I know the following facts. (Don't assume I know much more than the following facts.)


* The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
* The Atiyah-Singer index theorem can be proven using heat kernels.


This implies that both Riemann-Roch and Gauss-Bonnet can be proven using heat kernels. Now, I don't think I have the background necessary to understand the details of the proofs, but I would really appreciate it if someone briefly outlined for me an extremely high-level summary of how the above two proofs might go. Mostly what I'm looking for is physical intuition: when does one know that heat kernel methods are relevant to a mathematical problem? Is the mathematical problem recast as a physical problem to do so, and how? 


(Also, does one get Riemann-Roch for Riemann surfaces only or can we also prove the version for more general algebraic curves?)


**Edit:** Sorry, the original question was a little unclear. While I appreciate the answers so far concerning how one gets from heat kernels to the index theorem to the two theorems I mentioned, I'm wondering what one can say about going from heat kernels **directly** to the two theorems I mentioned. As Deane mentions in this comments, my hope is that this reduces the amount of formalism necessary to the point where the physical ideas are clear to someone without a lot of background.



What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial\_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle: 


$u(t,\theta) = \sum\_n a\_n(t) \exp(in\theta) \implies a\_n(t) = a\_n(0)\cdot \exp(-tn^2)$


For a sufficiently nice case the solution of the heat equation is $u(t,\cdot) = \exp(-t\Delta) \* u(0,\cdot)$.


The hard part is building the HK: we have to compute the eigenstuff of $\Delta$ (this is the Hodge theorem). But once we do that, a miracle occurs and we get the 



> 
> Atiyah-Singer Theorem: The supertrace
>  of the HK on forms is constant: viz. 
> 
> 
> $\mathrm{Tr}\_s \exp(-t\Delta) = \sum\_k (-1)^k \,\mathrm{Tr} \exp(-t\Delta^k) = \mathrm{const}.$
> 
> 
> 


For $t$ large, this can be evaluated topologically; for small $t$, it can be evaluated analytically as an integral of a characteristic class.


**Edit per Qiaochu's clarification**


[This article of Kotake](http://books.google.com/books?id=bw4-P-Pm3FMC&pg=PA137) (really in [here](http://books.google.com/books?id=dqkfAQAAIAAJ) as the books seem to be mixed up) proves Riemann-Roch directly using the heat kernel.


