Starts 3 Aug 2020

Ends 14 Aug 2020

Central European Time

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EAUMP-ICTP School: Topics in Concrete Mathematics | (smr 3465)

Starts 3 Aug 2020

Ends 14 Aug 2020

Central European Time

Kigali - Rwanda

This School offers the students concrete and effective mathematical tools from algebra, group theory and geometry that can be applied to any scientific field. Below is a description of the courses.

**Advanced Linear Algebra**

The course starts by reviewing some basic ideas from a first course in linear algebra: matrices up to similarity, eigenvalues, eigenvectors, characteristic polynomial and diagonalisation. It then moves on to the Jordan canonical form. The aim is to cover also bilinear forms, multilinear forms and tensor products. After which we introduce the tensor, symmetric and exterior algebra with a view to representations of the general linear group.

**Groups, Counting and FFT**

We introduce the basic definitions in group theory using finite groups of rotations in 3-dimensional Euclidean space as examples. We then go on to use it in counting problems culminating in counting the number of ways one may colour faces of a cube with m-colours up to rotational symmetry. All of these lead to Maschke’s theorem and basic character theory. The aim is to use them to calculate character tables of some basic examples of groups and decompose some natural representations. Some applications of representation theory of finite groups, like the Fast Fourier Transform FFT, will be discussed.

**Introduction to Lie Algebras**

Lie Algebras are mathematical objects which, besides being of interest in their own right, elucidate problems in several areas in mathematics. The classification of the finite-dimensional complex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are to introduce Lie algebras, develop some of the techniques for studying them, and describe parts of the classification mentioned above, especially the parts concerning root systems and Dynkin diagrams.

**Modular Forms**

Modular forms are special functions on the complex upper half-plane. They are characterised by a functional equation which depends on an integer k, called the weight. These functions have beautiful properties - for example, the space of modular forms of a given weight is finite dimensional. Modular forms are naturally connected with many areas in mathematics, including the theory of lattices and quadratic forms, and various problems in number theory. The course will give an elementary introduction to modular forms. We will see how the space of modular forms of any weight k can be calculated explicitly. We will also study concrete examples of modular forms and their properties in detail, including the so-called Eisenstein series and the discriminant function.

The course starts by reviewing some basic ideas from a first course in linear algebra: matrices up to similarity, eigenvalues, eigenvectors, characteristic polynomial and diagonalisation. It then moves on to the Jordan canonical form. The aim is to cover also bilinear forms, multilinear forms and tensor products. After which we introduce the tensor, symmetric and exterior algebra with a view to representations of the general linear group.

We introduce the basic definitions in group theory using finite groups of rotations in 3-dimensional Euclidean space as examples. We then go on to use it in counting problems culminating in counting the number of ways one may colour faces of a cube with m-colours up to rotational symmetry. All of these lead to Maschke’s theorem and basic character theory. The aim is to use them to calculate character tables of some basic examples of groups and decompose some natural representations. Some applications of representation theory of finite groups, like the Fast Fourier Transform FFT, will be discussed.

Lie Algebras are mathematical objects which, besides being of interest in their own right, elucidate problems in several areas in mathematics. The classification of the finite-dimensional complex Lie algebras is a beautiful piece of applied linear algebra. The aims of this course are to introduce Lie algebras, develop some of the techniques for studying them, and describe parts of the classification mentioned above, especially the parts concerning root systems and Dynkin diagrams.

Modular forms are special functions on the complex upper half-plane. They are characterised by a functional equation which depends on an integer k, called the weight. These functions have beautiful properties - for example, the space of modular forms of a given weight is finite dimensional. Modular forms are naturally connected with many areas in mathematics, including the theory of lattices and quadratic forms, and various problems in number theory. The course will give an elementary introduction to modular forms. We will see how the space of modular forms of any weight k can be calculated explicitly. We will also study concrete examples of modular forms and their properties in detail, including the so-called Eisenstein series and the discriminant function.

**DEADLINE: 05/04/2020**

Omololu Akin-Ojo (EAIFR), Celestin Kurujyibwami (Rwanda), Balazs Szendroi (Oxford), ICTP Scientific Contact: Fernando Rodriguez Villegas