This is the homepage for the 2024 Summer Minicourses, a series of week-long graduate student-run minicourses at UT Austin.

This summer, the minicourses are being organized by Jacob Gaiter and Luis Torres. You can contact us at SMC.Organizers@gmail.com.

What are summer minicourses?

Minicourses focus on tools, methods, and ideas that aren't usually covered in prelims but are useful in topics classes/research. The idea is that a week-long minicourse will remain engaging, be easier to schedule, and help provide focus. These courses are primarily for graduate students, but all are welcome to participate!

Past courses have included:

  • Review of classes that were taught in previous years.
  • Primers for classes that will be taught next year.
  • Examples of useful computational tools.
  • Introductions to a subject/research area.

The 2024 Summer Minicourses have been scheduled!

Please check the schedule for updated abstracts and minicourse times. Meeting links for the minicourses will be sent to the mailing list and/or on the appropriate Discord Channel

If you are interested in participating in a minicourse, click here to join the SMC Discord.

This week's course:

Characteristic Classes, Chern-Weil Theory, and Principal Bundles

Instructor: Jacob Gaiter

When and where: July 22–July 26,
Lecture times: 11AM-12:30PM,
Problem sessions:2PM-3:30PM on MWF
In Person: PMA 9.166,
Zoom:https://utexas.zoom.us/j/9420321271

Abstract. This course will be an overview of some topics in the theory of characteristic classes: axiomatic treatments of Euler and Chern classes, a few applications, Chern-Weil theory for complex vector bundles, and its generalization to principal G-bundles.

The tentative outline for the course is:
Lecture 1: Vector bundles and their philosophy.
Problem Session 1: Proving basic facts about vector bundles.
Lecture 2: Oriented Vector bundles, Euclidean Vector bundles, and the Euler class.
Lecture 3: Complex Vector bundles and Chern Classes.
Problem Session 2: The Spitting principle.
Lecture 4: Connections and Principal bundles.
Lecture 5: Chern Weil Theory.
Problem Session 3: A few computations and applications.

As a prerequisite, the first half of my course will require a decent familiarity with, cohomology, Poincare duality, and differential topology (specifically, concepts like transversality, the tangent bundle, and the tubular neighborhood theorem). The differential topology aspects aren't essential to understand the actual content, but most of my examples come from differential geometry/topology.
The second half of my course will need comfort with the differential topology of Lie groups and their actions on smooth manifolds, as well as some understanding of differential forms, de Rham cohomology, and integration of forms. Lee's book on smooth manifolds is a good reference for the Lie groups and differential topology/geometry content, while Bredon's Topology and Geometry, Hatcher's Algebraic topology, or tom Dieck's Algebraic Topology should be more than enough for the required algebraic topology.


These courses were inspired in large part by the ones held at University of Michigan, which were started by Takumi Murayama.

You can click here to be added to the email list and click here to join the Discord server.