Duration
24 Jun 2024 - 30 Jun 2024
The course will consist of four parts, with the topics being: introduction to Hilbert spaces, methods in spectral theory of operators in Hilbert spaces, introduction to algebraic geometry and introduction to algebraic topology.
The aim of the first part is to present some of more advanced topics of the theory of Hilbert spaces such as the direct sum of Hilbert spaces, linear forms and operators, adjoint operators, the lattice of orthogonal projections, and topologies on B(H).
The second part will be used to present topics from the standard spectral theory of self-adjoint operators and their applications in quantum mechanics. This will include spectral decomposition and spectral types.
In the third part an introduction of basic concepts and methods of algebraic geometry through the study of algebraic curves will be given. The main goal is a comparison of the topological and algebraic invariants of projective curves.
The final part will enable students to work with cell complexes, to understand the construction of the fundamental group of a topological space, to be able to use van Kampen´s theorem to calculate this group for cell complexes, and to understand the connection between covering spaces and the fundamental group.