specialisations can still be pursued under the new specialisations.

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.

Topology and Differential Geometry 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 is advised to 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 "Topology and Differential Geometry".

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

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.

### Prerequisites

Apart from standard mandatory courses in a Mathematics Bachelor's programme, it is strongly advised to be familiar with analysis and mathematical modelling at large.

### Mathematics courses that fit the specialisation:

- Introduction to Complex Systems
- Continuous Optimization
- Inverse Problems in Imaging
- Machine Learning Theory
- Stochastic Gradient Techniques for Optimization and Learning
- Numerical Methods for SDEs
- Nonparametric Statistics
- Forensic Probability and Statistics
- Scheduling
- Discrete Optimization
- Parallel Algorithms
- Advanced Linear Programming
- Coding Theory
- Algorithms Beyond the Worst Case
- Quantum Information Theory
- Mathematical Neuroscience
- Systems and Control
- Multiscale Methods with Application to Climate
- Seminar High-Dimensional Probability Theory in Data Science
- Seminar Machine Learning
- Seminar Mathematical Epidemiology
- Seminar Random Graphs
- Statistical Learning
- Seminar Neural Networks and Finance
- Computational Finance

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 - Advanced Algorithms
- INFOEA - Evolutionary computing
- INFOMGP - Physics of motion

Physics courses:

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

Biology courses:

- B-MBIEG06 - Bioinformatics and Evolutionary Genomics
- B-MBIOINR - Introduction to R for Life Sciences

Chemistry course:

- 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 (for applied mathematics, complex systems) or Dr. Chiheb Ben Hammouda (for mathematical finance).

Mathematical analysis is one of the fundamental branches of mathematics that involve the study of functions, their properties and their relationships using tools such as limits, integrals and differential equations. Differential equations describe relationships involving rates of change, possibly across multiple dimensions, heat flow and wave propagation and fluid dynamics. They have applications in areas such as physics, biology and engineering.

Real analysis forms the mathematical foundation of modern probability theory. Probability theory is the science of chance, in the sense that it aims at describing mathematically a specific random phenomenon. In fact, stochastics can be viewed as the branch of mathematics concerned in developing an appropriate formalism for describing the principle of randomness. Real analysis and measure theory provide the mathematical foundations of modern probability theory. According indeed to Mark Kac probability is measure theory with a soul, where the soul is the pivotal notion of independence. Stochastic processes are collections of random variables that evolve over time. They are used to model and analyze dynamic and random systems. Common types of stochastic processes include Markov processes, Poisson processes, Brownian motion, and random walks. Moreover, stochastic processes are another example of the deep link of probability theory with Analysis: the theory of Markov processes is strongly connected with the theory of partial differential equations and harmonic analysis.

### Prerequisites

Students are expected to have a solid knowledge in analysis and/or probability, for instance by having completed their bachelor programme in mathematics. In particular:

#### Prerequisites for analysis:

After the mandatory analysis courses --in Utrecht these are Calculus and linear algebra (WISB107, WISB108), Analysis (WISB114) and Introduction to analysis in several variables (WISB213)-- you will have followed second year courses like Functions and series (WISB211), Differential equations (WISB231) and/or Analysis in several variables (WISB212) and third year courses like Functional analysis (WISB315), Measure theory and integration (WISB312) or Differentiable manifolds (WISB342). Topics that you might be familiar with include holomorphic functions and Fourier series, existence and classification of solutions of ordinary differential equations and methods to compute them, as well as the inverse and implicit mapping theorems.

#### Prerequisites for probability:

You need to have a solid knowledge of basic probability theory which is taught in an introductory course on probability (for example the course WISB161 at UU). Topics that you shoud be familiar with include discrete and continuous random variables, distribution and moments of random variables, generating functions, characteristic functions, law of large numbers, central limit theorem. You have followed at least one additional course in your bachelor’s programme from the following list:

- Measure theory and integration (for example the course WISB312 at UU)
- Stochastic processes (for example the course WISB362 or WISB373 at UU) or a course on martingales
- Mathematical statistics (for example the course WISB263 at UU)

### Advised courses

For the elective courses (60 EC, next to the 15 EC of compulsory courses) you can take any course from the national Mastermath programme or from the local Utrecht courses in mathematics. It can also be arranged to take local mathematics courses at other Dutch universities. You will also have the opportunity to take courses in other departments (physics, medicine, sociology, economics, etc.) that relate to your interests as secondary electives. You can furthermore choose up to 2 courses out of the Utrecht bachelor programme to remedy deficiencies as secondary electives. Your personal list of courses will also include at least one seminar and during your master programme you will discuss your list with your tutor.

#### Advised courses for analysis:

When compiling your personal list, most courses will be from the following list of courses:

- Dynamical Systems
- Partial Differential Equations
- Functional Analysis
- Calculus of Variations
- Introduction to Numerical Bifurcation Analysis of ODEs and Maps
- Nonlinear Partial Differential Equations
- Symplectic Geometry
- Differential Geometry
- Lie Groups
- Group theory and dynamical systems
- Ergodic Theory
- Statistical Mechanics
- Operator Algebras
- Riemann Surfaces
- local analysis courses (these change from year to year)

#### Advised courses for probability:

The course Measure Theoretic Probability is a prerequisite for a lot of other courses in the specialisation. We strongly advise to follow this course. Further we strongly recommend to choose at least three (including one seminar) from the following list of courses:

- Ergodic Theory
- Stochastic Integration
- Stochastic Processes
- Probabilistic and Extremal Combinatorics
- Queueing theory
- Numerical Methods for Stochastic Differential Equations
- Seminar Random Graphs
- Seminar Analysis: Quantization
- Seminar Ergodic Theory

### For more information about this specialisation, please contact Dr. Heinz Hanßmann (for analysis) or Dr. Cristian Spitoni (for probability).

The specialisation "Mathematics of physical structures" explores in depth the mathematical foundations of modern physical theories. It provides the mathematical understanding and tools underpinning an extensive range of contemporary science including statistical mechanics, relativity, string theory and the quantum theory of many body systems. It is structured into two broad thematic blocks: mathematical relativity and mathematics of large-scale systems. A combination of courses from both blocks is also possible.

Students who undertake a master's in "Mathematics of physical structures" will acquire the mathematical background essential to understand and describe nature at its most fundamental levels. The specialisation will not only prepare students for research in mathematical physics and related disciplines; it will foster creativity and develop high-level skills in critical and analytical thinking, paramount in problem solving.

This specialisation is particularly well suited for students who did a bachelor's degree in mathematics and who have a curiosity for physics, and are interested in developing broad mathematical skills in order to understand physical phenomena.

### Prerequisites

As prerequisites we require a bachelor's degree in mathematics or related areas with enough mathematical background. It is recommended if the student has basic knowledge of:

- Banach and Hilbert spaces (WISB315 Functional analysis)
- analysis in several variables (WISB213 Introduction analysis in several variables, WISB212 Analysis in several variables)
- group theory (WISB124 Introduction groups and rings)
- ordinary differential equations (WISB231 Differential equations)
- complex analysis (WISB211 Functions and series)
- [mathematical relativity] differentiable manifolds (WISB342 Analysis on manifolds)
- [large-scale systems] graph theory or combinatorics (INFOB3DW Discrete mathematics)

**Recommended mathematics courses that fit the specialisation:**

- Differential Geometry
- Partial Differential Equations
- Functional Analysis
- Quantum Information Theory
- Riemann Surfaces
- Operator Algebras
- Lie Algebras
- Symplectic Geometry
- Seminar Differential Geometry
- Seminar Analysis: Quantization

You can take 15 EC of secondary electives from the physics programme or the mathematics bachelor programme to remedy deficiencies in consultation with your tutor.

### For more information about this specialisation, please contact Dr. Michał Wrochna or Dr. Wioletta Ruszel.

The first major achievements of Logic are rigorous definitions of the notions "proof" and "computation". Such definitions enable one to study the boundaries of these notions, and consider statements that cannot be proved, functions that cannot be algorithmically computed.

In the "level 3" bachelor course Foundations of Mathematics (which we recommend you incorporate in your master's curriculum if you did not already complete it in your bachelor studies), students are familiarized with the notions of an abstract language for mathematical structures, the appropriate notion of truth for statements in this language, and the logical connections between these statements: a statement can be a "consequence" of a "theory" (a set of statements) if it is true in every model of the theory. Moreover, they are introduced to formal proofs and Gödel's Completeness Theorem: for every consequence of a theory there is a formal proof which concludes it from assumptions which are elements of the theory.

Modern logic has several subdisciplines: Model Theory (pioneered by Tarski) studies the properties of the class of models of a theory from an abstract point of view. It has several applications outside logic, and its most pronounced heirs: stability theory and the theory of o-minimality, try to formulate their results independently of any logical basis. The theory of computable functions (Recursion Theory, or Computability Theory) studies the functions and relations between discrete structures (usually built up from the natural numbers) which can be characterized in terms of Turing computability. Closely related is Complexity Theory, which analyzes algorithms in terms of the resources they use). It is these fields that have the most direct connection to Computer Science. Set theory is concerned with the ingeniously constructed models which (among other things) tell so-called "large cardinals" apart. Proof Theory (which started with Gödel's famous Incompleteness Theorems) was built up by Herbrand and Gentzen in the 1930's and deals with combinatorial properties of formal proofs.

A precursor to Computability Theory is the Lambda Calculus which is of paramount importance to Computer Science.

In the Mathematics department, several of these strands are represented in research and teaching. Categorical Logic generalizes the standard definition of models, and views logical operations as universal constructions. Two areas which build on this field are Topos Theory and Homotopy Type Theory. An area which is quite fashionable nowadays, and which received a large stimulus from Homotopy Type Theory, is the field of automated proof verification: special tools help you to construct formal proofs for mathematical statements. A combination of Topos Theory and Computability Theory is the field of Realizability, which studies structures connected to toposes defined using Turing machines or very similar ideas.

### Logic in the master's programme Mathematical Sciences and prerequisites

We assume that you have a bachelor with enough pure mathematics in it (a reasonable dose of abstract algebra and topology). A basic course in Logic is an advantage but not strictly necessary. It should be emphasized that the program is a program in Mathematics, not a Logic Master (such as, for instance, is offered by the University of Amsterdam). We recommend that in your curriculum, you choose at least one focus outside Logic (this might be Algebraic Topology or Number Theory), so that in your master's thesis you can do research on the connections between two fields, or use techniques of one field in another.

### Available courses

Every year, we teach in Utrecht the basic course Foundations of Mathematics and the Logic Seminar. The seminar has a different theme every year, so it can be followed twice. Topics of recent seminars include: Hilbert's Tenth Problem, Tame Topology and o-minimal structures, Boolean-valued models of Set Theory, Models of Intuitionism, Constructible Sets, Ultracategories, Algebraic Set Theory.

Apart from that, Utrecht participates in the Mastermath programme, which offers courses available to all mathematics students in the Netherlands: in the year 2024-2025 there will be the courses Category Theory, Topos Theory, Homotopy Type Theory and Formal Methods in Mathematics.

Then, there are local courses at other universities, which students are usually allowed to incorporate into their curriculum (it is always advisable to discuss this with your tutor): in Amsterdam there are yearly courses in Model Theory, Proof Theory, and more; the Radboud University in Nijmegen offers (alternatingly) Computability Theory and Complexity Theory, and also an alternative Proof Assistants course (based on Coq); moreover, Lambda Calculus is strongly represented in Nijmegen. Finally, we mention courses at the boundary of Mathematics but which are still of interest to Logic students, such as Philosophy of Mathematics (lectured intermittently at the Philosophy Department) and courses offered by the History and Philosophy of Science group.

### For more information about this specialisation, please contact Dr. Paige North or Dr. Johan Commelin.

## History of Mathematics

Students may choose to do their master's research 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 master's thesis should have a substantial mathematical content in the area of the specialisation. In practice, the subject of a master's thesis in history of mathematics is often one in the 17th century or later, although exceptions are possible. Master's theses in history of mathematics are 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 the UU bachelor's course History of Mathematics. Courses in history of science are an advantage.

The student should take the courses History of Philosophy of Mathematics and Seminar History of Mathematics, subject to availability. Depending on the subject of the master's thesis, certain mathematics courses may be necessary. For example, a student who writes a master's thesis in the History of Elliptic Curves will also have to take mathematics courses in this field.

The Educational Profile complements coursework and a research project in the History of Mathematics well.

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