---
date: 2022-06-08 14:46
modification date: Wednesday 8th June 2022 14:46:55
title: "Talbot Talk Outline 1"
aliases: [Talbot Talk Outline 1]
status: complete🤠
---

Last modified date: <%+ tp.file.last_modified_date() %>

---


> Tags: #projects/talbot-talk [@Zak17b]
> See [2022 Talbot MOC](Projects/2022%20Talbot/2022%20Talbot%20MOC.md), [Talbot Syllabus](Zotnotes/LitNote-Campbell%20and%20Zakharevich-2022-Talbot%202022%20Syllabus-22TalSyl.md), and [Talbot Talk Advice](attachments/talk-advice.pdf)


> Possibly relevant papers:
>
> [LitNote-Braunling et al.-2021-The standard realizations for the K-theory of varieties-BGN21](Zotnotes/LitNote-Braunling%20et%20al.-2021-The%20standard%20realizations%20for%20the%20K-theory%20of%20varieties-BGN21.md)
> [LitNote-Sarazola and Shapiro-2021-A Gillet-Waldhausen Theorem for chain complexes of sets-SS21](Zotnotes/LitNote-Sarazola%20and%20Shapiro-2021-A%20Gillet-Waldhausen%20Theorem%20for%20chain%20complexes%20of%20sets-SS21.md)
> [LitNote-Hoekzema et al.-2022-Cut and paste invariants of manifolds via algebraic K-theory-HMM+21](Zotnotes/LitNote-Hoekzema%20et%20al.-2022-Cut%20and%20paste%20invariants%20of%20manifolds%20via%20algebraic%20K-theory-HMM+21.md)

---

# Talbot Talk

## Todos

- [ ] Finish talk outline
- [ ] Try working out some spectral sequences
- [ ] Write down some diagrams and computations

## Misc Notes

- Advice to remember:
	- What is the overall narrative of the workshop, and how does this talk fit in?
	- If theorems are needed from prev talks, coordinate with speaker.
	- Foreshadow future talks.
	- What theorems are needed for later talks?
	- To every idea, attach: how it fits into the bigger picture, and a concrete example.
	- Focus on key ideas for proof (which could reasonably be used to reconstruct the details).
	- Attach examples and non-examples to definitions, try to motivate.
	- Attach to theorems how they apply to examples.
	- Give an outline of the talk, try to give indicators of where in the outline we are at various points in the talk.
		- Also works for more computational arguments.
	- For proofs: what is the history? Are there general heuristics? Is there an easier "fake" proof with a reasonably way to recover the actual argument?
- Previous Talks
	- Theme: Part 2 on "Scissors Congruence as K Theory"
	- Talk 6: Assemblers.
		- Definition of assembler.
		- Definition of $\K$ for assemblers.
		- The cofiber theorem.
		- Total (and classical) scissors congruence
	- Talk 7: $\K(\mcv)$.
		- Definition of $\K(\mcv)$
		- Motivic measures
		- Discuss of scissors congruence of varieties
		- Intro to the annihilator of $\LL$
		- Motivic zeta functions
		- Borisov's result (Bor18?)
- My talk: $\Ann(\LL)$.
	- Larsen-Lunts' question about scissors congruence of varieties
	- Borisov's construction of an element in $\Ann(\LL)$.
	- Use higher $\K$ to show this always happens
- Next talks
	- Part 2:
		- Talk 9: SW-categories
			- Using Waldhausen categories to capture geometric decompositions
			- $\EE_\infty$ structure on $\K(\mcv)$.
		- Talk 10: Derived motivic measures
			- Use SW categories to construct algebraic motivic measures, e.g. local zeta functions
		- Talk 14: Squares $\K\dash$theory
			- Generalizes assemblers and "subtractive" $\K$
			- Used to define cut-and-paste groups and invariants of smooth compact manifolds
	- Part 3:
		- Talk 15: Cathelineau and $\K_M$.
			- Goncharov's conjecture
		- Talk 17: CZ21
			- Analyze the Goncharov complex, relate it to $\K$ and homology of $\GL_n$?
- Remarks:
	- [@Zak17a] shows a tight link between $\K(\mcv)$ and birational geometry.
	- Question: what arithmetic information does $\K_{\geq 1}(\mcv)$ encode?
	- Larsen-Lunts 03 show that Kapranov's motivic zeta function $\sum_{i\geq 0} [\Sym^i(X)]t^i$ from $\K_0(\mcv\slice k) \to W(\K_0(\mcv\slice k))$ is not a *rational* motivic zeta function. 
	  Key tool: $\mu_{\mathrm{LL}}: \K(\mcv\slice \CC) \to \ZZ\adjoin{\SB\slice \CC}$. Since $[\PP^1]_{\SB} = [\pt]_{\SB}$, we have $\mu_{\mathrm{LL}}(\AA^1) = 0$. So inverting or localizing at $\AA^1$ may yield an appropriate modification where it is rational.
		  - Zak17a lifts $\mu_{\mathrm{LL}}$ to a map of $\K$ spectra, but it would be desirable to have a direct construction of an SW-category that encodes stable birational equivalence of varieties.
	- We can still localize at zero divisors, it's just that the map $R \to R\invert{\LL}$ where $x\mapsto {x\over 1}$ may not be injective. It can even happen that the images of zero divisors are no longer zero divisors!
	  ![](attachments/Pasted%20image%2020220609212424.png)
![](attachments/Pasted%20image%2020220609212801.png)
## Talk Outline V1

- Motivation
	- What is the ring structure on $\K(\mcv)$?
	- Why are piecewise isomorphisms of varieties interesting/useful?
	- When do birational automorphisms lift to piecewise isomorphisms?
	- What properties does $\LL$ have?
- Conceptual review of items from previous talks:
	- ?
- Main theorems:
	1. $\mcv$ is a filtered category, inducing $\Fil\, \K(\mcv)$ with 
	   $$\graded_n \K(\mcv) = \bigvee_{[X] \in B_n} \Sigma^\infty_+ \B \Aut_k \, k(X)$$
	2. $\psi_n$ has a nonzero kernel for some $n$ iff there exists nonzero differentials between columns 0 and 1 of a spectral sequence.
	3. For convenient fields, if $\LL$ is a zero divisor in $\K_0(\mcv)$ then $\psi_n$ is not injective for some $n$.
	4. For convenient fields, if $\chi \in \ker(\times \LL)$ then $\chi = [X] - [Y]$ where $[X\times \AA^1] = [Y\times \AA^1]$ but $X\times \AA^1, Y\times \AA^1$ are not piecewise isomorphic.
	5. There is an isomorphism of groups $\K_0(\mcv)/\LL \to \ZZ\adjoin{\SB}$ where $\SB$ are iso classes of varieties up to stable birational isomorphism.
- Things coming up in later talks:
	- ?