applied topology

595TA: Tuesdays-Thursdays 12:30-1:50pm

We meet in Transportation Building 204.

The course is an introduction into the tools of algebraic topology applicable to problems of engineering and science. The emphasis will be on operational side of the theorems, not on proving them.

Lecture plans and other materials will be posted here.

Potential topics for presentations and course projects are posted here.

To contact regarding this course, either comment on the course posts, or email.

We will be using some combination of recent textbooks (such as Rob Ghrist’s text available online and Hatcher’s textbook, also available online) and research papers. A computational project on one of the covered topics and/or a short presentation are expected from the participants.

Necessary prerequisites include good grasp of multivariate calculus and linear algebra.

Syllabus:

  • Useful topological spaces: manifolds; simplicial complexes; examples. (1 week)
  • Overview of tools of algebraic topology: homotopy equivalence, homologies (singular); elements of de Rham and Hodge theorems. Basic algebraic tools will be addressed as needed.  (2 weeks).
  • Bjorner-Lovasz-Yao theory on lower bounds of decision trees. (1 week)
  • Definable sets, constructible functions and Integrals with respect to Euler characteristic and applications (2 weeks)
  • Topological data analysis: shape of data, persistence, cyclicity (2 weeks)
  • Alexander duality and applications (caging in robotics) (1 week)
  • Classical configuration spaces; their cohomologies, Arnold-Brieskorn relations. Configuration spaces on graphs; hard disk configuration spaces (2 weeks)
  • Spaces of directed paths, their topology and applications (2 weeks)