Stable homotopy theory, Winter 2026

Summary

This course is an introduction to stable homotopy theory. After taking this course, students will be able to work with spectra in a practical manner, perform computations, and engage in research projects in stable homotopy theory or related fields (such as algebraic geometry or geometric topology) that use stable homotopy techniques.

A potential list of topics includes Spectra; Stable homotopy groups; Extraordinary homology and cohomology theories; Stable homotopy category; Smash product; Associative and commutative ring spectra; Spectral sequences; Operads; Homotopy limits and colimits; Spectra and manifold theory.

A draft syllabus will be available soon, but subject to change. We will draw from a variety of sources. The syllabus provides more detail, but beware that it is subject to change.

Participant expectations

This course will be taught in a flipped-classroom format, with all course meetings online. Before each course meeting, students are expected to either read from a reference or watch a short video. Classes will consist of a short recap of the material by the instructor, followed by group work on problem sets. These problem sets will be due for a grade at the beginning of the following week.

Each student is expected to participate fully in the course, just as if it were an in-person course at the student’s home institution. Attendance at all live meetings is required, and active participation will sometimes be expected. Students are expected to spend additional time each week engaged with the course material on their own.

As a general rule, students are expected to obtain 2-3 independent study credits from their home universities for their participation in the course. At the end of the semester, a grade of A/B/C/Fail will be reported to each student’s home institution advisor.

Prerequisites

Students should have taken the equivalent of two semesters of algebraic topology at the graduate level, including homology, cohomology, and higher homotopy groups. Students should have a working knowledge of category theory, including but not limited to: natural transformations, limits, colimits, adjunctions, abelian categories, and symmetric monoidal categories. Exposure to K-theory, characteristic classes, cobordism, and other topics in algebraic topology is not required, although it may be helpful.

Instructors

Kiran Luecke (University of Illinois Urbana-Champaign) is the lead instructor. Graduate teaching assistants are Mattie Ji and Jesse Keyes.

Application

Please submit the application form to apply. Applications received by December 5 will receive full consideration. Unfortunately, space is limited, and we may not be able to accept all qualified applicants. We will respond to applicants by December 12.

Schedule

The course meets on Tuesdays and Thursdays at 1:00-2:00pm eastern time, starting 20 January and ending on 30 April. The course will not meet on 17 March and 19 March. Applicants should tentatively reserve these times in their schedule.