Structure of the upcoming report

Published: October 16, 2023

Part 0: t-structures (Talk about this in Day 1?)

Slogan:

t-structure is a “refinement” structure on triangulated categories. We are concerned about this because it helps describe the convergence of spectral sequences as well as simplifying $(\infty,1)$-categories (by considering its heart).

What we need in our lectures:

Brief introduction of applications of t-structures. (HA 1.2) (About 5 minutes)
The definition of t-structures, filtered objects and the correspondence spectral sequences. (HA 1.2) (About 20 minutes)
- Compare it with the classical Adams spectral sequences (in the proof part)
- Explain the definition by showing the geometric meaning of these definitions in $Sp$ as examples.
(Optional) $\infty$-category version of Dold-Kan correspondence.

Cautions:

Remember to talk with ZSH about the cooperation. Better to finish this part on Day 1.

Slogan:

What we need in our lectures:

Refer to “the motivic Adams spectral sequence”, Daniel Dugger and Daniel C. Isaksen.
Brief introduction of motivic Adams spectral sequences. (About 20 minutes, or longer)
Compare it with classical Adams spectral sequences. (i.e. what structures we enrich, some differences in calculation)
- Some easy calculations as examples.
Clarify what have we known and what we haven’t about motivic Adams spectral sequences
Its relation with algebraic Novikov spectral sequence.
The case of non-algebraicly algebraically closed fields. (i.e. Show its difficulty, and what properties we lose in that case. This part may have a strong relation with Chow t-structures.)
(Optional) Some other motivic versions of spectral sequences, such as the May spectral sequence.

Problem:

What will happen in real motivic homotopy? (The influence of tri-graded spheres.)
Update some problems after I finish my reading note about this paper.

Slogan:

(Guess:) Chow t-structures help us understand motivic Adams spectral sequences in general situations.

What we need in our lectures:

The definition of Chow t-structures.
Compare with ordinary t-structures.
Show why does it have such benefits on:
- Valid over stable motivic homotopy category than just the p-completed sphere spectrum.
- Valid for the integral case.
- Valid for other base field than just $\mathbb{C}$.
(Optional) Brief introduction about real motivic homotopy theory as an example

Problem:

Slogan:

What we need in our lectures:

Brief introduction of those spectral sequences we need.
- Some calculations as examples.
- Translate category languages back to some explicit examples
Show how the information flows in those categories by diagram. (About 20 minutes) (With diagrams)
- Milnor square
What’s new in BKWX
- Show why we need Chow t-structure.
- （Optional) Some calculations.

Problem:

I can’t write a better version of the introduction of GWX, but just reading the introduction of GWX for audiences seems not so good. Same problem for BKWX.

Motivic Adams spectral sequence is important.
For GWX, understanding the Milnor square is the most important thing. Instead, introducing every category that appeared in that paper is unnecessary. This paper still stands without the $\infty$-category part.
Focus on the relations between those spectral sequences, and figure out how information of homotopy flows in them.
Ask Prof. Wang for some pieces of advice about the outline. His suggestions about which parts are important are very instructive.
Refer to the overview articles of GWX (ICM2022) and BKWX (ICBS2023). Ask Prof. Wang for these articles.
Try to translate category language into some explicit calculation that may be helpful to our understanding.
The estimated time is not meaningful, but it partly shows the amount of knowledge we need to introduce at that part.
Talk with LTT about the following part. Building a working group with FDU and UCSD is a good choice.