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I'm not sure if anything has "happened to the world" - 30 years ago data science didn't exist, and your only choice was finance. Also, data science and finance are both huge fields, and I have to wonder how hard you've actually looked for beauty (mathematical or otherwise) in them. How much do you know about the theory of stochastic differential equations? Information theory? Bayesian inference?
@SebastianGoette Good catch - I missed that point. The nLab page points to Freed's notes on Dirac operators, and so in that context there is probably a fixed choice of Spin or Spin$^c$ structure lying around - but the nLab page itself should probably be corrected.
When you write "The point, for applications of such results..." what applications are you talking about? Or at least: what field of mathematics do they belong to?
I think the historical origin of the distinction between "provable" and "proved" is Godel's first incompleteness theorem. Very roughly, this encodes the observation that a consistent formal system F cannot prove the statement "This statement is not provable in F". But the fact that F cannot prove the statement means that the statement is true in some broader sense.
Well, the functional analysis is bound to be quite involved no matter how you proceed. KK-theory is convenient because it wraps up all of the hard analysis into a few nontrivial theorems, like the fact that the Kasparov product exists and is associative, and that the KK-class of a product is the product of KK-classes. But it should still be possible to work out the analysis by hand if desired.
Well, I don't have a reference handy, but this should be pretty straightforward using Kasparov products. A family of fiberwise elliptic operators determines a class in KK-theory, and the families index map is the Kasparov product of this KK-theory class with the K-theory class of a vector bundle over the base. But the KK-theory class of the product of two families of operators is the Kasparov product of the KK-theory classes of the families, so the result follows from associativity of Kasparov products. Maybe this is worked out in Blackadar's book?
Infinity categories show up in Gaitsgory and Lurie's proof that the Tamagawa number of a simply connected simple algebraic group over a function field over a finite field is equal to 1. This is still quite far from a statement that you could explain to a high schooler, but for all I know it could imply something nontrivial about some elementary diophantine equation. It's sort of an accident that Fermat's last theorem is connected to elliptic curves and modular forms, and if there's an answer to your question then it's probably also an accident of that sort.
Well, I for one feel more enlightened having read your answer than mine. It also explains why differential entropy isn't coordinate invariant - you could work relative to different reference grids.
The ordinary Gaussian distribution is a probability distribution on $\mathbb{R}$, not a probability distribution on the tangent spaces of $\mathbb{R}$. The standard way to generalize to manifolds is via integration against a top degree differential form (just the standard volume form in the case of $\mathbb{R}$). If you truly want probability distributions on the fibers of a vector bundle $E \to M$, take a section of the top degree exterior algebra of the bundle of vertical forms.
