# NMAK23001U Applied Algebra and Geometry

Volume 2023/2024
Education

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject

Content

The aim of this course is to introduce the sstudent 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, and root classification for polynomials in one variable.

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.

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 Maple, Singular, Sage, Julia). 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 especially suited to students with an interest in combinatorics.

Learning Outcome

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, mixed volume.

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 references will be used:

- Cox, Litlle, O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra"

- Cox, Litlle, O'Shea, "Using Algebraic Geometry"

- Joswig, Theobald, "Polyhedral and Algebraic Methods in Computational Geometry"

Knowledge in analysis and linear algebra as covered in a BSc degree in mathematics. Ring theory e.g. as obtained in Algebra 2. Basic familiarity with programming is useful but not necessary.
3 hours of lectures and 4 hours of exercises for 7 weeks.
Exercise sessions combine theoretical exercises with practical exercises using mathematical software.
The students must bring their own laptop to the exercise classes.
• Category
• Hours
• Lectures
• 21
• Preparation
• 97
• Theory exercises
• 28
• Exam
• 60
• Total
• 206
Written
Oral
Individual
Continuous feedback during the course of the semester

The teachers will provide written or oral feedback to the written assignments, and feedback during the exercise classes.

Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Two 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.
Aid
All aids allowed
Marking scale