advanced_tools:stacks
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advanced_tools:stacks [2017/12/04 08:01] |
advanced_tools:stacks [2017/11/09 09:52] (current) jakobadmin [History] |
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| + | ====== Stacks ====== | ||
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| + | <tabbox Why is it interesting?> | ||
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| + | <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”. | ||
| + | |||
| + | [...] | ||
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| + | 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.** | ||
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| + | <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> | ||
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| + | * See section 6 in [[https://arxiv.org/abs/0802.0999|Mirror Symmetry, Hitchin's Equations, And Langlands Duality]] by Edward Witten | ||
| + | | ||
| + | <tabbox Examples> | ||
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| + | --> Example1# | ||
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| + | |||
| + | <-- | ||
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| + | --> Example2:# | ||
| + | |||
| + | |||
| + | <-- | ||
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| + | <tabbox FAQ> | ||
| + | | ||
| + | <tabbox History> | ||
| + | The idea of using stacks goes back to a manuscript titled Pursuing Stacks by Alexander Grothendieck in 1983. For some more information, see https://ncatlab.org/nlab/show/Pursuing+Stacks | ||
| + | </tabbox> | ||
| + | |||
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