lipshits-continuous

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“what do you do in algebra”

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Something that was really enlightning for me was learning that S_6 has a nontrivial outer automorphism and that none of the other S_n do.

That made me realize that S_n is not a "real" family of groups. They have a lot in common, but they also have lots of differences. It just so happens that they fit the same description changing one parameter, but that doesn't mean that you can meaningfully compare them.

And once you see this it's really hard to unsee it. Finite fields are like this too. They have a lot in common, but also those of different characteristics have huge differences.

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youve heard of twin primes now get ready for twin odd numbers. its when there are 2 odd numbers separated by a non odd number. how many are there? will another pair be discovered? is there an explanation for this phenomenon?

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Gosh the Picard group is so scrumptious. Pic me up any time Picard group.

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Try intuitively visualising spinors, again.

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I can't stop thinking about you

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Love killing stuff in maths. "Kill the kernel", "kill the second cohomology group", "kill the commutators", "kill the constant term", "kill the Lie bracket." Yes. Hell yeah even.

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Favourite short exact sequence?

Great question! I gotta give this one to

0 -> P(ℝ³)/P(ℝ²) -> ℝ ⊗ ℝ/πℤ -> Ω¹(ℝ) -> 0,

where P(ℝ³) is the abelian group of polyhedra in ℝ³ modulo scissors congruence, P(ℝ²) is its subgroup generated by the prisms, and Ω¹(ℝ) is the group of absolute Kähler differentials on ℝ. The first map is the Dehn invariant, and the second maps a generator r ⊗ (θ + πℤ) to (r/cos θ) d(sin θ) if cos θ ≠ 0 and to 0 if cos θ = 0. This is actually a short exact sequence of real vector spaces, but note that the tensor product is a tensor product of abelian groups.

The injectivity of the left map shows that volume and the Dehn invariant are necessary and sufficient to characterize scissors congruence of polyhedra :) The fact that not all edge length-angle tensor combinations can show up reflects a fundamental relationship between polyhedra and the kinds of edges that they can be made up of, and the difference between the ones that can and can't show up is precisely measured by the space of Kähler differentials!

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Who is on y’all’s shortlist for most important mathematicians of all time?

I think mine (in chronological order) probably include:

Pythagoras, Euclid, Descartes, Leibniz, Euler, Gauss, Gödel, Turing

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are there any books and/or online resources you’d recommend for learning category theory?

i’ve been looking over some of the basics and think it looks really interesting :>

Borceaux's Handbook of Categorical Algebra is for me the definitive reference. It may be a bit much to recommend at a beginning level, but on the other hand it has absolutely every detail carefully spelt out, in contrast to other resources that leave things as exercises to the reader.

For something a bit more streamlined, I would say Riehl's Category Theory in Context is an excellent option.

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a general concept in maths that i love is when the standard rules of a simple version of a structure fail, but the extent to which they fail is measurable with some other structure. e.g. number rings dont necessarily have unique factorisation, but the extent to which it fails is controlled by the ideal class group, a finite abelian group. in R^n, closed forms of nonzero degree are exact (e.g. u may know that having zero curl makes you conservative) but this isnt true for spaces with stranger topology. the degree to which this fails is measured by the de rham cohomology, closed forms mod exact forms !! great stuff

[lipshits-continuous]

a general concept in maths that i love is when the standard rules of a simple version of a structure fail, but the extent to which they fail is measurable with some other structure. e.g. number rings dont necessarily have unique factorisation, but the extent to which it fails is controlled by the ideal class group, a finite abelian group. in R^n, closed forms of nonzero degree are exact (e.g. u may know that having zero curl makes you conservative) but this isnt true for spaces with stranger topology. the degree to which this fails is measured by the de rham cohomology, closed forms mod exact forms !! great stuff

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algebraic-duellist : galois :: algebraic-dualist : gellois

correct, +10 points for you

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category theory emergency? dial 00よ, now!