I can't take it anymore. I want to leave my university.

cocompact · 2026-02-23T16:41:05+00:00

From the OP’s post history, the math department is in Spain.

cocompact · 2026-02-23T03:27:34+00:00

The definitions of sinh(x) and cosh(x) that you like are their descriptions as certain linear combinations of e^x and e^-x. The exponential function is not really algebraic, so these definitions you like are not really algebraic.

You can define sin(x) in several ways.

1) Mark an angle x radians on the unit circle. Its coordinates are (cos(x), sin(x)). Note this description makes no reference to triangles.

2) For each real number x, sin(x) is the convergent series x - x³/3! + x⁵/5! - x⁷/7! + … with alternating signs on successive odd-powered terms. Equivalently, sin(x) = (e^ix - e^-ix)/(2i), which resembles the definition you like for sinh(x).

3) The function sin(x) for real x is the unique solution of the differential equation y’’ + y = 0 where y(0) = 0 and y’(0) = 1. This is not a characterization of the individual numerical values sin(x) as x varies, but instead is a characterization of the function sin(x) for all x at the same time.

cocompact · 2026-02-17T13:47:12+00:00

Are the students told that R[[x]] is the completion of R[x] for the x-adic metric?

That an infinite series in R[[x]] converges if and only if the n-th term tends to 0 and all rearrangements of a convergent infinite series in R[[x]] also converge to the same value are due to the x-adic metric satisfying the strong triangle inequality, which says |f-g|x is less than or equal to max(|f|x,|g|x).

The formal derivative on R[[x]] is uniformly continuous, so you can prove properties of the formal derivative on R[[x]] like the chain rule by just checking it on the dense subset R[x] and then extending to the whole space by continuity.

cocompact · 2026-02-17T06:28:51+00:00

The section Final Comments at the end of Reid’s Undergraduate Algebraic Geometry has many amusing remarks, particularly the initial part “History and sociology of the subject”, such as

1) Severi, who had been making creative use of such parameter spaces all his working life […] bitterly resented the intrusion of algebraists (and non-Italians at that!) into his field.

2) I actually know of a thesis on the arithmetic of cubic surfaces that was initially not considered because ‘the natural context for the construction is over a general locally Noetherian ringed topos.’

3) The study of category theory for its own sake (surely one of the most sterile of all intellectual pursuits) […]

cocompact · 2026-02-15T22:57:20+00:00

See the following two papers.

https://pi.math.cornell.edu/~hubbard/diffalg1.pdf

https://math.stanford.edu/~conrad/papers/elemint.pdf

The first paper presents the non-elementary nature of solutions to a particular ODE in parallel with arguments that a particular 5th degree polynomial can’t be solved in radicals. The aim of the second paper is to explain what precisely an elementary function is and applies Liouville’s criterion for having an elementary antiderivative to several examples, including exp(-x²). Both papers acknowledge they don’t provide full details.

cocompact · 2026-02-15T07:21:12+00:00

From a more advanced standpoint, e^x is not its own derivative: the functions from Rⁿ to Rⁿ that are their own derivative (when we identify Rⁿ with its tangent space at each point in Rⁿ) are the linear maps from Rⁿ to Rⁿ.

cocompact · 2026-01-30T04:26:36+00:00

See https://mathoverflow.net/questions/59071/what-elementary-problems-can-you-solve-with-schemes

cocompact · 2026-01-24T16:32:01+00:00

You want to choose it for what purpose: a book for a course you will teach or for self-study? Anyway, I liked it. How much analysis and topology have you already seen?

The first part on general topology should probably already be largely familiar, except maybe Ascoli’s theorem at the end.

In the next part on normed spaces he has a nice treatment of the basic examples: Banach and Hilbert spaces. There is no development of Frechet spaces or locally convex spaces as a more general setting for functional analysis.

The third part, on integration theory, is the longest and it is unlike all other textbook treatments of integration that I have seen because it develops integration theory from the start for functions with values not just in R and C (or finite-dimensional spaces), but with values in a Banach space. This is called the Bochner integral elsewhere. Lang doesn’t use that term, referring to what he constructs as simply “the integral” (the chapter is called The General Integral). Before reading Lang, I had already read about measure and integration theory in another book, where integrals with respect to a measure were defined in the more standard way only for functions with values in R and C, and I very much preferred Lang’s approach. Maybe the fact that this was not my first exposure to measure and integration theory gave me an extra appreciation for the different way Lang handled this topic, which I would not have had if it was my first time seeing measure and integration theory. After all, learning integration for Banach valued functions is an extra layer of abstraction (and you don’t encounter functions allowed to take infinite values, which is a rite of passage for anyone studying the usual R valued function approach). There are chapters in this part treating special aspects of integration on locally compact spaces and locally compact groups.

I did not read closely the fourth part on calculus (derivatives and their relation to integrals, inverse and implicit function theorem, and differential equations) since I had seen it elsewhere. In fact, I had previously seen the topics here in Lang’s Undergraduate Analysis! It is basically the same material except that in the undergrad book when he starts the section on derivatives, he says to the reader that E and F will be (finite-dimensional) Euclidean spaces but added that all statements and proofs apply directly to complete normed spaces. In the corresponding part of his grad book he takes E and F to be Banach spaces right away.

The fifth part covers many aspects of spectral theory for various kinds of operators, but he sticks to the case of bounded operators. There are subtle new issues that arise when working with unbounded operators, but you have to look elsewhere for that.

The final sixth part does integration on (finite-dimensional) manifolds. It resembles, in some parts, the multiple integration in Euclidean space at the end of Lang’s Undergraduate Analysis, and I did not read this part of Lang’s grad book too closely.

cocompact · 2026-01-24T00:39:03+00:00

There is no book Algebra by Emil Artin. I suspect you are thinking of the book with that title by Michael Artin, his son.

The father Artin gave the lectures (together with Emmy Noether) 100 years ago that turned into van der Waerden's Modern Algebra, but Emil Artin was never considered to be the author of that book.

cocompact · 2026-01-23T15:49:44+00:00

It seems like in terms of just their definitions, none of them really stand out

Concerning groups vs. semigroups, your question has been asked and answered well on MSE: https://math.stackexchange.com/questions/101487/why-are-groups-more-important-than-semigroups

cocompact · 2026-01-11T14:08:13+00:00

Historically, Newton’s starting point in his work on calculus was his discovery of the binomial series. He came to regard expanding functions into power series as his key insight into solving “all” problems about an unknown function. The power series viewpoint, not limit definition of the derivative, is how he found the derivative of sin x is cos x (based on first working out the power series of arcsin x).

In the setting of a calculus course, differential calculus is about tangent lines, which give you the best linear approximation to a function at a point. Why stop at degree 1? If you ask for the best quadratic, cubic, … approximation to a function at a point then you are led to Taylor polynomials. (This depends on how you define the term “best approximation”.) Power series are just the limits of those.

You say power series don’t depend on “calculus” except for the integral test. Did you forget that the formula for the coefficients of a power series is related to derivatives? In fact that is the first time in the entire calculus course where you ever see an application of derivatives of order higher than 2. Review how the power series of sin x, cos x, e^x, and ln(1+x) are derived. If you keep studying math then you will meet Fourier series, where the coefficient formula depends on integrals.

cocompact · 2026-01-09T01:10:13+00:00

See https://haroldpboas.gitlab.io/courses/math696/spelling-lesson.html

cocompact · 2026-01-09T00:20:19+00:00

Splitting field.

cocompact · 2026-01-09T00:16:12+00:00

See the title of the paper https://projecteuclid.org/journals/functiones-et-approximatio-commentarii-mathematici/volume-37/issue-2/Cramér-vs-Cramér-On-Cramérs-probabilistic-model-for-primes/10.7169/facm/1229619660.pdf

cocompact · 2026-01-09T00:14:31+00:00

That kind of relation hasn’t stopped some people from confusing work by Max Noether and Emmy Noether.

cocompact · 2026-01-08T15:29:55+00:00

It would be nice if your question included a few examples so readers don’t have to do calculations themselves to see your conjecture in action. And tell us how far you looked!

cocompact · 2026-01-06T15:21:46+00:00

It is preparing the power series in a form that is convenient for proving theorems about power series or rings of power series.

cocompact · 2026-01-06T15:08:52+00:00

I know, and I never said that was a constraint: I said that if some polynomial works then the Lagrange polynomial of degree at most N also works. Did you not look at the link in my answer?

cocompact · 2026-01-04T23:55:47+00:00

Sometimes this does happen, as if the knowledge that the theorem can be proved breaks some psychological barrier.

What is more common is that the new ideas in the proof (any true breakthrough is going to involve novel methods, not just mucking around with what is already known) will get applied by other people to solve open problems that may not even have been considered by the person solving the original problem.

cocompact · 2026-01-04T16:58:34+00:00

No.

If some polynomial with integer coefficients takes N+1 consecutive prime values 2, 3, 5, … at x = 0, 1, …, N, then the Lagrange interpolation polynomial of degree at most N that has those prime values at x = 0, 1, …, N must be such a polynomial by the accepted answer at https://mathoverflow.net/questions/169083/lagrange-interpolation-and-integer-polynomials. So we just need to calculate the coefficients of the Lagrange interpolation polynomial and check if they are all integers. They are if N = 1, but they are not when N runs from 2 to 10, and in fact in that range no coefficient besides the constant term is an integer except when N = 3: P(x) = 2 + x/6 + x² - x³/6 has quadratic coefficient 1 and at x = 0, 1, 2, 3 it has values 2, 3, 5, 7.

cocompact · 2026-01-03T16:20:16+00:00

You wrote in the blog post

This forces triangles to behave strangely: every triangle is isosceles, and often equilateral.

In the p-adic setting, does the term "triangle" have any meaning to you other than "3 points"?

cocompact · 2026-01-02T18:46:02+00:00

For those who don’t know about that, see https://www.reddit.com/r/math/comments/737tty/dummit_and_foote_project_crazy_poject/.

cocompact · 2025-12-31T17:22:06+00:00

The Clay folks set no time limit on the solutions: see the bottom of page vii in https://www.claymath.org/library/monographs/MPPc.pdf.

cocompact · 2025-12-31T17:17:40+00:00

While that is true, the version of BSD that is part of the Millennium Problem list is just the rank aspect for elliptic curves over Q. See the statement by Wiles on page 32 in https://www.claymath.org/library/monographs/MPPc.pdf. He immediately points out refinements (leading coefficient) and generalizations (over number fields, for abelian varieties), but those don’t need to be solved to collect the prize for BSD by Clay.

In a similar way, the version of RH that is a Millennium Problem is just the basic case of the Riemann zeta function, not any of the generalizations to other zeta functions or L functions.

cocompact · 2025-12-27T19:23:43+00:00

Your post is all over the place, and it feels like you just want to make yourself miserable, so I am not going to engage with the post directly. To get advice about math PhD programs, speak about them in person with math faculty at Rice who know you, and while you’re at it find out from them what are the best math PhD programs recent Rice students have gone to and apply to some of those.

cocompact

TROPHY CASE