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.
Schedule

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

The course treats Galois theory for finite field extensions.

Schedule

The focuses of the course are on the theory of convex sets and functions and its connection with a number of topics that span a broad range from continuous to discrete optimization.

Course Content

The course covers signed measure, Hahn decomposition, measures on metric spaces, Radon-Nikodym theorem, Lebesgue decomposition, dual spaces, weak topologies, Banach-Alaoglu theorem, adjoint operators, compact operators and their spectrum, Fredholm alternative, Hilbert spaces and operators on Hilbert spaces, spectral theory of self-adjoint operators in Hilbert space, Fredholm determinant, unlimited operators.