I just learned a remarkable theorem: Euclidean space can be characterized by the fact that the distance squared function is everywhere differentiable! 1/10 I found this in a paper by Franz-Erich Wolter called "Distance function and cut loci on a complete Riemannian manifold". Here's the statement of the result from the introduction: 2/10

An absolutely central concept in mathematics is that of (co)homology. In one of its earliest incarnations, the singular (co)homology of a topological space, it consists of a sequence of groups that measure the number of ‘holes’ in a space of various dimensions. 1/11 This can take an inscrutable geometric shape, and boil it down to a more understandable sequence of groups. 2/11

The Borel summation is an amazing method for summing divergent series which is based on the Laplace transform. But before I tell you about it, I can’t help but share a screenshot from the top of the Wikipedia page: 1/10 Borel summation is built using the Laplace transform, which is an integral transform: it takes in a function of a variable x, and spits out a function of another variable z: 2/10

What does geometric quantization have to do with (non-commutative) projective geometry? 1/21
arxiv.org/abs/2108.01658 Start with a symplectic manifold (M, w). Physically, this is the space of all possible states of a system: a point specifies both the position and momentum of a classical particle. Let’s be slightly unphysical here and assume that M is compact. 2/21

Back in the day, people believed that the earth is surrounded by celestial spheres that hold the planets and the stars. 1/13 Today we no longer believe this, but we can still make sense of the celestial sphere as the sphere which is 'out at infinity' in our field of view. When we look out at the stars in the night sky, it's easy to see why people thought it was a spherical shell surrounding earth. 2/13

Linear ordinary differential equations is a topic that you might have first encountered in a calculus course, but they are secretly hiding topological structure! This idea goes back to Riemann, and is the subject of one of Hilbert’s famous questions.  1/n Linear ordinary differential equations are defined over the real line, or the complex plane. But they can be generalised to gadgets living over any smooth manifold. These generalised differential equations are called flat connections. 2/n

The tropical numbers are defined as the real numbers with minus infinity added in, along with the operations of tropical addition and multiplication. 1/n The tropical numbers give an example of a semi-field. This means that they satisfy all the field axioms except for existence of additive inverses.
The tropical 0 is minus infinity. Can you see why there are no inverses? 2/n