theorems:atiyah-singer
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| ====== Atiyah-Singer Index Theorem ====== | ====== Atiyah-Singer Index Theorem ====== | ||
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| + | <tabbox Intuitive> | ||
| + | |||
| + | <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 Concrete> | ||
| + | |||
| + | <note tip> | ||
| + | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | ||
| + | </note> | ||
| + | |||
| + | <tabbox Abstract> | ||
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| + | <note tip> | ||
| + | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| + | </note> | ||
| + | |||
| + | | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| <blockquote> | <blockquote> | ||
| - | **The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of topology.** Topology is that part of mathematics that deals with those aspects | + | **The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of [[advanced_tools:topology|topology]].** Topology is that part of mathematics that deals with those aspects |
| of geometrical objects that don't change as one deforms the object | of geometrical objects that don't change as one deforms the object | ||
| (the standard explanatory joke is that 'a topologist is someone who | (the standard explanatory joke is that 'a topologist is someone who | ||
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| beautiful topological formula for the number of their solutions. [...] Atiyah became interested [in Yang-Mills theory], and | beautiful topological formula for the number of their solutions. [...] Atiyah became interested [in Yang-Mills theory], and | ||
| they quickly realised that **their index theorem could be applied in | they quickly realised that **their index theorem could be applied in | ||
| - | this case and it allowed a determination of exactly how many solutions the [[quantum_field_theory:notions:self_duality_equations|self-duality equations]] would have**. | + | this case and it allowed a determination of exactly how many solutions the self-duality equations would have**. |
| <cite>page 118 in Not Even Wrong by Peter Woit</cite> | <cite>page 118 in Not Even Wrong by Peter Woit</cite> | ||
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| </blockquote> | </blockquote> | ||
| + | ---- | ||
| - | See also: http://physics.stackexchange.com/a/1865/37286 | + | * See also: http://physics.stackexchange.com/a/1865/37286 |
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| - | <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> | + | |
| - | + | ||
| - | <note tip> | + | |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
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| - | <tabbox Examples> | + | |
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| - | --> Example1# | + | |
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| - | + | ||
| - | <-- | + | |
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| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
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| - | <tabbox FAQ> | + | |
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| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
theorems/atiyah-singer.1513762404.txt.gz · Last modified: 2017/12/20 09:33 (external edit)
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