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.

$Tr_s \exp(-t\Delta) = \sum_k (-1)^k Tr \exp(-t\Delta^k) = const$

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 (really in here as the books seem to be mixed up) proves Riemann-Roch directly à la McKean and Singer. The book of Rosenberg I mentioned in the comments illustrates using the key points of McKean-Singer. See also hereheat kernel.

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.

$Tr_s \exp(-t\Delta) = \sum_k (-1)^k Tr \exp(-t\Delta^k) = const$

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

Supposedly (I don't have ready access to this) an article of Kotake in here proves Riemann-Roch directly à la McKean and Singer. The book of Rosenberg I mentioned in the comments illustrates the key points of McKean-Singer. See also here.

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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.

$Tr_s \exp(-t\Delta) = \sum_k (-1)^k Tr \exp(-t\Delta^k) = const$

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