[gigatexal]

For anyone frustrated that the article doesn’t say what specific part of math has the most to gain it’s here:

“ The crucial point of condensed mathematics, according to Scholze and Clausen, is to redefine the concept of topology, one of the cornerstones of modern maths. A lot of the objects that mathematicians study have a topology — a type of structure that determines which of the object’s parts are close together and which aren’t. Topology provides a notion of shape, but one that is more malleable than those of familiar, school-level geometry: in topology, any transformation that does not tear an object apart is admissible. For example, any triangle is topologically equivalent to any other triangle — or even to a circle — but not to a straight line.

Topology plays a crucial part not only in geometry, but also in functional analysis, the study of functions. Functions typically ‘live’ in spaces with an infinite number of dimensions (such as wavefunctions, which are foundational to quantum mechanics). It is also important for number systems called p-adic numbers, which have an exotic, ‘fractal’ topology.”

[dr_kiszonka]

I know nothing about topology. If you have time, could you please explain this sentence?

"For example, any triangle is topologically equivalent to any other triangle — or even to a circle — but not to a straight line."

Is it because triangles and circles are "2D" and lines aren't?

[yongjik]

Crudely speaking, topologists consider spaces as if they're made of rubber - a mathematically perfect rubber that can be made infinitely thin or stretch to infinity. So, a circle can be made with an infinitely thin circular rubber ring, and you just pinch three points and stretch, and you get your triangle, in any shape.

But you can't get a straight line - to do that you need scissors to cut one point of a circle (to be precise, remove a single point) - and then you can stretch the remainder to infinity and now you have your line.

[QuesnayJr]

You can give a function that maps the points of the triangle to the points of the circle in such a way that this function is continuous. (Continuous basically means "no jumps".)

You can give this function explicitly. Let's assume that the triangle is drawn on a piece of paper with an x and y axis, such that the intersection of the axes (the origin) is inside the triangle. For each point on the triangle, draw a line from the origin to the point. Take the angle between that line and the x-axis. Use that angle to map it onto a circle.

[hopfenspergerj]

If you remove a point from a line, it breaks into two pieces.

[77pt77]

No. Both are 1-d (lines).

It's because triangles are loops (closed) and straight lines aren't.