I don't think differential topology is a dying field.

I'll interpret this as the classification of smooth
manifolds and, more broadly, maps between them
(immersions, embeddings, diffeomorphism groups).
Also, I'll restrict to the finite-dimensional case.

There are related topics which are very active,
usually studying smooth manifolds with extra structure,
e.g. exterior differential systems, foliations and
contact structures, symplectic and Riemannian geometry.
I won't comment much on these areas.

The classification of smooth manifolds was quite
successful in the 60s with the h- and s-cobordism
theorems framing many classification problems in terms
of surgery problems. The classification of exotic
spheres was more-or-less reduced to problems in
homotopy theory, the stable homotopy groups and Kervaire
invariant problems. The study of these invariants is
still active, but the
techniques are more algebraic. Moreover, there is
still an industry of studying Riemannian metrics on
exotic spheres.

Maybe one of the biggest open problems now in differential
topology is the cobordism hypothesis, originally formulated
by Baez-Dolan, but reformulated by Lurie. This is formulated
as a classification of "fully extended topological field theories"
in terms of $(\infty,n)$-categories. His sketch of proof
is regarded as incomplete, and a few groups are trying to fill
in the details. From discussions I've had with experts,
a big issue here is foundational results in differential topology.
Lurie's outline relies on results about manifolds with corners,
and I think that Schommer-Pries has filled in some details,
but I think that the proof of the cobordism hypothesis is still
incomplete.

Another (very special) problem that has received some attention
is the Hirzebruch Prize Question:

Does there exist a 24-dimensional compact, orientable,
differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, \hat{A}(X) = 1$, and $\hat{A}(X, T_C) = 0$?

Here $\hat{A}$ is the A-hat genus. The twisted Witten
genus is supposed to be related to certain modular functions
(McKay-Thompson series) associated with Monstrous Moonshine.
I believe that Hopkins proved that a manifold with the right
properties exists, but only in the topological category,
and without the action of the Monster group. Daniel Allcock
is working on constructing this manifold.

Shmuel Weinberger has championed the study of decidability
questions in differential topology.

The Novikov conjectures would imply that $\mathcal{L}$-classes
(certain combinations of Pontryagin classes) are invariant
under homotopy equivalence of smooth aspherical manifolds. See a recent survey.

There is still active study of diffeomorphism groups.
An active topic here is the study of homological stability
for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.

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