4 hours of lectures per week.
Lecturer
Niels Lauritzen
Content
Maybe all of modern algebra springs out of attempts to formalize the process of solving equations. In linear algebra one learns how to solve systems of linear equations in several variables. In the theory of Gröbner bases these methods are extended to solve systems of polynomial equations in several variables. This extension of linear algebra leads to some breathtaking mathematics. The workhorse of the teory is in some sense the division algorithm in several variables. Compared to the classical division algorithm for polynomials of one variable, this is a relatively new method invented by Hironaka and Buchberger in the 1960s. The strange new phenomenon is that the remainder in polynomial division in several variables is not unique.
The first week will be a course on the theory covered in Algebra 1. From here we will move on to more advanced topics like the connection to convex geometry (term orders are parametrized by a certain convex polytope called the state polytope) and algorithms to find best term orders. Then we will enter toric ideals and toric geometry with applications to integer linear programming and statistics. Finally we will look at the world of algebraic geometry and commutative algebra through the (toric) Hilbert scheme.
The course will consist of a variety of highlights reflecting the young and unpolished nature of the subject. Many of the results presented are not more than 1 - 15 years old. Participants are expected to work out projects (which may be used as projects covering 2 out of the 8 points at the "2.delseksamen").
Prerequisites
Algebra 1.
Literature
Cox, Little and O'Shea : Varieties, Ideals and Algorithms, Springer, UTM. Sturmfels: Gr.obner bases and Convex Polytopes, AMS, University Lecture Series. Selected papers on current topics.