# Topological Cyclic Homology

In this seminar, we will learn about Topological Cyclic Homology. However, the goal of the seminar is not so much “learning” but “doing” by jumping into the deep end. Here’s a sort of a guide, and a set of notes.

The literature we will need to get started are:

1. Dundas, Goodwillie, McCarthy.
2. Nikolaus, Scholze.
3. Notes from Thursday Seminar at Harvard (will be distributed to participants)

Background that one needs to participate fruitfully in the seminar (though of course all are invited):

1. Familiarity with rudiments of stable homotopy theory (spectra, the Steenrod algebra, power operations).
2. Familiarity with elements of homotopical thinking (basics of $\infty$-category and how to manipulate them, e.g., limits/colimits, unstraightening/straightening, $\infty$-operads).
3. Familiarity with notions from algebraic geometry (sheaves, sites, cohomology, the cotangent complex, the deRahm complex).
4. Familiarity with algebraic $K$-theory (Waldhausen $K$-theory and its basic properties).

The format:  We will meet on Fridays after happy hour around 5pm and continue indefinitely. One feature of this seminar is that we will learn on the fly – perhaps the only skill one needs in graduate school – so even though there’s a lot of prerequisites, we will try as much as possible to impart participants with the required knowledge. The other feature is that we will delve into detailsone the one hand this can be excruciating, but we will do it as a group.

In practice, someone will be assigned to lead a discussion. This means that

1. he/she must have have at least 45 minutes of material to talk about,
2. think about concrete questions he/she would like to see being discussed.
3. Suggest (along with the other participants) an exact topic for the next session.

Piotr and I will moderate the discussions and try to keep them as relevant as possible. However, digressions are not unwelcomed – this is the point of the indefinite timeline. In any event, after the first lecture (which will be given by Piotr), we will be attempting to add on (more likely) or address (less likely) the following questions/problems:

Questions:

1. What is $TC$ and friends ($TP$, $TC^-$, $THH$) (after Nikolaus-Scholze)?
2. What is the deRahm-Witt complex?
3.  For what rings/ring spectra is TC known? (E.g. the integers, finite fields, group rings) How does it look like?
4. What is the relation to $TC$ to the chromatic program?
5. How is $latex TC​$ the “derived deRham algebra” (even for classical schemes/rings)?
6. How does $TC$​ “get rid of denominators” present in classical algebraic geometry (Bhatt-Morrow-Scholze)?
7. How does $TC$​ and the trace methods help in computing algebraic K​-theory?

Problems:

1. Descent for $TC$ – does it have étale descent?
2. Develop some tools to compute $TC$​ for nonconnective ring spectra (people say that there are “no tools”).
3. Develop iterated $TC$​ and see if it’s good for anything (trace to secondary $K$-theory)
4. Look at structural features of $TC$​ – wrong way maps/transfers/extra functorialities/power operations.
5. Other contexts (outside of classical schemes/ring spectra) where $TC$​ can be useful.
Meeting 1: (Sept 22)
1. Piotr: An Invitation to $TC$.
2. Elden: Beyond $\mathbb{A}^1$-invariance: the DeRahm-Witt complex.

Mailing List:

1. Elden
2. Piotr
3. Yajit
4. Paul
5. Viktor
6. Aydin
7. Jānis
8. Ozgur
9. Joel
10. Micah
11. Rahul
12. Kyle
13. Grigory
14. Aron
15. Mingyi