Métodos de Álgebra e Geometria em Engenharia e Física

Objectives

To introduce topics of algebra, geometry and topology and to give examples of their applications in physics and engineering. This course is adequate to physics and engineering students interested in learning more mathematics with applications to e.g. information security, particle physics, data science, robotics, structural stability, etc. Useful also to mathematics students interested in recent applications of mathematics.

Syllabus

Topics of algebra and applications: Relations in a set. Groups, actions and representations of finite groups. Applications: information security. Vibrations of symmetric structures. Topics of geometry and topology and applications: elements of topology. Topological spaces and metric spaces. Fundamental group. Simplicial complexes and homology. Manifolds and tensor fields. Riemannian manifolds. Forms and integration. De Rham cohomology. Flows of vector fields. Lie derivatives and symmetry group of a tensor field. Applications: data science and persistent homology. Lie algebras, Lie groups and applications: Lie groups and Lie algebras. Compact Lie groups and their Lie algebras. Root systems. Elements of the theory of representations. Lie group actions on manifolds. Applications: Cosmological models. Applications of grassmanians and flag manifolds to statics and dynamics of robots. Particle physics and theories of unification.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.