Directed Reading Program – Fall 2025

This is the Fall 2025 webpage for the Directed Reading Program.

Course Information:

Mentor: Junaid Aftab
Email: junaida@umd.edu
Mentee: Kelin Zhu
Meeting Times: Weekly

Description: Topological K-theory is a generalized cohomology theory built from vector bundles over topological spaces. It encodes stable isomorphism classes of vector bundles and has deep connections to algebraic topology and geometry. Central results include Bott periodicity and computations of K-groups for familiar spaces such as spheres and projective spaces. It also serves as a bridge to algebraic K-theory, index theory, and has applications in mathematical physics.

Deadlines: The following are the key deadlines for the semester:

  • September 26 – Submit project prospectus.
  • October 17 – Submit a brief project update (4–5 lines).
  • December 1 – Submit tentative talk title.
  • December 5 – Submit the final presentation to drpumd@gmail.com.
  • December 9 & 10 – Student talks scheduled.

Suggested References:

  • Allen Hatcher [Hat]. Vector Bundles and K-Theory (2003). PDF.
  • Max Karoubi [Kar]. K-Theory: An Introduction. Springer-Verlag, 1978. PDF.

Tentative Schedule:

Week Topic Reading
1 Vector Bundles: Definitions and Examples Hat 1.1, Kar 1.1–1.2 & 1.4–1.5
2 Classifying Vector Bundles: Clutching Functions, Classifying Spaces Hat 1.2, Kar 1.3 & 1.7–1.8
3 Exercises and Review
4 Grothendieck Completion & \(K^0\) Hat 2.1, Kar 2.2
5 Generalized Cohomology Theories TBD
6 Bott Periodicity & \(K\)-Theory as a Generalized Cohomology Theory Hat 2.2, Kar 2.3–2.5 & 3.1
7 Computations & Applications: Spheres, Projective Spaces, Division Algebras Hat 2.3, Kar 3.2
8 Fredholm Operators & Clifford Algebras Kar 3.3–3.5
9 TBD TBD
10 TBD TBD
11 TBD TBD
12 Presentation N/A