Admission requirements
Students following this course should have some understanding of discrete mathematics, logic, linear algebra and calculus.
Description
Mathematical structures appear everywhere throughout computer science, be it sets of states in automata, groups in cryptography, models in logic, graphs in algorithms; or vector spaces in machine learning. However, these structures are often only looked at in isolation from each other or as only single instances are considered. The aim of this course is to instead introduce students to thinking in terms of structures and their relations, and how these can help to solve concrete problems in computer science. This enables students to apply pure mathematical thinking in applications.
We will concretely focus on applications in continuous optimisation, simulation, and logic each with constraints that have to be modelled as so-called configuration spaces. For example, the tip of a robot arm cannot be in an arbitrary position, rather it is constrained by the rotation angles of its joints. These angles form the configuration space. In order to describe and solve problems in such applications, we will cover elements of abstract linear algebra, vector calculus, topological spaces, smooth manifolds, and vector/covector fields. The arising structures and their relations will be organised with basic category theory. We will pay special attention to structural thinking, learning how to write rigorous proofs and applying the gained insights in concrete projects in the three application areas.
Course objectives
Can explain the roles of configuration and phase spaces in modelling continuous control problems, and how these are realised as smooth manifolds.
Can calculate standard inner products, norms and derivatives (partial and total) on Euclidean spaces.
Understands the role of inner products, norms and distances in geometry and topology, and how they relate.
Can construct vector spaces, function spaces, norms, topological spaces and smooth manifolds, and can apply definitions to classify maps as linear, continuous or smooth.
Can prove that a structures and their maps form a category, and that mappings between structures yield a functor.
Can model continuous behaviour with ordinary differential equations, and continuous and differentiable paths on Euclidean spaces.
Can model continuous behaviour with on smooth manifolds with fields, smooth paths and trajectories.
Can program an optimiser/simulator for continuous problems with constraints modelled via configuration and phase spaces.
Schedule
In MyTimetable; you can find all course and programme schedules; allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.
Additionally; you can easily link MyTimetable to a calendar app on your phone; and schedule changes will be automatically updated in your calendar. You can also choose to receive email notifications about schedule changes. You can enable notifications in Settings after logging in.
Questions? Watch the video; read the instructions; or contact the ISSC helpdesk.
Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.
Teaching method
The course comprises
Weekly lectures
Weekly tutorials, where assignments are worked on in preparation for individual homework assignments
Weekly practical class, during which students can work on and receive help with their team project
Supplemental reading, when necessary
The projects are worked on in teams of two. Several possibilities are offered and students pick depending on their interest a project from the offering. These consist of an implementation and theoretical elements, like proofs of correctness for the implementation.
Assesment method
The assessment comprises three parts:
- Homework sets (individual grade): 10%, where each assignment has the same weight.
- Team project in three parts (group grade): 30%, where each part weighs 10%.
- Oral exam on project and course (individual grade): 60%
Instead of the oral examination a written examination may be held, should the number of participants be too high. Each part must have an average grade higher than 5.5 to pass the course.
It must be possible to evaluate every submission as the students' own work. Cases of plagiarism and fraud, including AI generated submissions, will be brought to the board of examiners.
Resit, review & feedback
The team project can be improved once, if the average grade is 5.5 or lower and for each of the three parts a serious attempt has been submitted by the deadline. The oral exam can be retaken. No retake is available for the homework assignment.
Feedback publication and review takes place on Brightspace.
Reading list
[LNS16] I. Lankham, B. Nachtergaele, and A. Schilling; Linear Algebra as an Introduction to Abstract Mathematics. World Scientific, 2016. doi: 10.1142/9808. Available: https://www.math.ucdavis.edu/%7Eanne/linear_algebra/ [Mor24] S. A. Morris; Topology Without Tears. 2024. [Online]. Available: https://www.topologywithouttears.net/ [Run05] V. Runde; A Taste of Topology. in Universitext. New York, Springer, 2005. doi: 10.1007/0-387-28387-0. [Lee12] J. M. Lee; Introduction to Smooth Manifolds, vol. 218. in Graduate Texts in Mathematics, vol. 218. New York, Springer; 2012. doi: 10.1007/978-1-4419-9982-5. [Tu11] L. W. Tu; An Introduction to Manifolds. in Universitext. New York, Springer, 2011. doi: 10.1007/978-1-4419-7400-6. [Jän01] K. Jänich; Vector Analysis. in Undergraduate Texts in Mathematics. New York, Springer, 2001. doi: 10.1007/978-1-4757-3478-2. [Rie16] E. Riehl; Category Theory in Context. in Aurora: Dover Modern Math Originals. Dover Publications, 2016. [Online]. Available: http://www.math.jhu.edu/\~eriehl/context/
Registration
Contact
Education coordinator LIACS bachelors