Editing Atiyah–Singer index theorem (section)

===Heat equation===
{{harvs|txt=yes|last=Atiyah |author2-link=Raoul Bott|last2=Bott|author3-link=Vijay Kumar Patodi|last3=Patodi|year=1973}} gave a new proof of the index theorem using the [[heat equation]], see e.g. {{harvtxt|Berline|Getzler|Vergne|1992}}. 
The proof is also published in {{harv|Melrose|1993}} and {{harv|Gilkey|1994}}.

If ''D'' is a differential operator with adjoint ''D*'', then ''D*D'' and ''DD*'' are self adjoint operators whose non-zero eigenvalues have the same  multiplicities. However their zero eigenspaces may have different multiplicities, as these multiplicities are the dimensions of the kernels of ''D'' and ''D*''. Therefore, the index of ''D'' is given by 
:<math>\operatorname{index}(D) = \dim \operatorname{Ker}(D) - \dim \operatorname{Ker}(D^*) = \dim \operatorname{Ker}(D^*D) - \dim \operatorname{Ker}(DD^*) = \operatorname{Tr}\left(e^{-t D^* D}\right) - \operatorname{Tr}\left(e^{-t DD^*}\right)</math>

for any positive ''t''. The right hand side is given by the trace of the difference of the kernels of two heat operators. These have an asymptotic expansion for small positive ''t'', which can be used to evaluate the limit as ''t'' tends to 0, giving a proof of the Atiyah–Singer index theorem.  The asymptotic expansions for small ''t'' appear very complicated, but invariant theory shows that there are huge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.