# Modules of the mathematics track

## linear algebra

This course focuses on Linear Structures and its application in coding theory. For example the theory about vector spaces and the linear images between them belong to Linear Structures. Often vectors are seen as columns with real numbers and linear maps are indentified with matrices. But even continuous functions can be interpreted as vectors, and so can matrices. By looking at vectors and linear maps in a general way more, seemingly different, cases can be treated and understood in one go. A really nice application of this general treatment can be found in the coding theory, namely the theory of linear codes. Coding theory and, in particular, linear codes have wide applications. In the most divergent situations where data has to be sent from a transmitter to a receiver via an unreliable channel, codes are used to correct errors that may occur in the data. This is done by incorporating redundancy in the data in a smart and efficient way. In that way erroneous data, that were received, can often be correctly. One of the most appealing applications is the compact disc. Due to scratches and fingerprints the signal read by the laserhead contains almost always errors, but by the use of coding these errors can be corrected, if not too numerous. As a result Beethoven’s ninth can still be enjoyed without the disturbing cracks that are characteristic for good old vinyl records.

### General information

This course will take place in the third quartile of your first year in the mathematics track. The course starts with a general introduction to the basic notions in coding theory. Subsequently we shall focus on linear codes. As an intermezzo linear structures will be treated in a general context. There will be special attention to finite vector spaces Finally, we shall apply the theory of finite vector spaces linear coding theory. It will turn out that it is very handy and useful to think in a more abstract way about linear spaces and vectors.
Learning goals:

• Understanding the general basic theory about vector spaces and linear images.
• Recognizing the situations in which vector spaces can be used and the actual application of the theory
• Knowing the main problems in the coding theory
• The application of the theory of vector spaces in linear codes
• Drafting linear codes that satisfy the given specifications and decoding these codes
• Recognizing situations in which codes can be used

Prerequisites: basic linear algebra

#### teachers

Jan Willem Polderman graduated cum laude in 1983 at the university of Groningen on a topic in algebraic number theory. After his study Mathematics he was a research assistant at the Center for Mathematics and Computer Science in Amsterdam. In 1987 he defended his PhD thesis on adaptive control systems at the Rijkuniversiteit Groningen.. His promotor was prof. J.C. Willems, who sadly passed away recently. With professor Willems he published a book about mathematical system theory. With I. Mareels (Melbourne) he wrote an introductory book on adaptive systems. Since 1987 Jan Willem Polderman has been with the department of Applied Mathematics at the University of Twente, First as an assistant professor and from 2003 as an associate professor. His main research interest is adaptive systems and system theoretic approach tp coding and decoding. Recently his research interest has shifted towards stability and control of hybrid systems. Since 2008 he is the program director in the department Applied Mathematics and the 3TU master Systems and Control. Jan Willem Polderman likes to play tennis and ice skating, be it at a very modest level. He listens to classical music and plays the piano, again at a questionable level. Together with Ella de Jong he has two daughters, Lotte (1996) and Jorieke (1998) with whom he loves to go camping, for example on the beloved island of Vlieland.

## finding vs verifying

For many problems we can find solutions quickly with help of computers. This class of problems is called P. But there are also a large number of problems, where even the fastest computers are not able to solve them. A special class is NP: for these problems we are able to verify whether a solution is correct, but we are not able to find new solutions efficiently.
Even after 40 years the question remains whether for all problems in NP also solutions can be found efficiently, in other words: P equals NP. This question is one of the millennium problems of the Clay Mathematics Institute. We look at the question whether P equals NP, have a look at why many researchers do not believe that P equals NP and we look at possible consequences of P equals NP and P is unequal to NP.

### General information

This course will take place in the fourth quartile of your first year.

#### teachers

Bodo Manthey earned his PhD in computer science from the University of Lübeck. After postdoctoral positions at Saarland University and Yale University, he joined the Department of Applied Mathematics at the University of Twente in 2009 as an Assistant Professor.

## geometry

Geometry can be build up in different ways. In an axiomatic way it was Euclides (300 BC) who introduced geometric figures that can be constructed by straightedge and compass. This is also our start of the course, but (linear) algebra smooths the later stages of geometry. The resulting analytical geometry gives us the generalisation of planar and solid geometry to higher dimensions. The origin of projective geometry lies in perspective paintings, Italian artists discovered in the 15th century how to draw three-dimensional scenes in correct perspective. All the different types of geometry can be defined by invariants of groups of transformations (Klein, 1872).
A list of different topics that will be dealt with:

• Euclid’s Elements
• Constructions with straightedge and compass
• Theorems ascribed to (famous) persons
• Vector geometry
• Perspective
• Group theory in relation to geometry
• Non-euclidean geometry
• Topology
• Spherical geometry

### GENERAL INFORMATION

This course will take place in the first quartile of your second year.
Prerequisites : Linear Algebra (Math C1)

#### teachers

Gerard Jeurnink (1954) obtained his master degree in pure mathematics at Radboud University Nijmegen. Within the field of Functional Analysis he defended his PhD thesis ‘Integration of functions with values in a Banach lattice (1982, supervisor A. Van Rooij). Several years he practised geometrical phenomena in his math lessons at senior highschool, whereas during the last fifteen years he is the lecturer of the master course Geometry (University of Twente, Mastermath Utrecht). Besides his position at the department of Applied Mathematics he is also affiliated to the teacher training group ELAN. He organises refresher courses in Analytical Geometry. His main interest lies in the interaction between geometry and algebra.

## signals with information

#### Fast Fourier Transform (FFT)

The impact of this in 1965 invented linear operation is unbelievable big. Each laptop calculates for example 250 thousand FFTs per second! And if you would like to multiply two large numbers you also use FFT. JPEG uses FFT as well.

#### Wavelets

With Fourier you write the signal as sum of everlasting harmonic functions. That is quite strange, when the signal (e.g. a piece of music) is finitely long. You can look at wavelets as an extension of Fourier, but then one that is closer to the musical notation: the building blocks are of finite length, but we still have limited frequencies. We illustrate wavelets for images and we shall see that spectacular compression ratios can be achieved.

#### Information theory

This is maybe the most beautiful example of the force of mathematical modeling. A file with only zeros is easy to compress and if the zeros and ones are alternating often, the compressing becomes more difficult. But how can we understand this mathematically? In this part we provide the basis of the information theory of Claude Shannon. We will see that there is a natural measure for the lack of structure called entropy and that this entropy is equal to the optimal compression ratio.
You would also like to send files, say, from your router to a laptop and you have a limited amount of information that you can send per unit time. That is what we call (channel)capacity. How do you model this and optimize it? This are we going to cover too and we make the connection with entropy.

### General information

This course will take place in the second quartile of your second year.

#### teachers

Gjerrit Meinsma was born on the 29th of January 1965 in Opeinde, a little village in the northern part of The Netherlands. There he attended the nursery school ``De Kindertuin'' and elementary school. In the off-school hours he spent many hours making and throwing boomerangs at and with his friends, and in the weekends he used to play korfball. From the age of 12 till the age of 18 he attended the grammar-school ``Het Drachtster Lyceum'' in the nearby town Drachten. Somewhere in the middle of this period the fascinating world of music caught his attention. Listening to music and playing the piano have been his major hobbies since. At the same time Gjerrit developed a keen interest in mathematics, and he decided to pursue his career in this direction. In 1983 he went to Enschede to study applied mathematics at the University of Twente. At the end of 1988 he finished his master's thesis entitled ``Chebyshev approximation by free knot splines'' and in March 1989 he received his master's degree. After a break of five weeks he returned to the faculty he graduated with, only this time as a research assistant (AiO) with the Systems and Control Group. Half a year later he bought himself a brand new Klug & Sperl upright piano, and a few years later, in 1993, he finished his thesis on H-infinity control. Not long thereafter he dumped his winter coat in the garbage bin and left for sunny Australia were he worked as a post-doc for three years with the Electrical Engineering Department at the University of Newcastle. In January 1997 he returned to Enschede to take up a two-year post-doc position with his former group. He quickly acquired a much needed winter coat. Since 1999 he holds a permanent position at this group.
Gjerrit likes all sorts of music, but he is fanatic only about classical music. His favorite composers are Bach, Beethoven, Schubert, Chopin, Grieg, Ravel, Skriabin and Janacek. He is a fan of the pianist Sviatoslav Richter, and also likes very much Sofronitsky, Lipatti, Rubinstein, Oborin and Henkemans and many others.

## complex networks

Networks play an increasing role in society. Examples are the internet, social networks, phones networks and electricity and energy networks. Many of these structures have fascinating common properties. In this module we will have a look at the underlying mathematical theory that describes these structures. This theory, random graph theory, studies the collective behaviour of big sets of graphs and can be compared with the ordinary graph theory as statistical physics to the physics of Newton.
We will look at methods that will show that big networks have peculair properties. Furthermore, we study the time that is needed to let the network perform a certain task (like helping customers in a post office), look at strategies to search for information like Google does and look at how you can effectively spread information (or viruses) in networks. The underlying mathematics is that of random graphs and networks of queues.

A list of the different topics that will be covered:

• Complex networks in practice: different manifestations, common structures. (Maurits de Graaf)
• A first theoretical model, with far-reaching consequences (Maurits de Graaf)
• Wait-and time calculation for logistic network processes (Nico van Dijk)
• Two most simple network structures: complex? (Nico van Dijk)
• Preferential attachment, or: how the rich are getting richer in complex networks. The impact of preferential attachment on robustness and spreading of information and viruses in networks. (Maurits de Graaf)
• Why the world is 'small', search graphs and the basis of the Google PageRank algorithm (Maurits de Graaf)
• Realisation of 'smart dust' networks: very small sensors in space: how we distribute energy and how we deal with problems involved with that (Maurits de Graaf)

### General information

This course will take place in the third quartile of your second year.

#### teachers

After a Ph.D. in Applied Mathematics and a visiting position at an American Business School, professor Nico van Dijk has been responsible for an Operations Research (OR) and Management (OM) Program at the University of Amsterdam for two decades. Since April 2012 he is affiliated to the University of Twente. He has a strong research interest in the area of stochastic OR, particularly Queueing. He also aims to popularize the potential of OR.  He has been involved in a variety of practical projects, among which for the Dutch railways, the Dutch airport Schiphol and KLM , the Dutch Triple A, the Dutch Ministry of Health and various hospitals, Dutch financial institutes (banks) and for over several years still is for the Dutch Bloodbank Sanquin

Nelly Litvak received her PhD from Eindhoven University of Technology (EURANDOM) in 2002. Last years she has been working on the analysis of complex networks, in particular, algorithms for node ranking, efficient detection of network structure, and analysis of network correlations. Her other research interests are in stochastic processes, queueing theory, and probabilistic solutions for combinatorial problems. Nelly has received several grants and awards including the Google Faculty Research Award 2012. She is a member of programme committees and invited speaker at many top conferences in mathematics and computer science and a managing editor of the Internet Mathematics journal.

## dynamic systems

Most of the concepts used by studying dynamic systems are derived from Birkhoff7. The ideas of Poincaré, Lyapunov and Birkhoff, first developed in finite much dimensions, are generalized to infinite dimensional equations. This type of equations we meet in numerous fields like the fluid dynamics, optics, mathematical biology et cetera. For mathematicians it is a constant challenge to explain and describe patrons that occur in nature by analyzing non-linear dimensional systems.

### General information

This course will take place in the fourth quartile of your second year.

Stephan van Gils