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Not to shit on all of your middlebrow dismissals but mathematicians are known to steal language from other fields as metaphors for mathematical phenomena. Somet
by danharaj 6y ago
Not to shit on all of your middlebrow dismissals but mathematicians are known to steal language from other fields as metaphors for mathematical phenomena. Sometimes these metaphors are very rigorous and precise and sometimes they're fast and loose. Here, look, I found all these papers which use "repulsion" as a mathematical metaphor for distances between zeroes, eigenvalues, and other special values of a geometric object:
Random matrices: tail bounds for gaps between eigenvalues
Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications."
https://arxiv.org/abs/1504.00396 https://arxiv.org/abs/1504.00396
Real roots of random polynomials: expectation and repulsion
https://arxiv.org/abs/1409.4128 https://arxiv.org/abs/1409.4128
Zero repulsion in families of elliptic curve L-functions and an observation of Miller
https://academic.oup.com/blms/article-abstract/45/1/80/297678 https://academic.oup.com/blms/article-abstract/45/1/80/29767...
Integral Points on Elliptic Curves and the Bombieri-Pila Bounds
Let C be an affine, plane, algebraic curve of degree d with integer coefficients. In 1989, Bombieri and Pila showed that if one takes a box with sides of length N then C can obtain no more than O_{d,\epsilon}(N^{1/d+\epsilon}) integer points within the box. Importantly, the implied constant makes no reference to the coefficients of the curve. Examples of certain rational curves show that this bound is tight but it has long been thought that when restricted to non-rational curves an improvement should be possible whilst maintaining the uniformity of the bound. In this paper we consider this problem restricted to elliptic curves and show that for a large family of these curves the Bombieri-Pila bounds can be improved. The techniques involved include repulsion of integer points, the theory of heights and the large sieve. As an application we prove a uniform bound for the number of rational points of bounded height on a general del Pezzo surface of degree 1.
https://arxiv.org/abs/1301.4116 https://arxiv.org/abs/1301.4116
- LolWolf 6y agoAbsolutely agreed—it's a little funny to see dismissals of the article (which, as I mentioned elsewhere in the thread is actually quite good), even though, as a mathematician [0] we use these analogies (and sometimes really the whole idea) all the time. To add to your list (for less number-theory specific topics): - Lyapunov functions? Energy in physics (just a mathematical surrogate) - Exponential families? Canonical ensemble in physics - Convex duality? Lagrange duality in classical mechanics and the list continues. ----- [0] And, admittedly, a member of a physics lab, even though I don't really do any physics.
- danharaj 6y agoYea! And here's a metaphor to... baking!! The moment this metaphor clicked for me reading a musty textbook with yellowing pages was delightful, I can go back to that moment just by thinking about it. https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneadi...
- LolWolf 6y agoI love it! I had heard of some of these results in relation to Persi Diaconis's work (e.g., [0]). I'm curious, what is the textbook? Would love to take a quick glance at it at least! :) EDIT: I also really like the fact that even the original paper mentioned in the quanta article references other (quite pictorially fun) things, such as the aptly named Hedgehog spaces [1] (see Theorem 3, for example). Perhaps it's violating the prime directive to be mentioning this, but it's a little bit of a shame to see what I think is a good article so quickly dismissed by what is essentially just bikeshedding (that isn't even justified!). ---- [0] https://link.springer.com/article/10.1007/s10955-011-0284-x https://link.springer.com/article/10.1007/s10955-011-0284-x [1] https://en.wikipedia.org/wiki/Hedgehog_space https://en.wikipedia.org/wiki/Hedgehog_space
- danharaj 6y agoThought about it, and it was definitely the original paper in some journal volume. I remember the pictures, and the ones I drew myself :) Found it here: http://www-personal.umich.edu/~kochsc/MilnorThurston.pdf http://www-personal.umich.edu/~kochsc/MilnorThurston.pdf