African Mathematical School
Algebraic Structures, Cryptography, Number Theory and Applications
Praia, Santiago, Cape Verde
April, 13th to 28th, 2015

Universidade de Cabo Verde

Courses in Algebraic Structures and Applications

Finite Groups (6 lectures), by Natalia Furtado and Tetyana Goncalves (Universidade de Cabo Verde, Cape Verde):

Examples of groups, cyclic groups, Lagrange Theorem. Fundamental Theorem on finitely generated abelian groups. Symmetric groups. Simple groups, classification of simple finite groups. Group actions, orbits and stabilizers. Sylow theorems. Groups of order pq.

Caracters of Finite Groups (2 lectures), by Iryna Kashuba (Universidade de São Paulo, Brazil):

Abstract to be given later

Invariant Rings of C[X,Y] under Finite Subgroups of SL_2(C) (4 lectures), by Patrick Le Meur (Université Paris Diderot - Paris 7, France):

Link to abstract.

Introduction to Lie Theory (4 lectures), by Vyacheslav Futorny (Universidade de São Paulo, Brazil):

We will give an Introduction to the representation theory of Lie algebras. We start with basic examples and definitions of Lie algebras, their connections with Lie groups. We will briefly discuss representations of solvable and nilpotent Lie algebras, Theorems of Lie and Engel. Main focus of the course will be on the classification of finite dimensional simple Lie algebras via Dynkin diagrams. Finally we will discuss the clasical classification of simple finite dimensional representations of the above algebras and more modern topics related to the category O, quasi-hereditary algebras and Kazhdan-Lusztig Conjecture.

Modules over Non-Commutative Algebras (5 lectures), by Eduardo Marcos (Universidade de Saõ Paulo, Brazil):

This course introduces the basic concepts of noncommutative algebras and their modules. It can be done in English or Portuguese.

Octonions and Other Nonassociative Algebras (4 lectures), by Alberto Elduque (Universidad de Zaragoza, Spain):

After reviewing the process of the construction of the complex numbers from the real numbers, this process will be iterated to produce the algebra of quaternions. Applications of this algebra to the study of rotations in the Euclidean spaces of dimension 3 and 4 will be considered. A further iteration provides the algebra of octonions. Some geometrical applications of this algebra will be presented. Finally, other nonassociative algebras will be considered, mainly the exceptional simple Jordan algebra (the Albert algebra).

Algebra in Control Theory (3 lectures), by Zbigniew Bartosiewicz (University of Byalistok, Poland):

I will show how algebraic structures appear in mathematical control theory and how algebra is used to solve control problems. First lecture will be devoted to linear control systems described by linear differential equations. I will discuss stability, controllability and feedback stabilizability of control systems, and show how these properties can be characterized in the language of matrices, linear spaces and linear maps associated to the system. The second lecture will be devoted to nonlinear systems, described by analytic differential equations. I will study the problem of local observability where the rings of germs of analytic functions will be exploited. The properties of the ring and certain ideals of this ring will appear to be important in finding a characterization of local observability. In particular real radicals of ideals and corresponding geometric objects will be essential in this study.