Real Algebra and Geometry

General Information

In the field of Real Algebra, the main basic objects are orderings of algebraic structures. We will particularly focus on ordered fields (examples are the rational and real numbers) and see that the set of sums of squares plays a special role in this theory. In particular, we will discuss Artin's solution of Hilbert's 17th problem. We will also study quantifier elimination and basics in semi-algebraic geometry (an example of a tame geometry). Towards the end of the semester, we will also study questions in real algebraic geometry, in particular real questions in classical projective geometry.
Prior knowledge in abstract algebra is sufficient. Basics in algebraic geometry and commutative algebra are helpful but not necessary.

Materials

You will find materials for the course mainly on Moodle, the learning platform used by the University of Leipzig. Here is a direct link to the course. Members of the MPI MiS should be able to login using the DFN AAI Login button, which is on the right bottom side of the Moodle website (this is not the regular login for members of the university).

Sources

Here is a list of useful references that we will use throughout the course (see Moodle also).

Exercises

The course will start in the first week of classes (12. April) and run until the end of term (24. July).