Mathematics

The course introduces physical quantities and fundamental laws of electrical circuits. The basics of direct and alternating current networks are explained allowing the evaluation of complex electric networks.

In this hands-on course, students are introduced to the models and theory necessary to develop computational skills in the field of financial mathematics. Covering topics such as the Monte Carlo method, stochastic models, the binomial tree model, the theory of risk-neutral pricing, derivative pricing and the interpretation of random variables, students learn how computational methods can be used to evaluate different financial scenarios. During supervised programming sessions, which include an introduction to programming in Python, students have the opportunity to implement the computational methods introduced to students using relevant examples. 

This class addresses topics from network structure and growth to the spread of epidemics. The course studies diverse algorithmic techniques and mathematical models that are used to analyze such large networks, and give an in-depth description of the theoretical results that underlie them. Some topics are random graphs, giant components, power laws, percolation, spreading phenomena, community detection, basic algorithms for network science, lower bounds and advanced algorithms for polynomial-time problems, sampling algorithms, streaming algorithms, sublinear algorithms, and graph partitioning algorithms.

The course assumes basic skills in algorithms and mathematics: familiarity with basic graph algorithms (shortest paths, flows), and basic understanding of NP-completeness. Work with basic probabilities and some integrals in included.

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.