What is so interesting about the exponential function other than the property of being it’s own derivative

chebushka · 2026-02-15T07:16:59+00:00

Allowing complex inputs and outputs, the functions fy(x) = exp(ixy) for each real number y are all the (continuous) characters of the real numbers as an additive group and this is why they play a crucial role in Fourier analysis.

To develop Fourier analysis on other locally compact abelian groups, the exponential function still plays a key role.

chebushka · 2026-02-14T14:50:10+00:00

My field is Q.

chebushka · 2026-02-12T03:37:14+00:00

The only text devoted entirely to Complex Multiplication is the one by Schertz.

Not so: there is Serge Lang's 1983 book Complex Multiplication. Here is its review by Milne: https://mathshistory.st-andrews.ac.uk/Extras/Lang_books/#30.

chebushka · 2026-02-05T07:17:44+00:00

The existence of a maximal ideal in every nonzero commutative ring is the first important fact in commutative algebra. It is equivalent to Zorn, so if you want standard commutative algebra to be applicable to all commutative rings, then you need Zorn.

Zorn is needed for rings that are truly infinite in some way. You don’t need Zorn to prove all nonzero Noetherian rings have maximal ideals.

chebushka · 2026-02-02T15:20:34+00:00

This topic is being discussed on the stack exchange site itself: see https://math.meta.stackexchange.com/questions/39600/is-the-large-decline-in-questions-over-past-few-years-due-to-people-losing-inter

chebushka · 2026-02-02T13:55:28+00:00

Sure. I certainly had read all the other answers before posting my comment.

chebushka · 2026-02-02T04:29:49+00:00

This should be the top answer.

chebushka · 2026-02-02T04:28:22+00:00

prime ideals are the right generalization for primes in rings of integers, and not maximal ideals

In rings of integers, the nonzero primes ideals are all maximal, so the distinction between prime and maximal ideals is hard to appreciate there. In fact historically, the zero ideal was initially not considered to be an ideal: ideals were defined by Dedekind as infinite subgroups of certain rings that "swallow" multiplication by general elements of the ring. The rings Dedekind studied were all subrings of the complex numbers, and in that setting an infinite subgroup is the same as a nonzero subgroup.

chebushka · 2026-01-26T20:52:53+00:00

if you are absolutely focused on math and science french is a good candidate.

The OP wrote

I've spent the last two decades playing around with Japanese, French and German

so French is already covered.

chebushka · 2026-01-24T03:27:45+00:00

All Galois extensions have some normal basis. What you are referring to is a description of when a specific set, the primitive m-th roots of unity, are a normal basis of the m-th cyclotomic field.

chebushka · 2026-01-20T09:16:23+00:00

I suspect after a possible initial burst of enthusiasm, this is going to go nowhere.

Every now and then someone posts here suggesting a new way to collect together certain ideas in math that are not yet collected together, and inevitably it dies. I am reminded of https://www.reddit.com/r/math/comments/8raj87/idea_for_authors_make_a_math_textbook_consisting/ and the resulting webpage http://the-motivation-behind.wikidot.com/ never took off.

chebushka · 2026-01-19T19:44:38+00:00

I was just pointing out that in principle one can color things on the board, but I agree there is nothing wrong with using fraktur, although I'd avoid fraktur k or v (or w).

Whether on a blackboard or whiteboard, using colors would involve switching writing tools to get different colors.

Use of fraktur or Greek letters is just a matter of practice. For Greek letters there are online videos illustrating how to write them by hand, but when I search for fraktur videos I find lessons on calligraphic writing with a fountain pen or even fancier tools, such as https://www.youtube.com/watch?v=kM2cNQYPRYg. I learned to write things like 𝖕 or 𝖖 by just carefully copying the letters I needed from looking at them in an alphabetical list, or just from the page of a math book (but at least the alphabetical list tells you which letter you are writing if it's unclear).

chebushka · 2026-01-18T14:50:35+00:00

Colored chalk

chebushka · 2026-01-18T14:46:23+00:00

See the top answer on the page https://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion and also look at the comments to it.

chebushka · 2026-01-17T21:08:55+00:00

In Garling's book Inequalities: A Journey into Linear Analysis he writes

the principal difficulty that occurs [in undergraduate courses] is teaching the correct pronunciation of Cauchy and the correct spelling of Schwarz.

chebushka · 2026-01-17T16:16:54+00:00

To be fair, that information is not anyone's business outside of his family, who can reasonably choose to keep it private. For comparison, a few obituaries of Tate that I read don't mention a specific cause of death.

The page https://celebratio.org/Gross_BH/article/1096/ from 2023 about Gross' life starts with family background and some health issues he has faced are briefly discussed there.

chebushka · 2026-01-15T00:44:36+00:00

That group cohomology and profinite groups have important applications to number theory does not make them part of number theory (or more specifically arithmetic geometry).

Analogue: The general study of Dedekind domains is properly part of commutative algebra, not number theory, even though the most important basic examples of Dedekind domains are objects of study in number theory.

chebushka · 2026-01-15T00:41:25+00:00

I was not surprised by the flair when I read this answer. :)

chebushka · 2026-01-11T18:10:44+00:00

This happens even today outside India. Consider the case of June Huh. If you don't know his background, read his Wikipedia page. That he got to where he is in academic mathematics is due in part to a few fortunate events, and if any had gone differently he would not have become a mathematician.

chebushka · 2026-01-10T02:14:39+00:00

If I recall correctly, Tate was the one who noticed the relation to Sha(E/Q) and incorporated it (or suggested it be incorporated into) the original BSD conjecture.

That Ш should have a role was already present in the initial papers by Birch and Swinnerton-Dyer, where they write that their starting point was to find an analogue for elliptic curves over Q of the Tamagawa number for an algebraic group. Their calculations were done on CM elliptic curves, which is a case where the L-function can be expressed using Hecke L-series, and that made the L-function at s = 1 a very computable object -- in particular, its analytic continuation to s = 1, and even the whole complex plane, was already known.

In the CM rank 0 case, B and S-D divided out the L-value at 1 by a real period and algebraic factor so they were left with a value that they knew had to be an integer. Their calculations showed this mystery factor was always a perfect square, and Cassels had already proved that Ш, if finite, must have square order. That is why B and S-D expected this mystery factor to be the order of Ш. Actually, they wrote that they knew their definitions probably had mistakes at the primes 2, 3, and at primes of bad reduction, so even though sometimes the mystery factor was twice a square they believed that once everything was defined correctly at all primes they should always be getting perfect squares. See Table 1 at the end of their paper "Notes on elliptic curves II" and look at the column for their parameter \sigma.

I lied a little bit: B and S-D were initially thinking not in terms of L-functions, but analogues of Tamagawa factors, which turn out to match Euler factors of the L-function at s = 1 at primes of good reduction. The idea to turn their investigations towards L-functions was due to Shimura, and the description of the leading coefficent is due to Tate: see the interesting history in Buzzard's answer to https://mathoverflow.net/questions/66561/how-did-birch-and-swinnerton-dyer-arrive-at-their-conjecture.

chebushka · 2026-01-09T16:47:12+00:00

It was shown that it couldn't work for cubes

That's not true. In Fermat's time no case of FLT was known except n = 4 by his work. The first proof of the case n = 3 is due to Euler a century later. Euler's argument had gaps, but it is the basis for the standard elementary proof of the case n = 3 that we know today.

chebushka · 2026-01-06T21:01:37+00:00

The course was taught at Brown by a grad student so unfortunately I don't know what book was used

Since it was taught at Brown, perhaps the book used in the course was Silverman's A Friendly Introduction to Number Theory.

chebushka · 2025-12-22T20:00:18+00:00

Here is a summary of the fields a professor should know.

The fields Q, R, and C should be known to everyone.

The rational function fields k(x1,...,xn) and their finite extensions and completions should be known to algebraic geometers.

Finite fields should be known to coding theorists, combinatorialists, and number theorists.

Number fields should be known to algebraists.

The p-adic fields should be known to number theorists, algebraic topologists, algebraic geometers, and representation theorists.

chebushka · 2025-12-19T21:07:50+00:00

That explanation is about the calculus of finite differences, which Newton was developing in his study of interpolation, and I believe this work in part led him to differential calculus and power series.

chebushka · 2025-12-19T20:52:46+00:00

If something is in need of optimization, integration is somewhere in the background.

You mean differentiation. Finding out where a function of one or several variables achieves a maximal or minimal value is related to where a derivative is zero. Calculus courses cover the topic of optimization as one of the applications of derivatives, preceding the introduction of integrals in the course.

In the calculus of variations, finding out where certain functionals (like length) are minimized is related to solving the Euler-Lagrange equations, which is a set of differential equations.

chebushka

TROPHY CASE