The course treats curves in R3, curvature, torsion, Frenet’s formulae, surfaces in R3, the first and second fundamental form, geodesics, Gauss curvature and Theorema egregium, Gauss- Bonnet’s theorem, differential forms and Stokes’ theorem.

The course covers modern methods of cryptography, which form a basis for secure electronic communication, and methods to decryptize these. The focus is on the mathematical foundations in number theory, algebraic geometry and statistics and how these are used in cryptography. The course is of interest for those who work with security aspects of electronic communication, but also for those who want to see one of the more spectacular modern applications of mathematics.

Course contents 

The focus is on structure of linear transformations, quadratic forms and, if time allows, Hankel norm approximation by self study and student seminars. More precisely, it will cover the following topics, introduction to module theory, basic properties of the shift operator, circulant matrices, Hermit interpolation, duality reproducing kernels, cyclic transformations, the invariant factor algorithms, operators in inner product spaces, Sylvester's law of inertia, Bezoutians and their representation, Hankel matrices, orthogonal polynomials, Cauchy index, root location and Nevanlinna-Pick interpolation. 

Course description: The course treats mathematical tools which are common in economics modeling and applications. Topics include: ordinary differential equations, static and dynamic optimization, calculus of variations, optimal control and introduction to stochastic processes (particularly the Wiener process) including simulations and applications (e.g. option pricing)

Kursen behandlar linjära differentialekvationer med konstanta och variabla koefficienter, existens- och entydighetssatser, plana autonoma system, numeriska lösningsmetoder, Laplace-transform.

Foundations of number theory, quadratic reciprocity, unique factorization of ideals in algebraic number fields, the finiteness of the class number, Dirichlet's theorem on primes in arithmetic progressions.

The course treats the foundations of general set topolog (topological spaces, continuity, compactness, connectedness, identification topologies) , the fundamental group, classification of closed surfaces.