This is a 28-lecture course with each lecture lasting about 30 minutes.
This course will cover a miscellaneous collection of topics in combinatorics and closely related fields. What the topics have in common is that they all involve proofs that at one time surprised experts by their simplicity. Sometimes these were the first proofs of long-standing open problems, and sometimes they were new proofs of results that had previously been established by much longer arguments. Several of these arguments use ideas and techniques that have gone on to be used by many other people.
Another theme of the course is the sheer diversity of methods that are used in combinatorics. We shall see uses of probability, linear algebra, linear analysis, topology, entropy, multivariate polynomials, tensor rank, concentration of measure, and more. (There will also be one or two arguments that are completely elementary.)
In the lectures we will cover the basic concepts of modern differential geometry. Differential geometry studies smooth manifolds, that is, geometric objects that, roughly speaking, locally look like ℝn (and whose global topological properties are not too weird, see for example the so-called 'long line'). You already know examples such as the n-sphere Sn, or smooth surfaces in ℝ3 from analysis. On smooth manifolds we will study a number of constructions and structures, such as vector fields, metrics and various curvature concepts. In addition, smooth manifolds are suitable as spaces for ordinary and partial differential equations, which allow different global topological properties compared to regions in ℝn (e.g. PDEs on the Klein bottle or on the real-projective spaces ℝPn). A focus of this lecture will be submanifolds and induced geometric structures. We will also look at some topics from the perspective of the calculus of variations, e.g. geodesics as critical points of the energy functional. This lecture is also expressly suitable for students of physics courses, as differential geometry represents a fundamental theoretical basis for many modern theories in physics (especially ART, gauge theories such as Yang-Mills -Theory, SuSy, SuGra,...).
