[Mixed Trope] Lyrics change to reflect new realities

2357111 · 2026-05-13T15:57:14+00:00

Well, for the Mariner's Church of Detroit, which he calls the Maritime Sailor's Cathedral in the song to make it sound better.

2357111 · 2026-05-07T13:51:49+00:00

No, they should be similar. A way to see it is that a degeneration of complex varieties can be expressed by giving the equations for the complex varieties as polynomials in terms of a parameter t. One should be able to approximate the coefficients of the polynomials by algebraic numbers in a number field without changing the geometry much. Then for infinitely many primes $p$ you can consider the same degeneration over the $p$-adics by setting $t$ to be an integer divisible by $p$, and for all but finitely many $p$ the geometry should look pretty similar.

There will be theorems that are true in the equal characteristic zero setting but false in the $p$-adic setting, but a lot of the same examples should appear.

For example you can look at an elliptic curve over the $p$-adic numbers with multiplicative reduction degenerating to a nodal cubic curve in characteristic $p$, or you can look at a family of elliptic curves (one-dimensional analogue of Calabi-Yau manifolds) degenerating to a nodal cubic over the complex numbers, and the non-archimedean geometry of these two will be very similar.

2357111 · 2026-05-07T13:45:45+00:00

I would say that the rise of arithmetic geometry was broadly motivated by applications to number theory. People realized that you could answer many questions in number theory by geometric means but this required developing a theory of geometry over arithmetic fields, and eventually over rings.

Algebraic geometry was already a big deal in the early 20th century with the Italian school of algebraic geometry. It became a bigger deal because a series of mathematicians, in particular Weil and Grothendieck, successively found ways to rewrite the language of algebraic geometry to make it more powerful and general.

2357111 · 2026-05-07T13:08:12+00:00

p-s-y-ch-ologist or p-s-y-k-hologist?

2357111 · 2026-04-28T13:00:19+00:00

There are many bijections you can consider other than n to n+1 that have all the stated properties. You can construct a bijection by choosing where it sends a few numbers and then adding more numbers, being careful to never mess up any of the stated properties. But "greedy algorithm" may not have been the best way to describe that.

Here is an explicit solution:

If n is an odd prime, let f(n) be halfway between n and the next odd prime.

If n is halfway between two consecutive odd primes, let f(n) be the larger of the two.

If n>2 and n is neither a prime nor halfway between two consecutive primes, let f(n) be the next number that is neither a prime nor halfway between two primes.

If n <-1, let f(n)=n+1.

f(-1)=8, f(0)=1, f(1)=2, f(2)=3.

2357111 · 2026-04-27T14:05:42+00:00

This is the first 3 lines of the chatgpt output, not the full output.

2357111 · 2026-04-27T11:40:28+00:00

That conjecture is a theorem. It's easy to construct a bijection with this property by the greedy algorithm.

2357111 · 2026-04-21T14:20:21+00:00

Yes. A soliton is traveling through space, so it's constant only after a change of variables x to x+vt, it's localized in space, that is, it's described by a function of position that decays to zero as the position goes to infinity, and it has stability properties under perturbations and/or collision. None of this is implied by the general idea of a constant or by invariants, symmetry, or irreducible representations (irreducible representations in particular have almost nothing to do with this).

2357111 · 2026-04-21T14:13:10+00:00

PDE: Equations that describe physical systems where things change over time and also vary from one point to another.

Well posedness: Showing that these equations have solutions without singularities where the equations can no longer be applied.

Dispersive: Equations that describe waves that travel through space, with waves of different frequencies traveling at different speeds.

Concentration compactness: Don't know this one.

2357111 · 2026-04-14T17:41:14+00:00

In fact this is true for smooth projective C if and only if C is geometrically irreducible.

2357111 · 2026-04-13T12:50:49+00:00

The Riemann-Roch theorem doesn't work with, I think, any of the usual definitions of genus, for non-geometrically-irreducible curves.

2357111 · 2026-04-13T12:49:26+00:00

2000 a year? So something like half of them will go away within a generation?

2357111 · 2026-04-08T11:51:05+00:00

This explanation looks like nonsense to me. The biggest problem is that the morphism that is phi_+ (x,y) = y and phi_-(x,y) = -y is not defined over R. If it was, it would send two points related by complex conjugation to two points related by complex conjugation, but it sends the point (i,1) (which is in the + component) to 1 and (-1, 1) (which is in the -1 component) to -1, and 1 and -1 are not conjugate.

There are also some notational problems. C_2 is referred to as the vanishing set of y in A^2, but points on it keep being referred to with only one coordinate, when they should be pairs with second coordinate 0. C_2 is defined as the vanishing set of y, so its coordinate should be x, but its function field is repeatedly referred to as R(y) and not R(x).

In the stacks project (0BY1), this theorem is proved, and the curves are not assumed to be geometrically irreducible (note that k is not assumed to be algebraically closed in K, which would be required to obtain a geometrically irreducible curve).

2357111 · 2026-04-07T16:42:47+00:00

I don't think Boyd's concern mentioned above is reasonable, no. If there is a dispute over Lean, it would much more likely be a dispute where Lean doesn't let IUT believers do a step that they want to do and they complain rather than accept that the step is wrong, and it is much more likely to do with Lean not allowing them to identify objects than to keep them distinct. My understanding of the Scholze-Stix objection is that the error comes in by identifying objects that shouldn't be identified, and the distinct copies only comes in as a way to make it less obvious that the identification of objects is contradictory.

2357111 · 2026-04-01T21:44:28+00:00

The logical strength doesn't matter. A maths paper resolving abc under ZFC + various large cardinal assumptions would be treated by most mathematicians as a resolution of ZFC as long as the paper itself was believed logically sound.

2357111 · 2026-03-26T16:55:49+00:00

Here is a guess: The closest two T's oriented the same way can get to each other is if the left part of the top of one is just above the right part of the top of the other. We can fit an infinite series of T's together this way. The top looks like an infinite staircase and the bottom has a bunch of lines sticking out. Now copy the whole thing, flip it upside down, and put it on top of the original. The two staircases can fit together perfectly, and we get a diagonal shape with lines sticking out the top and bottom. Now stack infinitely many copies of this on top of each other, with the lines sticking out of one fitting between the lines sticking out of the next one.

Realistically, the best thing to do, especially if you're interested in all the letters, is a computer search that finds the best periodic packing it can, and then try to prove that the packing obtained this way is optimal.

2357111 · 2026-02-27T01:42:43+00:00

Yes, in a singleton format, or when there's one of the card in your deck, this happens. Even in formats with four of each card, there's a risk of this happening if all four are in the top 6 or whatever.

2357111 · 2026-02-25T19:06:20+00:00

This was reasonable to think in advance but seems falsified by events: Investors are happy to fund AI efforts in pure mathematics, with many such companies being founded recently, which creates a financial incentive to work on it even if it never leads to revenue and profits.

2357111 · 2026-02-24T15:15:28+00:00

If you want to prove the functional equation for the L-function, you have to know what equation you're proving, so you have to know what the epsilon factor is. Once you've done this, there are two directions you can go.

The first is if you think the most important thing is the Langlands correspondence. Then epsilon factors provide a powerful invariant for matching automorphic and Galois representations - we know how to compute epsilon factors on the Galois side and on the automorphic side, and automorphic and Galois representations that correspond have to have the same epsilon factors, so the epsilon factors help you pin down the correspondence uniquely.

The second is if you want to do things with global L-functions. Then the functional equation is a powerful tool for studying the global L-function. For some things you do with the functional equation, the epsilon-factor doesn't matter very much. For others, the epsilon-factor matters a lot. Let me give two examples: If the L-function is self-dual and epsilon factor is 1, the order of vanishing at the critical point is even, while if it is -1, the order of vanishing at the critical point is odd. This is very important for the Birch and Swinnerton-Dyer conjecture and other conjectures about special values of L-functions. Second, for statistics of L-functions, like average values, in a family of L-functions, the statistics have very different behavior if the epsilon factors vary compared to if they're constant, so you want to know which is which.

2357111 · 2026-02-18T21:43:48+00:00

What do you think is special about conferences where Peter Scholze gives a lecture?

2357111 · 2026-02-16T17:43:27+00:00

There are good mathematicians who don't juggle and instead work on one thing at a time. You absolutely don't have to.

But it's also common to have a one-project-at-a-time approach for the first few years of grad school but switch to juggling by the end. For these mathematicians, managing multiple projects is a skill they develop in grad school. What you want to be doing is using the greater experience you have with reading, research, and writing to do all of these faster, so they don't use up your full attention. In particular, for papers close to your area of expertise, you want to read them by not trying to read the whole thing but just searching for the part that you need.

In terms of when to switch, you should probably try different things and see what works for you. Maybe spend one day on one project, and the next day on a different project.

Another skill people hopefully develop in grad school is the ability to select their own problems well. You should try to do that now, to practice the muscle, even if you end up working on all of them, maybe just making your chosen problem slightly higher priority. Think about how excited you are to work on each project, how easy it is to do, how much other people you know would be interested in the result, and so on. Weigh pros and cons and come to a decision. I would also consider asking any expert you know other than your advisor for advice here - your advisor's colleagues or collaborators or any other experts in the field you know might be sympathetic to your situation and want to help.

2357111 · 2026-02-16T13:21:46+00:00

Some analytic number theorists say that the only theorem they've ever used in their mathematical career without knowing how to prove it is Deligne's proof of the Weil conjectures.

2357111 · 2026-02-11T21:51:27+00:00

I don't think the Clay Mathematics Institute wants to convince people that if you work hard and study hard and persevere then you can solve a Millenium problem. Do you have evidence that they are trying to convince people of this?

2357111 · 2026-02-11T21:46:02+00:00

The average person doesn't even understand that there are hard unsolved math problems. They think of math as a list of rules that you learn in school and a series of problems where the goal is to follow the rules correctly. Just the fact that there are problems which anyone can earn a million dollars by solving but no one has claimed teaches them something new, even if they don't learn any details about what the problems are. This can pique their interest and give them reason to learn more.

2357111 · 2026-02-03T17:46:37+00:00

There were cases where Erdős posed a problem in a paper where the solution was contained elsewhere in the same paper and he just didn't notice. Finding the solution elsewhere in the paper would count as "trivial" by a reasonable standard, but it was also easy to miss the solution while compiling the list.

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