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I ran a seminar last month on this stuff; I wrote detailed notes at http://www.mit.edu/~sanathd/iap-2018.pdf, which expand on the discussion above. Good sources to learn this stuff are Jacob Lurie's course from eight years ago and COCTALOS. For more references, check out this page. I hope this helps; let me know if there's something I should add/talk more about.

I ran a seminar last month on this stuff; I wrote detailed notes at https://sanathdevalapurkar.github.io/files/iap-2018.pdf, which expand on the discussion above. Good sources to learn this stuff are Jacob Lurie's course from eight years ago and COCTALOS. For more references, check out this page. I hope this helps; let me know if there's something I should add/talk more about.

But $KO$ and $TMF$ are not complex-oriented! Instead, they admit orientations from $MSpin$ and $MString$: there are $\Eoo$-maps $MSpin \to KO$ and $MString \to TMF$ that lift the Atiyah-Bott-Shapiro orientation and the Witten genus. This is in Ando-Hopkins-Rezk, but it's hard to work through. There's an overview in Chapter 10 of the TMF book (see here), and some notes in Appendix A.3 of Eric Peterson's book project.

But $KO$ and $TMF$ are not complex-oriented! Instead, they admit orientations from $MSpin$ and $MString$: there are $\Eoo$-maps $MSpin \to KO$ and $MString \to TMF$ that lift the Atiyah-Bott-Shapiro orientation and the Witten genus. This is in Ando-Hopkins-Rezk, but it's hard to work through. There's an overview in Chapter 10 of the TMF book (see here), and some notes in Appendix A.3 of Eric Peterson's book project.

A priori, there's no reason for $E$-theory to be a multiplicative cohomology theory (i.e., an $\Eoo$-ring spectrum). But Goerss, Hopkins, and Miller proved with what's known as Goerss-Hopkins obstruction theory (I livetexed notes from this year's Talbot workshop here, which was on obstruction theory, but you should check the Talbot website for the official and edited notes) that $E_n$ really is an $\Eoo$-ring spectrum! (It seems appropriate to remark here that Lurie has recently given an alternative moduli-theoretic proof of this result; see here.) They also proved something more: if $\mathbf{G}_n$ denotes the profinite group of automorphisms of the geometric point, then there is a lift of the action of $\mathbf{G}_n$ to an action on $E$-theory via $\Eoo$-ring maps. Moreover, $\mathrm{Aut}(E_n) \simeq \mathbf{G}_n$. (For instance, at height $1$, the group $\mathbf{G}_1 \simeq \mathbf{Z}_p^\times$, and the action of $\mathbf{G}_1$ on $E_1 = KU^\wedge_p$ is given by the Adams operations.)

But $KO$ and $TMF$ are not complex-oriented! Instead, they admit orientations from $MSpin$ and $MString$: there are $\Eoo$-maps $MSpin \to KO$ and $MString \to TMF$ that lift the Atiyah-Bott-Shapiro orientation and the Witten genus. This is in Ando-Hopkins-Rezk, but it's hard to work through. There's an overview in Chapter 10 of the TMF book (see here), and some notes in Appendix A.3 of Eric Peterson's book project.

A priori, there's no reason for $E$-theory to be a multiplicative cohomology theory (i.e., an $\Eoo$-ring spectrum). But Goerss, Hopkins, and Miller proved with what's known as Goerss-Hopkins obstruction theory (I livetexed notes from this year's Talbot workshop here, which was on obstruction theory, but you should check the Talbot website for the official and edited notes) that $E_n$ really is an $\Eoo$-ring spectrum! (It seems appropriate to remark here that Lurie has recently given an alternative moduli-theoretic proof of this result; see here.) They also proved something more: if $\mathbf{G}_n$ denotes the profinite group of automorphisms of the geometric point, then there is a lift of the action of $\mathbf{G}_n$ to an action on $E$-theory via $\Eoo$-ring maps. Moreover, $\mathrm{Aut}(E_n) \simeq \mathbf{G}_n$. (For instance, at height $1$, the group $\mathbf{G}_1 \simeq \mathbf{Z}_p^\times$, and the action of $\mathbf{G}_1$ on $E_1 = KU^\wedge_p$ is given by the Adams operations.)

But $KO$ and $TMF$ are not complex-oriented! Instead, they admit orientations from $MSpin$ and $MString$: there are $\Eoo$-maps $MSpin \to KO$ and $MString \to TMF$ that lift the Atiyah-Bott-Shapiro orientation and the Witten genus. This is in Ando-Hopkins-Rezk, but it's hard to work through. There's an overview in Chapter 10 of the TMF book (see here), and some notes in Appendix A.3 of Eric Peterson's book project.