Mathematics

The course providers an introduction to classic differential geometry, important for further studies in the subject and in relevant areas of physics. The course treats the geometry of curves and surfaces, especially in three dimensions. In particular, the concepts of curvature and torsion are studied. The course covers: The geometry of curves in Euclidean space, their curvature and torsion and how these determine the curves. The geometry of surfaces in Euclidean space, their first and second fundamental forms, the Gauss map, principal curvatures, Gaussian curvature and mean curvature. Theorema Egregium and a deep analysis of geodesics and their behavior both locally and globally. Gauss-Bonnet's Theorem: two different local versions and the famous global version.

This course clarifies relations between the fundamental groups and the Galois groups. As Galois groups can be seen as etale fundamental groups of the base field, the algebraic fundamental groups of algebraic curves (or even schemes) can also be regarded as an etale realization of more general objects, which is the point of view proposed by Grothendieck. The course investigates the algebraic fundamental groups from this point of view. Topics include infinite Galois theory and finite etale algebras of fields; Galois covers and monodromy actions; universal covers and local systems; riemann surfaces; algebraic curves; fundamental groups of algebraic curves.
 

This course provides the necessary mathematical skills for other physics courses. Topics include: complex numbers and hyperbolic functions; single-variable calculus; Taylor series; first order and second order ordinary differential equations; vectors and matrices; eigenvalues and eigenvectors; partial differentiation; multiple integrals; and physical applications. The course requires students to take prerequisites.

The objective of the course is to teach the student more advanced mathematical tools an d methods that are useful in physics, and to apply these methods on concrete physical systems. Topics include analytic functions, special functions, Fourier analysis: Laplace transforms, ordinary differential equations, partial differential equations, and Green's functions. 

The course covers the concepts and methods of mathematical language. The focus is more on the analytic and topological notions such as convergence and continuity, which are essential for a rigorous treatment of mathematical analysis. The ability to read and write mathematical proofs is also further developed in this module. Topics include real numbers, sequences and series of real numbers, metrics in Euclidean spaces, open and closed sets, continuous functions, compact sets, connected sets, sequences of functions. Major applications include intermediate value theorem, extreme value theorem.

This course is in the interdisciplinary field of icing in relation to aircraft. Ultimately, this course will draw from mathematics, physics, chemistry and engineering to provide attendees with a broad overview of the field of aircraft icing, and how the problem may be approached mathematically. This  involves understanding the problem, discussing the current state of engineering solutions, and study of how mathematics can help to improve, enhance and further this field. Modelling of this phenomena is a threefold approach. Firstly, the trajectory of particles within the fluid flow concerning an oncoming aircraft is calculated. Secondly, the behavior and mechanics of impinging particles (particles that make contact with the aircraft) needs to be understood. Thirdly, how ice builds up on a surface alongside the possibility of it shedding are important.
This course serves as an introduction to understanding this field and the analytical modelling of this problem.