## Arbeitsgemeinschaft 2020

The April 2020 Oberwolfach Arbeitsgemeinschaft will be on derived Galois deformation rings and the cohomology of arithmetic groups!

For those who don’t know, the Arbeitsgemeinschaft (“study group”) is different from usual Oberwolfach workshops (or workshops more generally) — the idea is that the participants learn the material and then teach it to each other. I have never actually been to one (please leave a comment on your experience if you have), so I’m not sure that I can describe it better than reproducing the official blurb here:

The Arbeitsgemeinschaften mainly address to non-specialists who want to broaden their outlook on mathematics and to junior researchers who wish to enter a field for future research. Experts are also welcome. The idea is “learning by doing” – similar to the Seminaire Bourbaki. Participants have to volunteer for one of the lectures described in the program of the Arbeitsgemeinschaft. After the deadline for application the organizers choose the actual speakers to give them enough time to understand the subject and to prepare for their lectures.

If you are interested in learning this material, please consider applying! There is currently an ambitious mix of material from the cohomology of arithmetic groups to the Taylor-Wiles method to to derived deformation rings. The expectation is certainly that you are not an expert on all of these topics. (The precise emphasis of the workshop will naturally depend on the exact mix of participants.) Click here to see an outline of how we have conceived what the lectures might be. When you apply, you can choose which lecture you (might be) prepared to give, assuming you are given enough advanced warning! All the links you might need (for applying and other information) can be found here:

2020 Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups

Note that the timing of this workshop is planned to immediately preceed the
Hausdorff Trimester in Bonn of which I have mentioned previously and will give more information about in an upcoming post. Consider going to both!

## Mathematische Zeitschrift (Part II: for authors)

In this post, I give some tips for authors considering submitting to Math Zeitschrift, especially a paper in algebraic number theory. The first suggestion is to read Part I. This should give you a good sense of the standards required. (Of course, it’s always hard to judge your own work without bias.) I do, however, have a few more specific tips for authors:

Submit the paper to the journal rather than email me directly: It’s certainly not a faux pas to send it to me directly, it’s just that it’s easier for me for various administrative reasons if you send the paper through the official channels. (I have a remark to that effect here but perhaps people find my email address directly though the editorial board listings.) There are two associate editors besides myself who deal with number theory papers at Math Zeit: Philippe Michel and Dipendra Prasad. Most of the time it will be clear which editor should be assigned what paper, but I believe you can also make a suggestion for which editor is appropriate when submitting. We can and do swap papers around if we believe another editor is more qualified to handle the paper.

Please don’t ask me before submitting if you think a paper is “of the right level” to submit to Math Zeit: Again this is not an unreasonable question, it’s just that it puts me in an awkward situation — if I say “yes,” then I will feel guilty if the paper is ultimately rejected, so I will usually just explain that as a matter of policy I refrain from giving any such opinions. If there is any situation in which I feel some conflict of interest with regard to a paper, I will usually ask Dipendra to take over (I think I have only done that once).

Don’t suggest a “suitable referee” unless you are doing so as a bluff and don’t want me to choose that person … except maybe I will call your bluff … unless maybe it is a double bluff!

Please don’t complain if your paper is rejected. It really isn’t doing you any favors. There is some cause for complaint if your paper gets rejected after six months without any indication that it’s been read, but that doesn’t happen with the papers I reject.

I have to admit (surprising though I’m sure this will be to regular readers) that I myself are not immune to mild frustration with journals, especially those for whom the evidence is considerable that the delinquencies in handling my paper may be due to editorial mismanagement. I very nearly wrote a blog post entitled “DUKE HOSTAGE CRISIS DAY NINE HUNDRED” documenting one of my recent experiences, but Richard Hain was apparently brought on board to tidy ship at the last moment and the situation was resolved a cool 54 days short of the intended posting date.

All that said, there is one context in which such an email may make a difference. If one of the authors is just about to go on the job market, then I do feel inclined (if possible) to make what ever efforts I can to expedite the process whenever that is reasonable. Actually I make such efforts unilaterally with younger people I have reason to believe may be applying for jobs (it doesn’t always work, of course).

When you get a referee report, please don’t rush to revise it as quickly as possible. It is exciting to get a report which indicates that your paper has a good chance of being published. There’s a temptation to rush through the referee report as quickly as possible addressing to the minimal extent the complains and sending it back immediately. You will do yourself (and everyone else) a favor by reflecting a little bit more both on what the referee says and on the paper itself. You probably spent a long time writing the paper; but it most likely has been a while since you looked at it and so the time when you get a referee report back is an excellent opportunity to return to the paper with fresh eyes and see how it can be improved. It’s true that referees often request pretty annoying things, but definitely sometimes those suggestions are good ones and should be taken seriously. Note that it is always a good idea to include in your resubmission a file indicating exactly how you responded to the referees comments.

Why should I submit to Math. Zeit. rather than other “similar” journals? How does one go about choosing a journal? It’s always been a tough question. I think that one reason to submit to a journal is that you come across interesting papers which are published in said journal. So take a look at the number theory papers published there recently and then think about submitting. I think that Math Zeit has some great papers and I’m happy with the papers that I get to accept. For example, during my tenure, I think there have been quite a few quality papers on Shimura varieties. I suspect that accepting one quality paper in subfield X often begets another submission by people in the same field. (In order to leverage that as an editor, of course, one then has to increase the standards of the journal in that particular subdiscipline.)

Have you made any blunders as an editor? There have definitely been one or two papers which were rejected but which I later came to think should have been accepted. At least the one paper in particular I am thinking of ended up in a nice home, so all’s well that ends well.

Posted in Mathematics | | 4 Comments

## Mathematische Zeitschrift (Part I: for reviewers)

I am an associate editor for Math Zeitschrift. I thought that here could be a good place to record a few useful comments that I often pass on to reviewers. It is my intention for future referee requests to include a link to this post. I have tried to keep to issues specifically related to Math Zeitschrift rather than issues common to all reviews, although much of this advice could be applied more broadly. In particular, I have avoided discussing problems like “how much of the proof am I supposed to check” because that is not really an answerable question (beyond noting that is is the authors who are ultimately responsible for the validity of their arguments).

This post should also be useful for authors as well, although I will have a few more specific comments from them in the next post. The first thing a reviewer needs to understand is:

What is the level of the journal? This is probably the most important question to answer. The usual way that journals position themselves is to say that they are looking for submissions “similar to the level of Duke” which ranges from accurate to ambitious at best but is not necessarily useful for the reviewer unless they take it at face value (which they shouldn’t). Another way to proceed is to discuss the acceptance rate. At Math. Zeit, I believe the acceptance rates are in the range of 20% or so. Again, this is not so useful unless you have access to all the papers which are submitted — which you don’t. Slightly more informative is the following experimental observation: early on in my tenure as associate editor, approximately 60% or so of papers had favourable “quick opinion” requests. So, most likely, the standards are higher than you think. This reflects the unfortunate fact that more decent papers are submitted than can be accepted. Perhaps the most useful measure of quality, however, is to give a list of papers I have managed which have been accepted.

This is by no means a complete list, but I believe it is a complete list from some contiguous period of time (so no cherry picking going on, slipping in Jack Thorne papers just to look good):

You may form your own judgement from this. More importantly, as the quality keeps increasing, it might be useful to compare the paper I send to you with the papers above in areas most familiar to you and ask: “is this paper better? is it at least as good?

What are you looking for from a quick opinion? At this point, I have decided not to convey any information from rejections in quick opinions to the authors. The reason is that enough authors respond with complaints that it is too much trouble, especially since (negative) quick opinions are mostly opinions concerning relative quality, not in depth reviews with useful suggestions. (Of course if the quick opinion comes with a useful suggestion I will pass it on.) So mostly I am looking for two things: is the paper at the level of Math Zeit, and (if so) do you have suggestions for possible referees for this paper? On the other hand, if it’s not up to snuff, could you give me a brief summary of why you came to that opinion (this is for my own benefit). Importantly, I am looking to reject most papers quickly at this stage. Mathematics has a problem with slow reviewing times, but it is especially painful to get your paper rejected (without any real useful feedback) after some especially long period of time. My dream is that all the papers that I reject will be rejected within a month. The journal process will always lead to good papers being overlooked and rejected; it’s hard for me to solve that problem but at least I can reject them quickly! I should say that I am pleased with how quickly I manage to reject most papers that are ultimately rejected (but by no means am I 100% compliant with my goal, unfortunately).

What if I don’t respond to your email asking for a quick opinion? Then you are added to the list of names destined for a sulfurous end, or, if you are lucky, merely a accidental prod from a poisoned umbrella next time we are at the same conference.

What if I write back but decline to offer an opinion, either with some excuse (busy, unqualified, uninterested, mortal enemies with one of the authors, or no excuse at all)? That’s perfectly acceptable.

What if I’m actually the advisor/mentor of one of the authors submitting the paper and don’t respond because I’m annoyed you didn’t check that before hand? Oops, sorry! An acceptable excuse but I would still prefer an email.

How long should a quick opinion take? Anywhere from a day to a month depending on exactly what you write back or where you are (I have one reviewer who always seems to get my emails while at the beach, for example.) That said, I have reviewers who always seem to write back several pages of detailed notes within 24 hours.

Should I summarize the paper? I mostly think that is not really necessary for any review. It is useful to explain what is novel or interesting about the paper, although the author should really have made an effort to do that as well.

Should I send my referee report as a pdf or a text file? I think I would prefer that, if it is a pdf file, the pdf file not contain the recommendation of the reviewer to accept the paper or not. Especially in cases in which the recommendation is not definitive, it puts me in an awkward position to reject the paper if the recommendation is generally (but not strongly) positive. So optimally send everything as a text file or send the summary as a text file and the detailed review (sans recommendation) in the pdf.

What happens if I always say no to quick opinion/referee requests? The immediate answer is that I will probably stop asking you to referee any more, which may sound like a positive to you. The down side is that the more people in your area do the same thing, the harder and harder it gets to find referees to do the job. At some point, the effect is going to be that I am just going to reject papers in that area unless they are obviously extremely strong, since it’s honestly a huge annoyance to email 10 people to get a single response, even when I am just asking for a quick opinion which could honestly take under 10 minutes. Some areas are definitely worse than others. Honestly, every time I get a paper related to the equivariant Tamagawa number conjecture I die a little, since it’s so hard to find anyone to review those papers.

How long should a review take? It would be nice if you could do it within about 3 months. At the point where I am finding people to review the paper, I am generally hopeful that the paper will be accepted. Under such circumstances, I am usually inclined to allow 6-9 months for the paper to be reviewed. I often ask referees to take a quick look early on to get a sense that they will be inclined to give a positive recommendation on the paper as well. Whether you think 6-9 months is way too long or relatively short probably depends on your own experiences. From my perspective, however, it’s such a difficult job getting people to agree to review that when I ask someone for a report in December and they say “I would be happy to, but I won’t get a chance to get to look at this paper in detail until the summer” I am almost always going to happily agree. Of course, some papers are harder to referee than others. Also, it may (and does) happen that, even after a positive quick opinion and a positive early indication from the referee the paper is rejected. The usual reason that this happens is that it is only when the reviewer looks at the paper in detail do they decide it is not as interesting as they first thought. The only way to avoid this is to only get referees who agree to write reports within 3 months. But if it takes me 6 months to find such a person, then it doesn’t really solve the problem.

What annoys you most about referee reports? Beggars can’t be choosers! But it does annoy me to wait 9 months for a review and get back a one sentence response “it is fine.” It’s honestly just embarrassing to me to have to send back such a report to the author after so long — can’t you at least find a few spelling mistakes/typos? I have received such reports on a number of occasions and then basically had to review the paper myself.

What pleases you most about referee reports? When referees time and time again selflessly put effort into helping the author make genuine improvements to their paper.

As editor, are you judge, jury, and executioner? The closer a paper is to my own expertise, the more likely it is that I have my own initial impressions of the paper. I don’t believe that my job is to be purely a functionary, and that I add value by having my own opinions. I have, on a few rare occasions, even solicited people to submit papers (and I would like to do so more frequently, although often by the time papers are on the arXiv they have already been submitted). In the end, I do believe that the process is impartial, although of course railway trains are impartial too, but if you lay down the lines for them that’s the way they go.

Posted in Mathematics | | 10 Comments

## The Ramanujan Machine is an intellectual fraud

There’s a lot that I like about how mathematics operates as a social discipline. We have a great respect for the history of the subject, which in particular includes acknowledging the work that has come before us. In the end, we ultimately agree that it is the mathematics which does the talking. Each of us has our own tastes (of course) and some of us are more prone to be excited about our own work than others, but we are remarkably free from bullshit (about the actual mathematics, at least). This is what all of science should aspire to.

Perhaps this is why I find the self-promotion surrounding the Ramanujan Machine so distasteful. (I wasn’t going to bother wasting any more time on this but here I am, last time I promise.) The idea of trying to automate methods for finding identities is an interesting one. But if want to claim that you have found something new, then some justification is required. For a start, you should at least be expected to do a cursory search of the literature. Perhaps you should even consult an expert? If the authors had been content to be more modest with their claims, merely explaining that automation was their main goal, and that they were merely hopeful to use these ideas to make new discoveries, I would have had no issue at all with their paper. Of course, nobody would have heard about the paper either. I already complained last time about the overblown rhetoric, but since then, my interactions with one of the authors indicated that the rot lies deeper still.

The author in question seemed happy (while listening to my previous complaints) to indicate that the novelty is not in producing new mathematics but in automatically generating formulas “without knowing Gauss’ work”. As I indicated above, that’s a reasonable and modest claim. But then, the same author will tweet out to the world the false claim that his program has discovered new and amazing mathematical conjectures. (A rather curious set of tweets to @elonmusk @yurimilner @stephen_wolfram @RHDijkgraaf; is the common thread people who have access to money that can be spent on math?) Don’t imagine for a minute that this is not a deliberate and conscious decision: the press stories generated about “automating the insight of Ramanujan” don’t happen on their own — they need the hook of “exciting conjectures” to be “newsworthy” and this is exactly what has been peddled via a concerted publicity campaign. The best way to describe this entire story is as follows: this is what happens when you import startup culture into mathematics. Maybe such Janus-faced interactions are commonplace in Silicon Valley where qualifiers are merely a distraction from a good sales pitch. But it’s an utterly abhorrent mindset that I think mathematicians must strive to banish at all costs. In particular, the choice to deliberately obscure the fact that the program has generated (as yet) nothing considered remotely new by an expert while simultaneously boasting of a triumph in automating intuition is not just absurd, but is an intellectual fraud. Ramanujan would roll over in his grave.

Caveats: The paper has a number of young authors who I consider completely blameless. The possibility of redemption still awaits in the next version of the paper.

Random Continued Fraction: I give you one of the author’s lastest tweets, in which one of their amazing conjectures has been generalized! (By a 15 year old apparently, well done to him, I hope he doesn’t waste any more time on this). I’ll spare you the amazing gif animation that builds up to this final climax:

Well … OK I guess? But, pretty much exactly as pointed out last time, not only is the proof only one line, but the nature of the proof makes clear exactly how unoriginal this is to mathematics:

(There is actually something vaguely interesting about how certain specializations of complicated identities are harder to prove than the original identities, but that is only tangentially relevant to this post.)

## En Passant V

(warning: today’s persiflage comes with possible extra snark due to sleep deprivation)

The Ramanujan Machine: I learnt from John Baez on twitter about The Ramanujan Machine, a project designed to “help reveal [the] underlying structure” of the “fundamental constants” of mathematics. It seems that more effort has been spent on hype rather than on learning anything about continued fractions, and there is nothing there that would be surprising to Gauss let alone Ramanujan. Despite the overblown rhetoric (sample nonsense from the website: Suggest a proof to any of the conjectures that were discovered by the Ramanujan Machine. Have a formula named after you!), the chances of finding anything novel by these methods seem slim. While I’m all for the use of computers in mathematics, at this point (for these type of identities) we are very much in a world where the guiding hand of human mathematical intuition is very much required. As for the “new conjectures” that are animated on their flashy website (screenshot below):

well, the less said the better. It’s harder to say which one is sillier to pretend is original. The latter is a specialization of a specialization of a specialization of a specialization of Gauss’ 2F1 continued fraction identity, sending the four parameters $a,b,c,z$ to $a = 0, b = 1/2, c = 5/2, z = -1.$ (Hint: when your result is a specialization of a formula on Wikipedia it does not count as original.) The former is even more elementary and proves itself by induction:

The rest of the paper is littered with mathematical trivialities and grandiose bombast. (All the other “new” identities are similar trivializations of Gauss’ hypergeometric continued fraction or just known infinite (or finite) sums in disguise via Euler’s continued fraction.) What would be a much more interesting project is to find a way of taking a continued fraction and recognizing it as a specialization of one of the (very many) known results. Lest I be considered a luddite, I should note that when it comes to infinite sums and integrals, this is something that Mathematica does amazingly well, so respect to the people who worked on that.

3Blue1Brown: I don’t really get the appeal, to be honest (yes, I know, I’m not the target audience). The presenter has a geometric perspective which gets shoehorned into everything whether it is appropriate or not. I watched two videos, both of which seemed to miss (or at least elide) at least one key underlying mathematical point. The first was a video on quaternions. The fundamental property of $\mathbf{H}$ is that it is (the unique!) non-commutative division algebra over the real numbers. But the video really only talked about the multiplicative structure, in which case you may as well talk about $\mathrm{SU}(2).$ Are you really “visualizing quaternions” when you only think about the multiplicative structure? Then there was this video on the Riemann hypothesis. The video does a reasonably good job of explaining analytic continuation in terms of conformal maps (not that I think of it that way, but this is a perfectly reasonable way to think about it). However, the entire video ignores (once again) the key point that what is amazing is that the zeta function has an analytic continuation at all rather than it is unique (time stamp 16m 45s):

The closest the video comes to acknowledging this is the quote “which through more abstract derivation we know much exist” which is somewhere between wrong (suggesting that the extension exists for formal reasons) and misleading (using “abstract” to mean something like “beyond the scope of this video” or something). How can you start to think about the Riemann Hypothesis without appreciating why the analytic continuation exists? I guess the video helps bridge the gap between mathematics and physics; popular accounts of physics have long since enabled people to think that they understand something about physics while actually not really having any real idea what is going on, now people can think that way about the Riemann zeta function as well! (The number of views of these videos is in the millions.)

Hatcher on Class Groups: I learnt that Allen Hatcher, author of a wonderful and free textbook on algebraic topology, is writing a textbook on the geometry of binary quadratic forms. I’m sure it will be a great read, and I don’t quite mean to lump it together with the two examples above, although I do notice that it does not appear to mention (in any way) class field theory. That seems to be a strange omission: couldn’t the book at least have a sentence or two indicating that two centuries of algebraic number theory has been built on generalizing Gauss’ work on the class group?

## The stable cohomology of SL(F_p)

Back by popular demand: an actual mathematics post!

Today’s problem is the following: compute the cohomology of $\mathrm{SL}(\mathbf{F}_p)$ for a (mod-p) algebraic representation.

Step 0 is to say what this problem actually is. It makes sense to talk about certain algebraic representations of $\mathrm{SL}_n(\mathbf{F}_p)$ as n varies (for example, the standard representation or the adjoint representation, etc.). For such representations, one can prove stability phenomena for the corresponding cohomology groups. But my question is whether one can actually compute these groups concretely.

The simplest case is the representation $\mathbf{L} = \mathbf{F}_p$ and here one has a complete answer: these cohomology groups are all zero in higher degree, a computation first done by Quillen and which is closely related to the fact that the $K_n(\mathbf{F}_p) \otimes \mathbf{F}_p = 0.$

Most of the references I have found for cohomology computations of special linear groups in their natural characteristic consider the case were p is very large compared to n, but let me remind the reader that we are exactly the opposite situation. One of the few references is a paper of Evens and Friedlander from the ’80s which computes some very special cases in order to compute $K_3(\mathbf{Z}/p^2 \mathbf{Z}).$

Note, however, that p should still be thought of as “large” compared to the partition which defines the corresponding stable local system(s).

In order to get started, let us make the following assumptions:

ANZATZ: There exists a space X with a $\mathrm{SL}(\mathbf{Z}_p)$ pro-cover such that:

1. The corresponding completed cohomology groups with $\mathbf{F}_p$ coefficients are $\mathbf{F}_p$ for i = 0 and vanish otherwise.
2. If $\mathbf{L}$ is the mod-p reduction of (an appropriately chosen) lattice in a (added non-trivial irreducible) algebraic representation of $\mathrm{SL}(\mathbf{Z}_p),$ then $H^i(X,\mathbf{L}) = 0$ for i small enough compared to the weight of $\mathbf{L}.$

Some version of this is provable in some situations and it may be generally true, but let us ignore this for now. (One explicit example is given by the locally symmetric space for $\mathbf{SL}(\mathbf{Z}[\sqrt{-2}])$ and taking the cover corresponding to a prime $\mathfrak{p}$ of norm p satisfying certain global conditions.) The point is, this anzatz allows us to start making computations. From the first assumption, one deduces by Lazard that

$H^i(X(p),\mathbf{F}_p) = \wedge^i M,$

where M is the adjoint representation. But now one has a Hochschild-Serre spectral sequence:

$H^i(\mathrm{SL}(\mathbf{F}_p), \mathbf{L} \otimes \wedge^j M) \Rightarrow 0.$

The point is now that one can now start to unwind this (even knowing nothing about the differentials) and make some conclusions, for example:

1. $H^1(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = 0.$
2. $H^2(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = H^0(\mathrm{SL}(\mathbf{F}_p),\mathbf{L} \otimes M).$

In particular, the first cohomology always vanishes, and the second cohomology is non-zero only for the adjoint representation where it is one dimensional. (One can see the non-trivial class in H^2 in this case coming from the failure of the tautological representation to lift mod p^2.) Note of course I am not claiming that the first cohomology vanishes for all representations, but only the “algebraic” ones, and even then with p large enough (compared to the weight). Note also that one has to be careful about the choice of lattices, but that is somehow built into the stability — for n fixed, the dual of M is given by trace zero matrices in $M_n(\mathbf{F}_p)$ and so (from the cohomology side) “$\mathbf{L} = M$” is the correct object to consider rather than its dual since the dual is not stable even in degree zero. But I think you can secretly imagine that p is big enough and the weight small enough so that you can choose n so that all these representations are actually irreducible).

The first question is whether 1 & 2 are known results — I couldn’t find much literature on these sort of questions (they are certainly consistent with the very special cases considered by Evens and Friedlander).

The second question is what about degrees bigger than 2? For H^3 things start getting a little murkier, but it seems possible that H^3 always vanishes. Beyond that (well even before that) I am just guessing. But one might hope to even come up with a guess the the answer which is consistent with the spectral sequence above.

Added: Some of the discussions in the comments below contain some minor inaccuracies, but in back and forth conversations with Will via email he has formulated a pretty convincing conjectural answer to my questions (and also my secret unasked questions). Hopefully I will come back to this post later when these are all proved!

Posted in Mathematics, Uncategorized | | 3 Comments

## Harris versus Buzzard

Michael Harris has a new article at quanta. The piece is (uncharacteristically?) coy, referring to the laments of two logicians without divulging either their names or their precise objections, making oblique references to a cabal of 10 mathematicians meeting at the institute, and making no reference at all to his own significant contribution to the subject. But that aside, the piece relates to one of themes from Michael’s book, namely, what is mathematics to mathematicians? In this case, the point is made that mathematics is decidedly not — as it is often portrayed — merely a formal exercise of deducing consequences of the axioms by modus tollens and modus ponens. More controversial, perhaps, is the question of what number theorists stand to gain by a massive investment in the formalization of mathematical arguments in (say) Lean. (I “say” Lean but I don’t really know what I am talking about or indeed have any idea what “Lean” actually is.) As you know, here at Persiflage we like to put words in people’s mouths which may or may not be a true reflection of their actual beliefs. So let’s say that MH believes that any thing produced by such programs will never produce any insight — or possibly not in anyway that would count as meaningful insights for humans (if a computer could talk, we wouldn’t be able to understand it). KB believes that without the promised salvation of computer verified proofs, modern number theory is in danger of shredding itself before your eyes like that Banksy. What do you think? Since everything comes down to money, the correct way to answer this question is to say what percentage of the NSF budget are you willing to be spent on these projects. Nuanced answers are acceptable (e.g. “as long as some really smart people are committing to work on this the NSF should get ahead of the curve and make it a priority” is OK, “better this than some farcical 10 million pound grant to study applications of IUT” is probably a little cheeky but I would accept it if you put your real name to it).

Let the battle begin!

(Photo credit: I went to Carbondale to see the solar eclipse, but all I saw was this lousy sign. The other is just a random web search for “vintage crazy pants”.)

Posted in Mathematics | | 24 Comments