What will I learn?

At secondary school you learn all kinds of mathematical techniques to calculate things. These include limits, differentiation, integration but also simple arithmetic such as addition, multiplication and division. But did you know that there's actually much more to it than that? During the Mathematical Structures course, you will learn, using definitions (agreements) and theorems, why certain calculations are allowed and how they are made possible. You will also learn how you can prove all of it by purely logical reasoning. This course is the foundation for many other mathematics courses. You will become familiar with important mathematical structures and symbols, such as how to formulate a proof and the distinctive characteristics of rows and series, including convergence.

During the Kaleidoscope module you will become familiar with a set of mathematical subjects such as Graph Theory, Complex Numbers, Counting and Probability. The module covers a whole range of subjects as a preview of the rest of the degree programme. You will also gain experience with three different software packages (LaTeX, Maple and MATLAB), which are important for the rest of your degree programme. LaTeX is a very useful word processor for mathematical texts that specialises in formulas. Maple and MATLAB are useful for mathematical calculations.

This is an important basic course of the programme as the knowledge acquired is applied in many other courses. In this course you will learn how you can calculate large systems of equations easily using matrices. The course also handles geometrical aspects that can be solved using Linear Algebra, e.g. determining points in spaces that are closest to other specified points. It teaches you new arithmetical techniques.

Translating a mathematical problem into a computer algorithm requires a basic understanding of programming. The Introduction to Programming course starts from scratch, building up the essential programming knowledge a mathematician needs. The programming language taught is Python. You will learn how to enter algorithms (a formula to perform a particular calculation) into Python language so that the computer will perform the calculation for you.

In this course you will finally recognise some of what you learned in secondary school: differentiating and integrating. You'll learn the theory behind this, such as when to differentiate or integrate a function. Analysis 1 builds on Mathematical Structures. It also covers concepts as continuity and limits. You will see how what you learned in secondary school is mathematically substantiated and soon move on to more complex subjects.

Modelling is very much an ‘applied’ mathematics course. You will be given a practical problem, which you will work on in groups in the form of a project. For example: 'Someone is drowning in the sea, and the lifeguard must get to them as quickly as possible. How can he best do that, given the current?' That may sound difficult, but during the Modelling course you will learn how to translate practical problems into mathematical problems. You will learn how to simulate the problem using software programs as MATLAB and Maple, taking all the variables into account. You do this by applying the knowledge you have acquired in the other mathematics courses. The fun aspect of this course is that it really gives you an idea of what you could do as a mathematician once you've graduated.

In Algebra 1 you will learn a very different, abstract way of looking at arithmetic, i.e. addition, subtraction, multiplication and division. The main subject of this course is group theory which includes examining symmetries, modular arithmetic (remainder calculation) and composing functions. This could result in calculating the number of cubes you could make with a certain number of coloured squares, for example. You must consider the fact that a cube is symmetrical and should therefore look exactly the same if you turn it 180 degrees. A very different application is an important method of cryptography (encryption) for data protection purposes.

While Analysis 1 handles functions with one variable, Analysis 2 deals with functions with multiple variables. You will learn, for example, how to determine the tangent planes of sloping surfaces and how to calculate the surface area of such a surface using a multiple integral. Once you have completed this course, you will be able to differentiate and integrate functions in 3D and higher dimensions, often without being able to give a representation of such a function.

Just like Analysis, Introduction to Probability is a course that deals with subjects you had in secondary school (Mathematics D). The course repeats the knowledge covered in secondary school and then moves on to new subjects. Applying probability theory can provide insight into situations that depend on randomness or where there is a level of uncertainty. You will learn to calculate the probability of a particular result occurring. You will also learn the different ways of mathematically describing different events and an appropriate calculation method. This course considers many familiar practical examples, such as tossing a dice or coin, as well as other more difficult, mostly practical situations.

In your first year you can choose to do a course that borders on a different field of study, namely electrical engineering, computer science or physics. Depending on your own interests, you can choose a course with less focus on mathematics. You can choose between the following electives: Electricity and Magnetism, Algorithms and Data Structures, or Mechanics and Theory of Relativity.

Building on the analysis courses Mathematical Structures and Linear Algebra, this course is an important foundation for later analysis and probability theory courses. The course is divided into two parts: metric spaces, and measurement and integration theory. At the end of the course, you will be able to understand, explain and apply the theory learned.

Are you interested in the mathematics underneath defining the shortest routes and matchings? In this course you will learn to view and solve these kinds of problems mathematically. For example: 'What are the best ambulance standby locations in a town (and how many are needed) to ensure that all areas of the town can be reached as quickly as possible?' This course deals with numerous algorithms, each of which solves a different general problem.

This course applies the knowledge acquired in Introduction to Probability Theory. Its components include making predictions and basing decisions on historical data. This involves being able to write an appropriate probability model in which the unknown parameters have to be estimated on the basis of the given data. Performing statistical analyses requires use of the statistical software package R, which you will learn to work with during this course.

Linear Algebra 2 takes up where Linear Algebra 1 left off and deals with sets of vectors. These sets have special properties. While in Linear Algebra 1 you learn about calculations and all kinds of rules, in Linear Algebra 2 you will also learn the theory and properties, as with Mathematical Structures.

This course is an introduction to differential equations. A differential equation is an equation for which the solution is a function. The equation involves both the function and its derivative. A differential equation is not as easily solved as a linear or quadratic equation. You will therefore learn the many different ways of solving different kinds of differential equations. There are also several laboratory modules that demonstrate how important differential equations are to mathematically describing and solving practical problems.

The structure of this course is similar to that of the first-year courses Modelling A and B. In the second year, part A involves working on a mathematical model for probability and statistics. In part B you will work on a mathematical-physical problem. One example of such a project could be creating a model of a flu epidemic. This would involve researching how the number of people affected increases and decreases over time as well as how the epidemic spreads in spatial terms.

Some mathematical problems cannot be solved exactly, or not easily. This is where numerical methods come into the picture. These are methods that approach the solution of a problem rather than solving it exactly. An important part of this is the laboratory module, during which you will implement these methods in MATLAB.

This course also concerns methods of solving differential equations, but a different type of equation. The problems discussed in this course are practical ones and include, for example, a simple model of traffic congestion. It is important to consider not only the mathematical solution but also its interpretation. When you calculate the heat distribution of a bar, for instance, it is impossible to determine an infinite temperature. It might be possible mathematically, but is physically impossible.

Complex Function Theory is actually another analysis course. In this case it concerns the application of functions to complex numbers and images that are also complex numbers. In effect, you will expand your understanding of the basic concepts of analysis to the complex domain. What's great about this is that these functions have very special properties. They can help you integrate functions that you were not yet able to integrate in Analysis 1 or 2, for example.

There is scope for a mathematical elective in the third quarter. The second-year electives are:

• Decision Theory (the application of probability theory and statistics to make decisions about problems with a degree of uncertainty),
• History of Mathematics (the history of mathematics is studied in work groups),
• Philosophy of Mathematics (the philosophy of mathematics is studied in work groups),
• Mathematical Models in Biology,
• Advanced Statistics (theory and application of generalised linear models, such as linear regression models),
• Applied Algebra: Codes and Cryptosystems (a course on the use of algebra to encrypt data and break codes, etc.) and
• Markov Processes (This is an introductory course on Markov chains, where time-discrete and continuous time Markov chains will be introduced, and their most fundamental basic properties studied).

Besides your minor, you also have freedom of choice in the form of electives. In the third year, you can choose two. The third-year electives are:

• Mathematical Physical Models (the application of the Partial Differential Equations course to such physical phenomena as heat conduction),
• Inverse Problems,
• Numerical Methods 2 (which follows on from Numerical Methods 1),
• Graph Theory (this course is about the mathematical theory of networks),
• Advanced Probability (theoretical treatment of analysis concepts that play a part in the calculation of probability),
• Fourier Analysis (theory and applications of Fourier series),
• Differential Geometry (The focus of this course will be on Riemannian geometry, the study of metric spaces in a smooth context),
• Topology (studies notions from previous subjects - such as open collections, convergence and compactness - in a broader context than that of metric spaces) and
• Mathematics Seminar.

The Bachelor Colloquium forms a relatively small part of the Bachelor's final project. Here, students develop skills in the verbal presentation of a mathematical subject.

Your Bachelor's degree programme concludes with a Bachelor Project, which involves working on a mathematical or practically oriented problem. You choose a problem from one of the various research groups of the Mathematics department. The next step is to search for the relevant background literature and to translate the problem into a mathematical form. You will then solve the mathematical problem and subsequently translate the solution back into the practical situation. The project concludes with the submission of a thesis and delivery of a presentation.