Program Overview

Program Overview

The University of Copenhagen offers a Master's program in Mathematics with a focus on Applied Algebra and Geometry. This program is designed to introduce students to the world of polynomials and polyhedral objects, and their many relevant applications in real life.

Program Description

The program covers two main topics:

• Foundations of applied algebraic geometry in the study of the zero-set of polynomial equations (an algebraic variety)
• Polyhedral geometry: polyhedra, cones, and polytopes (n-dimensional generalizations of polygons)

The program 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.

Learning Outcomes

By the end of the course, 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. The students will be 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
• Use and implement methods to find and describe solutions to polynomial equations using available mathematical software
• Identify main objects associated with polytopes and their relation to zero-sets of polynomial equations

Literature

The program will use material similar to the following references:

• Cox, Little, O'Shea, "Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra"
• Cox, Little, O'Shea, "Using Algebraic Geometry"
• Joswig, Theobald, "Polyhedral and Algebraic Methods in Computational Geometry"

Recommended Academic Qualifications

• Advanced Vector Spaces (AdVec)
• 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

Teaching and Learning Methods

The program includes 8 hours of exercises and discussion for 7 weeks. Exercise sessions combine theoretical exercises with practical exercises using mathematical software.

Workload

The program has a total workload of 206 hours, consisting of:

• Class Instruction: 56 hours
• Preparation: 90 hours
• Exam: 60 hours

Assessment

The program uses continuous assessment, with two written assignments counting for 30% of the grade each, and a final in-class problem set accounting for 40% of the grade. The final in-class problem set requires a laptop and is four hours long.

Exam Details

• Type of assessment: Continuous assessment
• Type of assessment details: Two written assignments and a final in-class problem set
• Aid: All aids allowed except Generative AI
• Marking scale: 7-point grading scale
• Censorship form: No external censorship, several internal examiners
• Re-exam: 27-hour take-home assignment

Course Information

• Language: English
• Course code: NMAK23001U
• Credit: 7.5 ECTS
• Level: Full Degree Master
• Duration: 1 block
• Placement: Block 2
• Schedule: A
• Course capacity: No limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student

Study Board and Department

• Study Board of Mathematics and Computer Science
• Department of Mathematical Sciences
• Faculty of Science

Course Coordinators

• Elisenda Feliu