# Background in scissors congruence

![image-20220930143323908](/home/zack/.config/Typora/typora-user-images/image-20220930143323908.png)



![image-20220930143346944](/home/zack/.config/Typora/typora-user-images/image-20220930143346944.png)

![image-20220930143408059](/home/zack/.config/Typora/typora-user-images/image-20220930143408059.png)

## Dehn invariant

![image-20220930143431895](/home/zack/.config/Typora/typora-user-images/image-20220930143431895.png)

![image-20220930143442730](/home/zack/.config/Typora/typora-user-images/image-20220930143442730.png)

## Goncharov's conjecture

![image-20220930143507553](/home/zack/.config/Typora/typora-user-images/image-20220930143507553.png)

![image-20220930143516767](/home/zack/.config/Typora/typora-user-images/image-20220930143516767.png)

![image-20220930143531193](/home/zack/.config/Typora/typora-user-images/image-20220930143531193.png)

![image-20220930143540883](/home/zack/.config/Typora/typora-user-images/image-20220930143540883.png)



# K theory of varieties



![image-20220930143642868](/home/zack/.config/Typora/typora-user-images/image-20220930143642868.png)

Constructing from assemblers:![image-20220930143909522](/home/zack/.config/Typora/typora-user-images/image-20220930143909522.png)



## Examples in this ring

![image-20220930143701875](/home/zack/.config/Typora/typora-user-images/image-20220930143701875.png)

![image-20220930143936774](/home/zack/.config/Typora/typora-user-images/image-20220930143936774.png)

![image-20220930143959813](/home/zack/.config/Typora/typora-user-images/image-20220930143959813.png)





## Ring structure

![image-20220930143839656](/home/zack/.config/Typora/typora-user-images/image-20220930143839656.png)









## Birationality and PW iso

![image-20220930143710931](/home/zack/.config/Typora/typora-user-images/image-20220930143710931.png)

![image-20220930144014798](/home/zack/.config/Typora/typora-user-images/image-20220930144014798.png)



![image-20220930144055974](/home/zack/.config/Typora/typora-user-images/image-20220930144055974.png)

![image-20220930144110017](/home/zack/.config/Typora/typora-user-images/image-20220930144110017.png)

![image-20220930144142874](/home/zack/.config/Typora/typora-user-images/image-20220930144142874.png)



## Motivating questions:

![image-20220930144156362](/home/zack/.config/Typora/typora-user-images/image-20220930144156362.png)



![image-20220930144213843](/home/zack/.config/Typora/typora-user-images/image-20220930144213843.png)

![image-20220930144228744](/home/zack/.config/Typora/typora-user-images/image-20220930144228744.png)

![image-20220930144301652](/home/zack/.config/Typora/typora-user-images/image-20220930144301652.png)



Result using mirror symmetry (Grassmannian-Pfaffian correspondence)

![image-20220930144322013](/home/zack/.config/Typora/typora-user-images/image-20220930144322013.png)





Inna's theorem

![image-20220930144433191](/home/zack/.config/Typora/typora-user-images/image-20220930144433191.png)