NMAK21001U Introduction to Applied Algebra and Geometry
The aim of this course is to introduce the beautiful world of polytopes, cones, polyhedra and polynomials, and their many relevant applications in real-life.
Specifically, the course covers two mains 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, and root classification for polynomials in one variable.
2- Polyhedral geometry in the theory of polyhedra and polytopes (n-dimensional generalizations of polygons) and their combinatorics.
The two topics of the course meet with the so-called Newton polytope, and relations between the zero-set of a system of polynomials and geometrical properties of an associated polytope beautifully emerge.
During the course, the relevant theory will be developed, and in the exercise classes, the student will put the theory in practice by using appropriate mathematical software (for example Maple, Singular, Sage). Additionally, the student will work with several applications to real life, such as chemistry, biology, robotics, optimization, coding theory, as well as computational theorem proving, among others.
The course is suitable for master students and last-year bachelor students. For the two written assignments, the student will be able to choose among several options, according to background and interest. In this way, master students familiar with algebraic geometry will be able to relate abstract concepts from algebraic geometry to the applied aspects of the course.
In particular, this course serves as a good complement to other master courses in algebraic geometry and is also especially suited to students with an interest in combinatorics.
Knowledge: The students are able to define, describe the main properties of, and use in practical situations the following: algebraic varieties, Gröbner bases, elimination theory, techniques for finding and classifying the roots of polynomials in one variable, polytopes, convex sets, Newton polytope.
Skills: By the end of the course the students are able to use and implement methods to find and describe solutions to polynomial equations using available mathematical software, and to identify main objects associated with polytopes and their relation to zero-sets of polynomial equations.
Competences: By the end of the course the students will have developed a theoretical and practical understanding of the main aspects and current trends in the field of applied algebraic geometry and polyhedral geometry, and be able to use this knowledge in theoretical contexts and in applications.
See Absalon. Material similar to the following two references will be used:
- Cox, Litlle, O'Shea, "Ideals, Varieties, and. Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra"
- Joswig, Theobald, "Polyhedral and Algebraic Methods in Computational Geometry"
Academic qualifications equivalent to a BSc degree in mathematics are recommended.
Exercise sessions combine theoretical exercises with practical exercises using mathematical software.
- Theory exercises
- 7,5 ECTS
- Type of assessment
- Continuous assessmentTwo written assignments count each 30% of the grade. A final in-class problem set (requires laptop), three hour long, accounts for 40% of the grade.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
Several internal examiners
27-hour take-home assignment.
Criteria for exam assesment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.