AGQ: Differential Topology MATHS5084

  • Academic Session: 2024-25
  • School: School of Mathematics and Statistics
  • Credits: 15
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Either Semester 1 or Semester 2
  • Available to Visiting Students: Yes
  • Collaborative Online International Learning: No

Short Description

The course will start with a refresher about manifolds, vector bundles and fiber bundles. It will then discuss in detail various aspects of calculus on manifolds, introducing the Lie derivative, the exterior derivative, connections, holonomy, curvature and the Stokes Theorem. This will be followed by an exposition of de Rham cohomology and its relation to Cech cohomology. Finally, the central part of the course is an introduction to Chern-Weil theory and characteristic classes. If time permits, and based on the interests of students, additional topics such as elliptic operators and Dirac operators may be touched.

Timetable

2 hour sessions weekly.

The AGQ program will be setting the timing.

 

Classroom as well as equipment for online delivery to transmit the course to Edinburgh.

Requirements of Entry

This is a non-required course for students in the AGQ programme. With permission from AGQ directors students not enrolled in AGQ may also take this course.

Excluded Courses

none

Co-requisites

none

Assessment

Normally students will complete course work.

Course Aims

This course covers several fundamental topics in Geometry & Topology, providing a solid and broad foundation to the subject and its interactions with algebra and analysis. The course discusses the vector bundles, connections, principal bundles, characteristic classes, etc.

Intended Learning Outcomes of Course

On completion of this course, the student will be able to: • Thoroughly understand the applications of differential topology to the local and global geometry of manifolds. • Solve problems involving characteristic classes, vector bundles, differential forms, etc. from first principles.

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.