Abstract: This lecture series on von Neumann algebras consists of four lectures. Von Neumann algebras were discovered by von Neumann (and further developed together with Murray) in the thirties and were intended to give a rigorous explanation behind quantum mechanics. In the first lecture we present the definition of von Neumann algebras and states and give some first properties. In the second lecture, we continue with the first classification of factors (which are building blocks of von Neumann algebras), due to Murray and von Neumann. We end with a generalization of the semi-direct product (as in group theory) to von Neumann algebras, the so-called crossed product. In the third lecture we further explore the crossed product, in particular the famous construction of Krieger, of crossed product of discrete abelian dynamical systems. The final lecture deals with the Tomita-Takesaki theory and the classification of type III factors due to Alain Connes.

Abstract: The theory of elliptic curves is a fascinating field with many connections to algebraic geometry, number theory, complex analysis and even computational problems. In this talk, we introduce these objects in a very elementary manner, describe some of their properties and as an application, we show how they can be used to prove special cases of Fermat's Last Theorem.