# Excellence Programme Modules Year 1

### Mathematical Structures

During Mathematical Structures we consider some basis concepts in mathematics. One of these basic concepts are the real numbers (numbers such as root(2), Pi, e and fractions and integers, but not the complex number i). During the regular course we discuss what the basic properties of the real numbers are that mathematicians always use, and we just assume that something like the real numbers exists. In the Excellence Programme we start with the natural numbers (1, 2, 3 etcetera), and consider the basic properties of those. We assume that these exist (in the end you have to assume something, but natural numbers are a more “natural” concept). Subsequently we construct the rationals and the real using these natural numbers. Thus we see that the idea that a set of numbers like the reals exist is not so strange.

### Linear Algebra 1

For the excellence module of Linear Algebra 1 we have chosen to let you meet three different aspects. One meeting is devoted to an application of Linear Algebra (coding theory), one meeting is devoted to a numerical problem: during the regular course students learn about so called eigenvalues of a matrix. The exact calculation of these values is difficult and often even impossible. In the past methods are developed to approach these eigenvalues as accurate as wanted. One of these methods will be studied.

During the other meetings the students are going to prove some deeper theorems that are stated during the regular course, but not proven there. For example the theorem that so called reduced echelonforms of a matrix are unique and the main theorem that symmetric matrices are orthogonally diagonalizable.

### Analysis 1

In the EP module for the course Analysis 1 we go deeper into the theoretical aspects of the course. The students will get additional, challenging reading material which they then have to present "in their own words" during the extra class sessions. Moreover, the students have to solve demanding problems at home and present their solutions, which are then discussed in class among all EP students to compare and check different solutions paths. The grade is determined by the amount of active involvement during the extra class hours as well as an oral final exam.

The extra material connects well with the regular courses of both Analysis 1 and Mathematical Structures. More specifically, we will cover the following topics:

• The exponential function: equivalence of definitions, additional properties, Euler’s formula, relation to trigonometry
• Irrationality of the Euler number e and Pi
• The Gamma function as an extension of the factorial function to positive real numbers
• Stirling's approximation of n! for large numbers is proved using the Gamma function and Wallis' product formula
• Riemann's rearrangement theorem

### Modelling 1A-1B

Students that do the excellence module for TW1050-A and TW1050-B will work in teams of at most two students (TW1050-A) or four students (TW1050-B) on one of the same assignments as in the regular practical. On top of the regular assignment, which includes modelling, analytical calculations, differential equations, optimization, numerical calculations, and writing a research report, they will have to produce:

• For each group of students, an extra substantial chapter on research related to the assignment, for example on:
• Theory/derivations of the methods used
• Numerical aspects
• Results for a substantial generalization of the problem in the assignment
The subject and scope of the additional chapter is determined in consultation with the tutor.
• Give a presentation on those extra results for the tutors of the course and interested students.

### Analysis 2

In Analysis 2 we will consider differentiation and integration on R^d. In the excellence module for Analysis 2, the theory will be complemented by an introduction into the topology of R^d and examples of Lie groups. The students will get reading material that they will have to present and explain in extra class sessions. The assessment will be kept the same as in the EP module of Analysis 1. We will in particular discuss:

• Homotopies
• Brouwer’s fixed point theorem
• Invariance of dimension and domain
• Jordan’s curve theorem
• Matrix Lie groups and Lie algebras

### Introduction to Probability

We will study more refined (than Chebychev) inequalities describing the deviation of the empirical mean from the expectation. First, we will study Cramér’s theorem, which is an elementary example of large deviations. Next, we will study concentration inequalities (Gaussian concentration bound, uniform variance inequality) and apply them to general functions of independent random variables. Examples from data science and from the study of networks are given. Finally, we will look at several forms of convergence (convergence in probability, almost surely, in distribution) of random variables and connections between them.