Cancelled Applied Algebra and Geometry

The aim of this course is to introduce the students to the beautiful world of polynomials and polyhedral objects, and their many relevant applications in real life.

Specifically, the course covers two main topics:

1. Foundations of applied algebraic geometry in the study of the zero-set of polynomial equations (an algebraic variety). It includes Gröbner bases of polynomial ideals, elimination theory, eigenvalue methods to solve polynomial equations, and homotopy continuation.
2. Polyhedral geometry: polyhedra, cones and polytopes (n-dimensional generalizations of polygons). Polyhedra arise naturally when studying systems of polynomial equations, and relations between the zero-set of a system of polynomials and geometrical properties of an associated polytope beautifully emerge.
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4. During the course, the relevant theory will be developed, and in the exercise classes, the student will put the theory into practice by using appropriate mathematical software (for example the package Oscar in Julia, Maple, Singular, Sage). Additionally, the student will encounter several applications to real life, such as chemistry, biology, robotics, optimization, statistics, coding theory, as well as computational theorem proving, among others.

The course is suitable for master students and last-year bachelor students.

Master students familiar with algebraic geometry will be able to relate abstract concepts from algebraic geometry to the applied aspects of the course, but this is not a requirement to follow the course. In particular, this course serves as a good complement to other master courses in algebraic geometry and is also suited to students with an interest in combinatorics.