Mathematics encompasses quantitative and geometrical reasoning in a wide range of manifestations, both in the abstract and as an essential tool across the natural, social, and applied sciences.

The University College Utrecht Mathematics track spans both pure and applied mathematics. In applied mathematics you master a variety of mathematical techniques, use them to model physical and social phenomena, and interpret and communicate the real-world significance of mathematical findings. In pure mathematics you study abstract structures in their own right as well as the foundations of concepts like infinity and the continuum, and you see the power of reasoning at a formal level using exact formulations and rigorous proofs.

Mathematics is singularly abstract, yet it describes the world. It is widely applicable with unreasonable effectiveness in an ever-growing number of areas. At the same time, many mathematicians are as likely to characterise the purpose of their enterprise in terms of “the desire for aesthetic perfection” or “the honor of the human spirit”.

A background in Mathematics opens the door for many potential career paths. For overviews of such possibilities, see for example: AMS, MAA, SIAM, weusemath.org, prospects.ac.uk, UU, UL.

UCSCIMAT11 introduces many broadly useful techniques and is therefore a recommended or required prerequisite for many courses in other subjects. UCSCIMAT22 is also rich in applications of an interdisciplinary nature. UCSCIMATL4 provides a mathematical view of statistical and data-driven techniques used in many fields. UCSCIMAT21 is an essential part of classical physics, and UCSCIMATL3 is an important subject for more modern theoretical physics.

In addition to the topics taught within the track, you can pick up interesting mathematical techniques in other science courses, including some topics that at some institutions are taught within a mathematics department. Courses of this nature include UCSCIPHY12, UCSCIPHY21, UCSCIPHY31, UCSCIPHY32, UCSCICHE22, UCSCIMATL2, and applied statistics courses offered within the academic core.

UCACCMAT01 is a basic mathematics course that covers some of the same topics as UCSCIMAT11 but at a more basic level. It is primarily intended for those who do not have a sufficient high school background to enter UCSCIMAT11.

The core courses required to complete the track are UCSCIMAT11, UCSCIMAT14, UCSCIMAT31 and at least one of UCSCIMAT21 and UCSCIMAT22. The choice at level 2 typically corresponds to which other fields you are interested in. UCSCIMAT21 is very useful in physics and chemistry and is also mathematically interesting. UCSCIMAT22 can very favourably be combined with the social sciences as well as various applications involving “big data”.

Greater breadth beyond the core of the track is provided by the summer and winter courses—known as “lab courses” although in the Mathematics track they differ from other content courses only in terms of scheduling.

For mathematically oriented students, it is advisable to complete UCSCIMAT14 quite early. Although it is a prerequisite requirement only for UCSCIMAT31, it is also a good preparation for UCSCIMATL3 and other courses (including off-campus courses).

UCSCIMAT01 is a course for non-science majors. It is generally not recommended for those who intend to go further in mathematics.

SCIMAT01 Mathematics for Poets
A non-technical introduction to mathematical thinking.

SCIMAT11 Basic Mathematics: Calculus
Alculus, differential equations, and linear algebra, with emphasis on applications.

SCIMAT14 Foundations of Mathematics
An introduction to proof writing and proof techniques.

SCIMAT21 Mathematical Methods
Partial differential equations and Fourier analysis, with applications to physics.

SCIMAT22 Mathematical Modeling: Networks
Mathematical modeling of network structures in fields such as sociology, ecology, economics, cognitive neuroscience, and computer science.

SCIMAT31 Advanced Mathematics
A rigorous study of real analysis in one and several variables, with applications to dynamical systems and Fourier analysis.

UCSCIMATL2 Computational Physics 
Introduction to scientific computation and numerical simulation.

SCIMATL3 Group Theory
Introduction to group theory.

SCIMATL4 Introduction to Probability and Statistics
Introduction to probability theory and stochastic reasoning, followed by project-driven applications.

SCIMATL5 Dynamical Systems
An introduction to dynamical systems — a subject with wide-ranging applications across the natural sciences and beyond.

SCIMATL6 Complex Analysis
An introduction to complex analysis, including both theoretical and applied aspects.

A University College Utrecht degree combining Mathematics with another relevant field is often a good preparation for Master's programmes of an interdisciplinary character. At Utrecht University, this could include some of the applied options within the Master in Mathematical Sciences itself as well as Master's programmes in adjacent fields.

To pursue a Master's programme strongly focused on mathematics itself ytou will typically need to supplement the track with a number of off-campus courses. In pure mathematics, Master's programmes will look especially for proficiency in working in the abstract, rigorous style characteristic of modern pure mathematics. Within the University College Utrecht track, courses of this type are especially UCSCIMAT14, UCSCIMAT31, UCSCIMATL3, but further depth and breadth is advisable to go further in pure mathematics. The entry requirements for the Utrecht University Master in Mathematical Sciences gives an indication of the core areas you may want to consider for off-campus courses.

To supplement a University College Utrecht degree, a common option is to do a one-semester pre-Master's programme, in which you take some mathematics courses at the advanced Bachelor level at Utrecht University. Such a programme can be individually put together with your interests and the intended focus of your Master's programme in mind.

Many options for off-campus courses in mathematics are available at Utrecht University. Many of these courses are taught in Dutch, but some in English. Coursework at University College Utrecht often does not align exactly the intended prerequisites within University Utrecht Mathematics Bachelor, but in many cases this can be resolved with a little additional reading and/or instructor permission. 

The following courses can be said to build on UCSCIMAT11 (which is arguably a bare minimum prerequisite, although additional preparatory coursework is sometimes preferable): WISB121 Linear algebra (some overlap with UCSCIMAT11 and other UCU courses), WISB137 Calculus B, WISB141 Introduction geometry, WISB152 Computer programming for Mathematics, WISB231 Differential equations (some overlap with UCSCIMAT11 and other UCU courses), WISB251 Numerical analysis, WISB272 Game theory, WISB382 History of mathematics.

The following courses are natural follow-ups to UCSCIMAT14 (a bare minimum prerequisite; often more preparatory coursework is preferable): WISB114 Analysis (some overlap with UCSCIMAT31), WISB243 Introduction to topology, WISB321 Basic number theory, INFOB3DW Discrete mathematics.

WISB211 Functions and series can be seen as a more theoretical follow-up or companion course to UCSCIMAT21. BETA-B1CS Introduction to complex systems can be a follow-up course to UCSCIMAT22.

WISB221 Group theory is an extended version of UCSCIMATL3, and WISB222 Rings and Galois theory is a natural follow-up course to this. WISB161 Introduction probability theory and statistics is an extended version of UCSCIMATL4.

WISB311 Complex analysis may also be suitable for advanced students. Students combining mathematics with economics may want to consider WISB377 Econometrics and WISB373 Introduction to financial mathematics.

Dr. Viktor Blåsjö is the Mathematics fellow at University College Utrecht. Office: Hans Freudenthalgebouw, Room 706, Budapestlaan 6, 3584 CD Utrecht.