Algebraic and Geometric Methods in Engineering and Physics — 1st Semester 2020/2021

Announcements

The seminars will be held on 15 February (from 14:30) and on 22 February (from 10:00 and from 14:30) on this Zoom session, and will have the duration of 25 minutes (+5 minutes for questions).

A PDF file containing the slides for the presentation must be handed in by the end of the day of the presentation (at the latest).

The subject may be any application of algebra, geometry or topology to physics or engineering, and should be explained at a level appropriate for students of this course (all of whom will be attending).

Here are some suggestions of possible seminar subjects:

Crystallographic groups;

Error-correcting linear codes;

Finite geometry and the card game Dobble;

Frieze groups;

Gauge theories and the Standard Model of particle physics;

Molecular symmetries;

Noether's theorem;

Representations of SU(2) and spin;

Representations of SU(3) and quarks;

Rubik's cube group

Spherical harmonics and atomic orbitals;

Topological data analysis.

Syllabus

Topics of algebra and applications: Rings, fields and modules. Groups, actions and representations of finite groups. Applications: Information security, vibrations of symmetric structures, structural optimization using representations of finite groups.

Topics of geometry and topology and applications: Elements of topology. Topological spaces and metric spaces. Fundamental group and coverings. 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. Morse theory. Aplications: data science and persitent homology, applications of Morse theory to big data, cosmological models.

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. Aplications: Statics and dynamics of robots, grassmanians and flag manifolds, particle physics and theories of unification.

Bibliography

We will use these lecture notes, which will be updated as the course progresses. See the bibliography in the notes for more references.

Grading Policy

Seminar: Students must prepare and present a seminar explaining some application of algebra, geometry or topology to physics or engineering. The classification obtained in this presentation will make up 50% of the final grade.

Exam: There will be an exam counting 50% towards the final grade. Students will be able to repeat this exam if necessary.

Presentations

• Application of Differential Geometry to study Mechanical Systems
• Braid Group and Anyons
• Crystallographic Groups I
• Crystallographic Groups II
• Entropy and Recurrence with applications to Statistical Physics
• Error-correcting linear codes
• Noether's Theorem
• Quaternions: theory and some applications
• Representations of SU(3) and quarks
• Spin-one Color Superconductor
• Stabilizer Codes
• The Rubik's Cube Group
• Topological Data Analysis

Final exams

• Practice Exam 1
• Practice Exam 2
• Exam 1
• Exam 2