(See the sibling comment with a "10." in it for context.)

5. Untriangulable spaces. Suppose you're doing topology: you're interested in what's left of geometry after you strip away notions like "distance" and "angle". The central notion here is that of "topological space", which is a set (of points) together with (in one formulation) a notion of "neighbourhood"; intuitively, a neighbourhood of a point x is a set that contains "all points close enough to x". Topological spaces can be pretty weird, so topologists are interested in various more restricted kinds of space that look more like the geometrical objects we're familiar with.

One of the most important is a "manifold". A d-dimensional manifold is a topological space with the property that all small enough bits of it look like ordinary d-dimensional Euclidean space.

Now, you might hope that that's a restrictive enough condition to allow you to put a lot of useful extra structure on any space that's a manifold. For instance, you might hope to be able to put a "differentiable structure" on it, which is equivalent (not trivially equivalent, given the usual definitions, but never mind) to saying that you can instantiate it as a subset of a nice Euclidean space, defined by saying that it's where some nice well-behaved functions on that space are zero. Or, kinda-oppositely, instead of heading in the direction of increasing smoothness you might head in the direction of increasing discreteness: you might want a "triangulation" of the manifold, meaning that you can build it out of points (0-dimensional) and line segments (1-dimensional) and triangles (2-dimensional) and tetrahedra (3-dimensional), etc., etc., glued together in obvious ways along their faces.

(Why would you care? Because if you have that extra structure you can use various bits of mathematical machinery designed to exploit it. For instance, if you have a triangulation then lots of things reduce to nice finite computations that you can do on a computer.)

Alas! It's been known for some time (though it was quite a surprise when first discovered) that some manifolds can't have a differentiable structure put on them (and others have more than one, which was also a surprise).

Well, it turns out that if a manifold has a differentiable structure then it can be triangulated, but perhaps not conversely. So maybe every manifold can be triangulated? Nope. In the 1980s, an example was found of a 4-dimensional manifold with no triangulation.

Still, things might be different in higher dimensions. (E.g., the Poincare conjecture -- resolved several years ago now by Grigori Perelman -- has versions in all dimensions; it's really hard in 3 dimensions but much easier in 4 or more.) And this is the question that Manolescu has resolved. It turns out that there are manifolds in 5,6,7,... dimensions that can't be triangulated.

He proved this using something called Floer homology. Homology is a technique for deriving algebraic structure from topological structure (or, actually, from other kinds of structure, and there's a bit of a connection here with the "univalent foundations" stuff); there are lots of different kinds of homology, all closely related, and this particular one turns out to have a lot to do with mathematical structures that arise in theoretical physics.

The relevant thing about Floer homology is that it derives algebraic structure from what are called "cobordisms". If you have d-dimensional manifolds A and B, a cobordism between them is a (d+1)-dimensional manifold-with-boundary (what does that mean? well, think of e.g. a disc; it's a 2-dimensional manifold, with 1-dimensional boundary, namely the circle around it) whose boundary consists of a copy of A together with a copy of B. (Example: suppose A and B are both spheres, like the surface of the earth. A cobordism between them is given by a hollow ball; one sphere is its outer surface, one its inner surface.)

I have no knowledge of the details, but it turns out (and was proved in the 1970s) that "all manifolds in dimension 5 and up are triangulable" is equivalent to a rather technical statement about homology and cobordisms (in, curiously, only 3 dimensions). And that's the statement that Manolescu proved was false.

I don't know whether anything interesting follows from the existence of untriangulable 5-manifolds as such, but a deeper understanding of Floer homology means a deeper understanding of cobordisms, which means a deeper understanding of topological quantum field theories, which may be useful in theoretical physics.