Summary

Hopf algebras arise naturally from the homology of spaces with multiplications (i.e. H-spaces, or “Hopf” spaces). In the language of category theory, the homology of such spaces is a group object in the category of coalgebras. When spaces have additional structure, this is reflected in their homology. For example, the spaces classifying generalized homology theories have a structure mimicking that of a graded ring. In turn, their homology has the structure of a Hopf ring, which is a ring in the category of coalgebras. Generalized homology theories lead to a generalization of the definition of a Hopf algebra to that of a Hopf algebroid, a cogroupoid object in the category of commutative algebras over a commutative ground ring.

The goal of this seminar is to study how Hopf algebras, rings, and algebroids lead to new computational techniques as well as elegant descriptions of answers in algebraic topology.

Here is draft seminar syllabus, subject to change.

Participant expectations

Participants will take turns giving presentations. The seminar organizers will make sure that presenters have written source material upon which to base their presentations. Participants are expected to attend regularly.

Participants are encouraged to skim the source material being presented each week. They are also encouraged to participate in the private shared online workspace.

As a general rule, students are expected to obtain 1-2 independent study credits from their home universities for their participation in the seminar.

Organizers

The seminar is organized by Sarah Petersen (University of Colorado). She will be assisted by Torgeir Aambø (NTNU) and Danika Van Niel (Michigan State University).

Application

Please submit the application form to apply. Applications received by December 11 will receive full consideration. We will respond to applicants by December 18. Contact Dan Isaksen for inquiries about applications.

Schedule

The seminar meets regularly on Tuesdays at 11:00am eastern time. The first meeting is on January 16, and the last meeting is on April 23. Informal office hours will be held on Thursdays at 12:00pm eastern time.