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advanced_tools:stacks [2017/11/08 18:05] jakobadmin [Why is it interesting?] |
advanced_tools:stacks [2017/12/04 08:01] |
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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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- | [...] | ||
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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> | ||
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- | <blockquote>locality principle + gauge principle = stack principle<cite>https://ncatlab.org/schreiber/files/SchreiberTrento14.pdf</cite></blockquote> | ||
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- | <tabbox Layman> | ||
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- | <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> | ||
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- | <tabbox Student> | ||
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- | <note tip> | ||
- | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | ||
- | </note> | ||
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- | <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 | ||
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- | <tabbox Examples> | ||
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- | --> Example1# | ||
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- | <-- | ||
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- | --> Example2:# | ||
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- | <-- | ||
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- | <tabbox FAQ> | ||
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- | <tabbox History> | ||
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- | </tabbox> | ||
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