Duration
15h Th
Number of credits
Lecturer
Language(s) of instruction
English language
Organisation and examination
All year long
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
This course is intended for students who wish to explore topological concepts as applied to polar and magnetic textures. It has a threefold objective:
(i) To introduce topological spaces and their relevance in various contexts of physics, with a particular focus on materials physics.
(ii) To develop the notions of homotopy and homotopy groups in order to describe singularities in vector fields, taking into account the dimensionality and topology of these fields.
(iii) To illustrate these concepts through the study of topological structures such as merons, vortices, and skyrmions, highlighting their implications in materials physics, their functional properties such as chirality, as well as their potential applications.
Learning outcomes of the learning unit
By the end of this course, students will have acquired a range of competencies categorized into two groups: generic competencies and specific competencies.
Generic Competencies:
- Application of Knowledge: Students will learn to apply abstract mathematical knowledge to their professional work or vocation effectively.
- Communication Skills: They will develop the ability to communicate ideas, problems, and solutions from the fields of mathematics and physics to both specialized and non-specialized audiences.
- Research Skills: Students will acquire skills in utilizing bibliographic and internet resource search tools.
- Scientific Reading: They will gain proficiency in reading and comprehending scientific texts written in English.
Specific Competencies:
- Mathematical Language Proficiency: Students will familiarize themselves with mathematical language and learn to abstract structural and fundamental properties, distinguishing them from purely incidental ones.
- Topology of Vector Fields: They will develop a solid foundation in the topology of vector fields and its application to the characterization of polar and magnetic textures.
- Introduction to Topological Spaces: Students will gain an introduction to topological spaces and homotopy theory, equipping them with essential tools applicable to various fields of condensed matter physics, including the study of topological insulators, the quantum Hall effect, and superconductors.
- Analysis of Topological Structures: They will develop the skills to classify and analyze complex topological structures such as merons, vortices, and skyrmions, gaining a deep understanding of their significance in materials physics, a field that has garnered significant attention over the past decade.
Prerequisite knowledge and skills
Basic concepts of solid-state physics.
Basic concepts of group theory and algebra: Concept of a group, equivalence relation and quotient group
Basic concepts of complex analysis.
Planned learning activities and teaching methods
The course will be delivered through a combination of face-to-face sessions led by the professor, integrating theoretical lectures, blackboard demonstrations, and slide presentations to facilitate understanding. To enhance learning, practical hands-on sessions will be included, allowing students to engage directly with the characterization of structures. These sessions will reinforce the theoretical concepts discussed in class and provide valuable experience in applying knowledge to real case scenarios.
Mode of delivery (face to face, distance learning, hybrid learning)
Face-to-face course
Further information:
The course will be delivered primarily in person, with the possibility of occasional exceptions via remote (online) sessions.
Course materials and recommended or required readings
Platform(s) used for course materials:
- eCampus
- MyULiège
Further information:
Munkres, J. R. (2000). Topology (2nd ed.). Prentice Hall.
Milnor, J. W. (1965) Topology from the differentiable viewpoint. The University Press of Virginia, Charlottesville.
Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
Junquera, J. al. Rev. Mod. Phys. 95, 025001 (2023). Topological phases in polar oxide nanostructures.
Material provided by the professor.
Exam(s) in session
Any session
- In-person
written exam ( open-ended questions )
- Remote
written exam ( open-ended questions )
Continuous assessment
Work placement(s)
Faculty of Science
Organisational remarks and main changes to the course
The course will be conducted primarily in person, with the possibility of a few remote sessions in exceptional cases.
Lecture materials and complementary resources will be provided by the professor through the university's learning platform.
Office hours will be scheduled weekly and available both on campus and online by appointment.
Contacts
Contact at the University of Liège: fgomez@uliege.be