**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 to (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 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 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?

Today’s problem is the following: compute the cohomology of 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 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 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

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

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 pro-cover such that:

1. The corresponding completed cohomology groups with coefficients are for i = 0 and vanish otherwise.

2. If is the mod-p reduction of (an appropriately chosen) lattice in a (**added** non-trivial irreducible) algebraic representation of then for i small enough compared to the weight of

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 and taking the cover corresponding to a prime 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

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

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.

2.

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 and so (from the cohomology side) “” 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!

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”.)

]]>]]>

From: Mathematics Editorial Office

Subject: [Mathematics] Review Request

Date: December 26, 2018 at 7:43:31 AM CST

Dear Professor Calegari,

Happy new year. We have received the following manuscript to be considered for publication in Mathematics (http://www.mdpi.com/journal/mathematics/) and kindly invite you to provide a review to evaluate its suitability for publication:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

If you accept this invitation we would *appreciate receiving your comments within 10 days.* Mathematics has one of the most transparent, and reliable assessments of research available. Thank you very much for your consideration and we look forward to hearing from you.

Kind regards,

[**name**]

Assistant Editor

###################################################

From: [**name**]

Subject: [Mathematics] Manuscript ID: Review Request Reminder

Date: December 27, 2018 at 9:48:02 PM CST

Dear Professor Calegari,

On 26 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

###################################################

From: [**name**]

Subject: [Mathematics] Manuscript ID: Review Request Reminder

Date: January 3, 2019 at 7:48:15 PM CST

On 28 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

###################################################

From: Frank Calegari

Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder

Date: January 3, 2019 at 11:08:03 PM CST

Dear [**name**],

Thank you for agreeing to enlist my professional reviewing services. My current rate is $1000US an hour. Please send me the contract forms and payment details. I estimate somewhere between 2-5 hours will be required to review this paper.

###################################################

From: Frank Calegari

Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder

Date: January 8, 2019 at 07:50:03 AM CST

Dear [**name**],

On January 3, I invited you to forward me the contract forms and payment details for my reviewing assignment.

However, if you are unable to provide payment because you are a predatory journal, please let me know quickly to avoid unnecessary reminders.

Do not hesitate to contact me if you have any questions about this request.

My previous message is included below:

…

Professor Francesco Calegari

###################################################

From: [**name**]

Subject: Re: [Mathematics] Manuscript ID: mathematics-412859 – Review Request Reminder

Date: January 8, 2019 at 9:25:25 PM CST

Dear Professor Calegari,

… Actually, the article process charge of this manuscript is only 350CHF. We need to invite at least two reviewers for each manuscript. We can’t bear the cost you proposed. So I will cancel the review invitation for you soon.

All the best to your work.

Kind regards,

###################################################

]]>However, a number of recent changes have been taken place. JNT has introduced “JNT Prime” which seeks to publish

a small number of exceptional papers of high quality (at the level of Compositio or Duke).

(I’m not sure if free two-day delivery is also included in this package.) My question is: why bother?

I have several points of confusion.

**It’s easier to start from scratch**. It is much easier (as far as developing a reputation goes) to start a new journal and set the standards from the beginning, than to steer a massive oil tanker like the Journal of Number Theory with its own firmly established brand. Consider the Journal of the Institute of Mathematics of Jussieu. This early paper (maybe in the very first issue) by Richard Taylor set the tone early on that this was a serious journal. Similarly, Algebra & Number Theory in a very few number of years became a reasonably prestigious journal and certainly more prestigious than the Journal of Number theory has ever been during my career.**The previous standards of JNT served the community well**. Not every journal can be the Annals. Not every journal can be “better than all but the best one or two journals” either, although it is pretty much a running joke at this point that every referee request nowadays comes with such a description. There is plenty of interesting research in number theory that deserves to be published in a strong reputable journal but which is better suited to a specialist journal rather than Inventiones. Journal of Number Theory: it does what it says on the tin. Before the boutique A&NT came along, it was arguably the most prestigious specialist journal in the area. It is true that it was*less*prestigious than some specialist journals in other fields, but that reflects the reality that number theory papers make up a regular proportion of papers in almost all of the top journals, which is not true of all fields. So where do those papers go if JNT becomes all fancy?**Elsevier**. Changing the Journal of Number Theory is going to take a lot of work, and that work is going to be done (more or less) by mathematicians. So why bother making all that effort on behalf of Elsevier? Yes, Elsevier continues to “make an effort” with respect to Journal of Number Theory, including, apparently, even sponsoring a conference. But (to put it mildly) Elsevier is not a charity, and nobody should expect them to start behaving like one.

So I guess my question is: who is better off if the Journal of Number Theory becomes (or heads in the direction of becoming) a “top-tier journal” besides (possibly) Elsevier?

]]>**Question:** Fix a prime p. Does there exist a non-solvable extension of unramified everywhere except for p?

There is a (very) related question of Gross, who (and I can’t track down the precise reference) was generous and allowed ramification at infinity. That makes the question easy to answer for big enough p just by taking the mod-p Galois representations associated to either the weight 12 or weight 16 cusp form of level 1. But what if you impose the condition that the extension has to be unramified at the infinite prime as well (so *totally real*) then you are completely out of luck as far as Galois representations from algebraic automorphic forms go, because for those, complex conjugation will always be non-trivial. (Things don’t get any easier if you even allow regular algebraic automorphic representations, as Caraiani and Le Hung showed). Except, that is, for the case when p = 2. There is a different paper by Lassina on this topic, which solved Gross’ question for p=2 by finding a level one Hilbert modular form over the totally real field for a 32nd root of unity with a non-solvable mod-2 representation. But (as he shows) this extension *is* ramified at infinity — in fact, the Odlyzko discriminant bounds show that to get a totally real extension (assuming GRH) one would have to take the totally real field to be at least as large as for a 128th root of unity. Is it even possible to compute Hilbert Modular Forms for a field this big?

Leaving aside the computational question, there is also a theoretical one as well, even for classical modular forms. Given a Hilbert modular form, or even a classical modular form, is there any easy way to compute the image of complex conjugation modulo 2? One reason this is subtle is that the answer depends on the lattice so it really only makes sense for a residually absolutely irreducible representation. For example:

**Question:** For every n, does there exist a (modular) surjective Galois representation

for a finite field of order divisible by which is also *totally real?* Compare this to Corollary 1.3 of this paper of Wiese.

I don’t even have a guess as to the answer for the first question, but the second one certainly should have a positive answer, at least assuming the inverse Galois problem. As usual, an Aperol Spritz is on offer to both the second question and to the first in the special case of p=2.

]]>To quote directly from the relevant blurb,

This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding five years.

Have you read any paper in either of these journals recently which you thought was really engaging, enlightening, or just neat? Perhaps something on a topic on which you were not so familiar which left you with the feeling that you gained some insight into what the key ideas or problems were? If so, please nominate them before June 30, 2019!

]]>For a smooth manifold M, de Rham’s Theorem gives an isomorphism

which can more naturally be phrased as that the natural pairing between (classes of) closed forms and (classes of) paths given by

induces a perfect pairing on the corresponding (co-)homology groups. The class of paths in homology has a very natural integral basis coming from the paths themselves. For a general M, the de Rham cohomology has no such basis. On the other hand, if M is (say) the complex points of an algebraic variety over the rational numbers, then there are more algebraic ways to normalize the various flavours of differential forms. To take an example which doesn’t quite fit into the world of compact manifolds, take X to be the projective line minus two points, so M is the complex plane minus the origin. There is a particularly nice closed form on this space which generates the holomorphic differentials. But now if one pairs the rational mutiples of this class with the rational multiples of the loop around zero, the pairing does *not* land in the rational numbers, since

In particular, to compare de Rham cohomology over the rationals with the usual Betti cohomology over the rationals, one first has to tensor with a bigger ring such as or at least with a ring big enough to see all the integrals which arise in this form. Such integrals are usually called periods, so in order to have a comparison theorem between de Rham cohomology and Betti cohomology over one first has to tensor with a ring of periods.

It is too simplistic to say that p-adic Hodge theory (at least rationally) is a p-adic version of this story, but that is not the worst cartoon picture to keep in your mind. Returning to the example above, note that the period is a purely imaginary number. This is a reflection of the fact that some arithmetic information is still retained, namely, an action of complex conjugation on the complex points of a variety over the rationals is compatible (with a suitable twist) with the de Rham pairing. A fundamental point is that, in the local story, something similar occurs where now the group generated by complex conjugation is replaced by the much bigger and more interesting group Very (very) loosely, this is related to the fact that p-adic analysis behaves much better with respect to the Galois group, for example, the conjugate of an infinite (convergent) sum of p-adic numbers is the sum of the conjugates. In particular, there is a Galois action on the ring of all p-adic periods. So now there is a much richer group of symmetries acting on the entire picture. Moreover, the structure of the p-adic differentials can be related to how the variety X looks like when reduced modulo-p, because smoothness in algebraic geometry can naturally be interpreted in terms of differential forms.

So now if one wants to make a p-adic comparison conjecture between (algebraic) de Rham cohomology on the one side, and etale cohomology (the algebraic version of Betti cohomology) on the other side, one (optimally) wants the comparison theorem to respect (as much as possible) all the extra structures that exist in the p-adic world, in particular, the action of the local Galois group on etale cohomology, and the algebraic structures which exist on de Rham cohomology (the Hodge filtration and a Frobenius operator), and secondly, involve tensoring with a ring of periods B which is “as small as possible”.

Identifying the correct mechanisms to pass between de Rham cohomology and etale cohomology in a way that is compatible with all of this extra structure is very subtle, and one of the fundamental achievements of Fontaine was really to identify the correct framework in which to phrase the optimal comparison (both in this and also in many related contexts such as crystalline cohomology). (Of course, his work was also instrumental in proving many of these comparison theorems as well.) I think it is fair to say that often the most profound contributions to mathematics come from revealing the underlying structure of what is going on, even if only conjecturally. (To take another random example, take Thurston’s insight into the geometry of 3-manifolds.) Moreover, the reliance of modern arithmetic geometry on these tools can not be overestimated — studying global Galois representations without p-adic Hodge theory would be like studying abelian extensions of without using ramification groups.

Two further points I would be remiss in not mentioning: One sense in which the ring is “as small as possible” is the amazing conjecture of Fontaine-Mazur which captures which *global* Galois representations should come from motives. Secondly, Fontaine’s work on *all* local Galois representations in terms of modules which is crucial even in understanding Motivic Galois representations though p-adic deformations, the fields of norms (with his student Wintenberger, who also sadly died recently), the proof of weak admissibility implies admissibility (with Colmez, another former student, who surprisingly to me only wrote this one joint paper with Fontaine), and the Fargues-Fontaine curve. (I guess this is more than two.)

Probably the first time I talked with Fontaine was at a conference in Brittany (Roscoff) in 2009. That was the first time I ever gave a talk on my work on even Galois representations and the Fontaine-Mazur conjecture, about which Fontaine had some very kind words to say. (One of the most rewarding parts of academia is getting the respect of people you admire.) I never got to know him too well, due (in equal parts) to my ignorance of the French language and p-adic Hodge theory. But he was always a regular presence at conferences at Luminy with his distinct sense of humour. Over a long career, his work continued to be original and deep. He will be greatly missed.

]]>They are both similar and very different at the same time — Lehrer is definitely the more acerbic of the pair, as evidenced by the following pair of quotes concerning satire (themselves satirical):

When Kissinger won the Nobel peace prize, satire died.

]]>The purpose of satire, it has been rightly said, is to strip off the veneer of comforting illusion and cosy half-truth. And our job, as I see it … is to put it back again!”