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.
Abstract to be given later
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.
This course introduces the basic concepts of noncommutative algebras and their modules. It can be done in English or Portuguese.
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).
Content:
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.