Jérémy Blanc: Birational geometry of surfaces and Cremona group
An algebraic variety is given by the zero locus of some polynomials. A rational map between two algebraic varieties is given by quotients of polynomials. It is defined on open sets of points and there are closed subset of indeterminacies. One can however compose these where they are defined and define birational maps as rational maps having a rational inverse. For curves, we may solve the indeterminacies and obtain maps defined everywhere by simply taking smooth projective curves. In dimension 2, this is no longer true, but birational maps are nicely describable via sequence of so-called blow-ups and their inverses. I will describe all this slowly and illustrate it via examples. I will then relate this to the Cremona group, which is the group of birational maps from the plane to itself. No specific knowledge of algebraic geometry is needed to follow the course, as everything will be introduced.
Blanc: Birational geometry of surfaces and Cremona group
Peter Feller: Topics in knot theory with a view towards classical algebraic geometry
We cover topics in low-dimensional topology with a focus on knot theory---the study of circles and surfaces in 3-dimensional and 4-dimensional space.
The material will be motivated by a topological perspective on problems from other fields. For example, we consider questions from the study of zero-sets of polynomials and polynomial maps between them:
- How does topology help in distinguish singularities of zero-sets of polynomials (following Newton, Oldenburg, Artin, Brieskorn, Milnor ...)?
- What is the topology of algebraic curves in complex 2D-space and ovals in the Euclidian plane (Harnack's curve theorem)?
- What are the polynomial automorphism of C^2---the polynomial maps with polynomial inverses (Abhyankar-Moh-Suzuki theorem and Jung-van der Kulk theorem)?
- What is the topology of algebraic surfaces in complex 3D-space?
With this focus, the lecture will allow for synergies with the graduate lecture by Jeremy Blanc at different points.
I will make an effort to illustrate all concepts with many pictures and examples. This lecture will not rely on specific prerequisites from topology or algebra.''