The technique of almost disjoint forcing was introduced in

MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. *Some
applications of almost disjoint sets*. In **Mathematical Logic and
Foundations of Set Theory** (Proc. Internat. Colloq., Jerusalem, 1968),
pp. 84–104, North-Holland, Amsterdam, 1970.

Fix an almost disjoint family $X=(x_\alpha:\alpha<\omega_1)$ of subsets of $\omega$, so $x_\alpha\cap x_\beta$ is finite for all $\alpha<\beta<\omega_1$. Given $s\subseteq\omega_1$, the poset $\mathbb P_{X,s}$ adds a real $r$ such that $$s=\{\alpha<\omega_1:r\cap x_\alpha\mbox{ is infinite}\}.$$
This shows that $s\in L[X,r]$; if $X\in L$ then in fact $s\in L[r]$. The poset $\mathbb P_{X,s}$ is ccc, and it follows from standard arguments that if $\mathsf{MA}_{\omega_1}$ holds and $\omega_1=\omega_1^L$, then $\mathcal P(\omega_1)=\bigcup_{r\subseteq\omega}\mathcal P(\omega_1)^{L[r]}$. Moreover, by taking as $X$ the first almost disjoint family in the standard enumeration of $L$, we see that $X$ is definable without parameters, and this gives us a simple parameter-free injection of $\mathcal P(\omega_1)$ into $\mathcal P(\omega)$ as long as there is, for instance, a parameter-free definable well-ordering of $\mathcal P(\omega)$. This is consistent with $\mathsf{MA}_{\omega_1}+\omega_1=\omega_1^L$.

(In fact, it is a consequence of $\mathsf{BPFA}+\omega_1=\omega_1^L$, see my paper with Friedman listed below and available at my page. For the consistency result just for $\mathsf{MA}$, this is a result of Friedman, see Theorem 8.51 in his class forcing book.)

In general, in the context of forcing axioms, to get a nice well-ordering of $\mathcal P(\omega_1)$, it suffices to find a nice well-ordering of $\mathcal P(\omega)$ and apply almost disjoint forcing. If we further arrange that the almost disjoint family we begin with is sufficiently nice, this reflects as well in the complexity and/or parameters of the well-ordering of $\mathcal P(\omega_1)$. The same holds for a nice injection of $\mathcal P(\omega_1)$ into $\mathcal P(\omega)$ and similar variants.

MR2895389 (2012m:03123). Caicedo, Andrés Eduardo; Friedman, Sy-David. *$\mathsf{BPFA}$ and projective well-orderings of the reals*. J. Symbolic Logic **76** (2011), no. 4, 1126–1136.

I like the description of the poset $\mathbb P_{X,s}$ in Jech's book: A condition is a function $f\!:d_f\to2$ whose domain $d_f$ is a subset of $\omega$ such that $d_f\cap x_\alpha$ is finite for all $\alpha\in s$ and $\{n\in d_f: f(n)=1\}$ is finite. The order is inclusion.

Note that if $p,q$ are incompatible conditions, then in particular $\{n:p(n)=1\}\ne\{n:q(n)=1\}$. This shows that $\mathbb P_{X,s}$ is actually Knaster.

For any $p$ and any $\alpha\in\omega_1\smallsetminus s$, we can extend $p$ to a condition $q$ whose domain contains $x_\alpha$ and satisfies $q(n)=0$ for all $n\in x_\alpha\smallsetminus d_p$ (since the $x_\beta$ are almost disjoint).

Similarly, for any $\alpha\in s$ and any $k\in\omega$, we can extend any condition to ensure that its domain meets $x_\alpha$ in a set of size at least $k$.

From this, standard density arguments give us the result.

By the way, it is consistent that $\mathsf{MM}$ holds and there is a parameter-free well-ordering of $\mathcal P(\omega_1)$, which naturally gives us a nice injection of $\mathcal P(\omega_1)$ into $\mathcal P(\omega)$. (I am using "nice" loosely here, but the definition of the well-ordering, and therefore of the injection, is actually not too bad.) See

MR2474445 (2009k:03085). Larson, Paul B. *Martin's maximum and definability in $H(\aleph_2)$*. Ann. Pure Appl. Logic **156** (2008), no. 1, 110–122.

localin a suitable sense. $\endgroup$