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What's the most difficult mathematical concept you've encountered and had to understand so far?

81 Answers

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Ring spectra. I can prove some theorems about them, but still have a really hard time getting used to them as a generalization of both rings and cohomology theories.
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Vertex operator algebras. Hopeless and misguided to explain it to a layman.
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I find probabilistic proofs of deterministic results unintuitive and hard to convince myself with. It just feels even more like a hat trick than most non-constructive methods to prove stuff.
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Quadratic variation. It’s like the total variation except you take a square in the defining sum and take a limit in probability instead.

 The definition is quite simple, but I have no idea why it encodes as much information as it does, or the intuition behind why it’s as important as it is.

Also honourable mention to curvature, entropy, and Ito integrals.
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Weyl group multiple Dirichlet series. I'm way more familar with them than I was two years ago, but damn.. Its basically an L-series of L-series but with special "correction factors" attached to make all the things you want work (analytic continuation, functional equations, etc.) Unless you understand where the correction factors come from (Fourier coefficients of Eisenstein series on metaplectic groups), they look like someone conjured Magic out of thin air!
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This isn't a specific theorem or anything, but some of the more advanced material on random graphs is terrifying. I was trying to read parts of Bollobás to understand a paper for a course and I just couldn't follow the arguments after a certain point. It is a very interesting subject though.
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Differential forms.

No matter how hard I try, I just can't make any sense of it. I know why it's amazing, I understand why it works, I know why it's important as a physicist to get a good understanding of it - but it just doesn't make "click".
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Probably a piece of cake for you guys, but Divergence/Stokes’ Theorems. It was one of the last topics we covered before the final and I just settled with learning that the former was div F dV and the latter curl F dS.

Tbh all of line integrals was a bit tough, but it got crazy near the end. If I recall correctly, the curve needed to be “piecewise, simple, smooth, closed,  positively oriented, and on an open region”. Almost gave up reading that particular mouthful.
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Universal properties. I can "understand" it in specific contexts and the intuition behind it but I still have a really hard time with the formal definition and technical details.
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K-theory. It's just papers citing constructions from other papers citing constructions from papers in French without any available translation. Wtf is anything. Try to explain it in layman's terms? uhhh.... if I could I think I would be published in a heartbeat. Nobody else seems to have such an explanation :(Maybe I could explain homotopy, and a splash of algebraic geometry, but this is not really beginning to cover the assignment of algebraic structures to all kinds of topological spaces that we call K-theory, it's only the beginning of homotopical and homological algebra, the setting in which K-theory reveals its necessity.

The layman's questions are: what is a hole? and who cares about a rigorous definition of a hole? Starting like 200 years ago a suitable definition of what constitutes a 'non-hole' was discovered. A 2-dimensional region contains no holes if you can draw a loop and guarantee that it can be shrunk to a point. A (number of) hole(s) is then a failure of this property. Similarly a 3 dimensional region contains no holes if you can paint a sphere and have essentially the same guarantee. This generalizes to higher dimensions, of course, as generalization to higher dimensions is any geometer's favorite thing to do.Now, who cares? Well the holes of a region determines some very nice properties of vector fields, specifically some properties that any physicist should care about. Holes of a region dictate how you can turn a crazy integral from vector calculus into a slightly less crazy integral from vector calculus using conservation laws (think Stokes theorem).

Now a splash of algebraic geometry: whether or not some loop \*can\* be shrunk to some other loop or some point, is, in general, a huge pain in the ass to determine sometimes. If instead of looking at such a geometric-topological property, we try to characterize this in terms of what functions exist over a region (be them complex functions, continuous functions, polynomial functions, or vector fields), we can turn something fundamentally useful, like conservation laws of vector fields, into a slightly less complicated question about polynomials for example, rather than directly looking at an underlying space. Basically we can use theories  to establish Vector fields <-> geometric nonsense, and geometric nonsense <-> algebraic nonsense, to then cut out the middleman and be left with Vector fields <-> algebraic nonsense and somehow this actually cleans up some arguments quite a bit.

Now, K-theory? who fucking knows, man. Not me.

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