COURSE GOALS:
 acquire knowledge and understanding of the pointset topology and algebraic topology
 understand the usage of these mathematical methods in physics
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.2 evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems;
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
5.2 remain informed of new developments and methods and provide professional advice on their possible range and applications
5.3 carry out research by undertaking a PhD
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Topology in Physics, the student will be able to:
1. Understand the basic concepts and tools in pointset topology
2. Understand the basic properties of manifolds and their usage in physics
3. Use the basic tensor calculus, including differential forms
4. Understand the meaning of the homotopy groups and their usage in physics
5. Understand the meaning of the cohomology groups and their usage in physics
6. Understand the geometrical picture behind the Lie groups and their important for physics
7. Understand the geometrical picture behind the fibre bundles and their usage in physics
COURSE DESCRIPTION:
The Fall semester (15 weeks)
1st week: Introduction to pointset topology (topological spaces, basis of topology)
2nd week: Connectedness and compactness
3rd week: Mappings between the topological spaces, homeomorphisms
4th week: Quotient spaces
5th week: Manifolds
6th week: Introduction to differential geometry (tensor calculus and its usage in physics)
7th week: Homotopy (fundamental group and higher homotopy groups)
8th week: Usage of homotopy in physics
9th week: Homology
10th week: Differential forms and their usage in physics
11th week: Cohomology and its usage in physics
12th week: Introduction to Lie groups
13th week: Fibre bundles
14th week: Examples of fibre bundles in physics
15th week: Student seminars
GRADING AND ASSESSING THE WORK OF STUDENTS:
During the semester each student gets several simpler problems which they have to solve and shortly present next week in front of the other colleagues. Students are encouraged to discuss these presentations and participate with additional questions. Also, each student gets one seminar topic which s/he has to present in a written and oral form at the end of the semester. The final grade is based upon the quality of this final seminar.
