====== Atiyah-Singer Index Theorem  ======



<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> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Why is it interesting?> 

<blockquote>
**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
(the standard explanatory joke is that 'a topologist is someone who
can't tell the difference between a coffee cup and a doughnut'). One
important aspect of the index theorem is that it can be proved by
relating the differential equation under study to a generalised version
of the Dirac equation. Atiyah and Singer rediscovered the Dirac equation for themselves during their work on the theorem. **Their theorem
says that one can calculate the number of solutions of an equation by
finding the number of solutions of the related generalised Dirac equation.** It was for these generalised Dirac equations that they found a
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
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>
</blockquote>

<blockquote>
The next major development in the story of the so-far-only-physicist’s Yang-Mills equations was one by mathematicians! The Atiyah-Singer Index Theorem said something about the dimension of the space of solutions to the Yang-Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics community began to change. It is here that the cooperation and cross-fertilization really began. I don’t know how to describe the tremendous difference this made. The entire position that mathematics held within the scientific community changed. The mathematicians actually had something to say!

<cite>http://www.ma.utexas.edu/users/lfredrickson/Attachments/TempleLectures.pdf</cite>
</blockquote>

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  * See also: http://physics.stackexchange.com/a/1865/37286

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