Talbot 2024

Mentored by Jeremy Hahn of MIT and Allen Yuan of Northwestern University

Topological cyclic homology is a rapidly developing subject sitting between homotopy theory and (via the work of Bhatt–Morrow–Scholze) p-adic arithmetic geometry. There are now many excellent introductions to topological cyclic homology that focus on discrete commutative rings and spherical group rings. We aim to give a computationally focused introduction to the topological cyclic homology of finite height ring spectra. The topic is also closely connected to (MU-based) synthetic spectra, and may help familiarize students with their use in computations.

Outline: The workshop will proceed by discussing THH and its concomitant structures: the motivic filtration, the circle action, and the Frobenius map. We will follow the discussion of each structure with computations in the examples of THH(Fp), THH(Z), and THH(ku), with the ultimate goal of understanding syntomic cohomology and Lichtenbaum–Quillen theorems. Time permitting, we will conclude with connections to the algebraic K-theory of ring spectra, chromatic redshift, the telescope conjecture, and prismatization.

Target audience: The workshop will be aimed toward graduate students with a basic understanding of stable homotopy theory, the Adams–Novikov spectral sequence, and ring spectra such as MU, BP and ku. We will also encourage attendees to familiarize themselves with the content of the lectures and notes of Achim Krause and Thomas Nikolaus.

The workshop discussions will have an expository character and a majority of the talks will be given by participants. Breaks will be built into the schedule for informal discussions and collaborations.  Here is some advice on how to give a good Talbot talk. The workshop will take place in a communal setting, with participants sharing living space and cooking and cleaning responsibilities. 

The workshop will be held entirely in-person. We cover all local expenses, including lodging and food. We also have limited funding available for participant travel costs. 

This workshop is supported by NSF grant DMS-2426392 and the K-theory Foundation.

Talbot is meant to encourage collaboration among young researchers, particularly graduate students. To this end, the workshop aims to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field. In particular, students at all levels of graduate education are encouraged to apply. Our decisions are based not on applicants' credentials but on our assessment of how much they would benefit from the workshop. As we are committed to promoting diversity in mathematics, we especially encourage women and minorities to apply.

In accordance with the Statement of Inclusiveness, this workshop will be open to everybody, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We are committed to ensuring that the Talbot Workshop is a supportive, inclusive, and safe environment for all participants, and that all participants are treated with dignity and respect.