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advanced_tools:stacks [2017/11/09 08:40]
jakobadmin [History]
advanced_tools:stacks [2017/12/04 07:01]
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-====== Stacks ====== 
- 
-<tabbox Why is it interesting?> ​ 
- 
-<​blockquote>​The development of modern physics in the first half of the 20th century 
-was closely related to the development of differential geometry, first 
-via Riemannian geometry in Einstein’s theory of gravity and then later 
-via Cartan geometry in Yang-Mills’s theory of gauge fields. But, as highlighted 
-by Grothendieck in the second half of the 20th century and as 
-witnessed by a multitude of modern developments,​ a more natural mathematical 
-description of many phenomena in geometry is obtained by refining from traditional geometric spaces to more refined kinds of spaces 
-known as “stacks”. 
- 
-[...] 
- 
-Our main motivation to consider sheaves and stacks is to provide a nonperturbative 
-framework in which we can do physics. Much of gauge theory is 
-done in perturbation theory, but in fact **non-perturbative effects** such as Dirac 
-monopoles and Yang-Mills instantons play a crucial role in fundamental physics 
-[5]. **The language of stacks is the natural language for these phenomena.** 
- 
-<​cite>​https://​ncatlab.org/​schreiber/​files/​Eggertsson2014.pdf</​cite></​blockquote>​ 
- 
-<​blockquote>​locality principle + gauge principle = stack principle<​cite>​https://​ncatlab.org/​schreiber/​files/​SchreiberTrento14.pdf</​cite></​blockquote>​ 
- 
-<tabbox Layman> ​ 
- 
-<note tip> 
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. 
-</​note>​ 
-  ​ 
-<tabbox Student> ​ 
- 
-<note tip> 
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. 
-</​note>​ 
-  
-<tabbox Researcher> ​ 
- 
-  * See section 6 in [[https://​arxiv.org/​abs/​0802.0999|Mirror Symmetry, Hitchin'​s Equations, And Langlands Duality]] by Edward Witten 
-  ​ 
-<tabbox Examples> ​ 
- 
---> Example1# 
- 
-  
-<-- 
- 
---> Example2:# 
- 
-  
-<-- 
- 
-<tabbox FAQ> ​ 
-  ​ 
-<tabbox History> ​ 
-The idea of using stacks goes back to a manuscript titled Pursuing Stacks by Alexander Grothendieck. For some more information,​ see https://​ncatlab.org/​nlab/​show/​Pursuing+Stacks ​ 
-</​tabbox>​ 
- 
  
advanced_tools/stacks.txt · Last modified: 2017/12/04 07:01 (external edit)