In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk.

It’s difficult to admit, but Amazon’s blurb lured me into reading Mr. Penumbra’s 24-Hour Bookstore by Robin Sloan:

“With irresistible brio and dazzling intelligence, Robin Sloan has crafted a literary adventure story for the 21st century, evoking both the fairy-tale charm of Haruki Murakami and the enthusiastic novel-of-ideas wizardry of Neal Stephenson or a young Umberto Eco, but with a unique and feisty sensibility that’s rare to the world of literary fiction.” (Amazon’s blurb)

I’m a fan of Murakami’s later books (such as 1Q84 or Killing Commendatore), and Stephenson’s earlier ones (such as Snow Crash or Cryptonomicon), so if someone wrote the perfect blend, I’m in. Reading Penumbra’s bookstore, I discovered that these ‘comparisons’ were borrowed from the book itself, leaving out a few other good suggestions:

One cold Tuesday morning, he strolls into the store with a cup of coffee in one hand and his mystery e-reader in the other, and I show him what I’ve added to the shelves:

Stephenson, Murakami, the latest Gibson, The Information, House of Leaves, fresh editions of Moffat” – I point them out as I go.

(from “Mr Penumbra’s 24-Hour Bookstore”)

This trailer gives a good impression of what the book is about.

Why might you want to read this book?

• If you have a weak spot for a bad ass Googler girl and her tecchy wizardry.
• If you are interested in the possibilities and limitations of Google’s tools.
• If you don’t know what a Hadoop job is or how to combine it with a Mechanical Turk to find a marker on a building somewhere in New-York.
• If you never heard of the Gerritszoon font, preinstalled on every Mac.

As you see, Google features prominently in the book, so it is kind of funny to watch the author, Robin Sloan, give a talk at Google.

Some years later, Sloan wrote a (shorter) prequel Ajax Penumbra 1969, which is also a good read but does not involve fancy technology, unless you count tunnel construction among those.

Read it if you want to know how Penumbra ended up in his bookstore and how he recovered the last surviving copy of the book “Techne Tycheon”.

One of the nicer tools around is bookworm arXiv which ‘is a collaboration between the Harvard Cultural Observatory, arxiv.org, and the Open Science Data Cloud. It enables you to explore lexical trends in over 700,000 e-prints, spanning mathematics, physics, computer science, and statistics’ posted on the arXiv.

One possible use is to explore the popularity of certain topics. Below is the graph of the number of papers submitted monthly to the arXiv in noncommutative geometry, quantum groups, cluster algebras and symplectic reflection (algebras).

The default gives the graphs in the percentage of all papers submitted, but it is better to change this to the number of papers (I think). Sadly, at present one can only search for one- and two-word phrases.

Extremely useful is that it gives you the full list of papers (with direct links to the papers) containing the search terms when you click on that months point in the graph. For example, there are 4 sheets of papers in noncommutative geometry for october 2011

Clearly, there are plenty of other fun uses for this bookworm. For example, you can graph the number of papers in a topic in function of the nationality of the submitter. Here are the papers in noncommutative geometry, submitted by people from the US, France, the UK and Italy.

Or, you can use it for vanity reasons, giving you the list of all papers containing a reference to your work, which may not always be a good idea, blood-pressure wise…

Last time we’ve seen that on June 3rd 1939, the very day of the Bourbaki wedding, Malraux’ movie ‘L’espoir’ had its first (private) viewing, and we mused whether Weil’s wedding card was a coded invitation to that event.

But, there’s another plausible explanation why the Bourbaki wedding might have been scheduled for June 3rd : it was intended to be a copy-cat Royal Wedding…

The media-hype surrounding the wedding of Prince William to Pippa’s sister led to a hausse in newspaper articles on iconic royal weddings of the past.

One of these, the marriage of Edward VIII, Duke of Windsor and Wallis Warfield Spencer Simpson, was held on June 3rd 1937 : “This was the scandal of the century, as far as royal weddings go. Edward VIII had just abdicated six months before in order to marry an American twice-divorced commoner. The British Establishment at the time would not allow Edward VIII to stay on the throne and marry this woman (the British Monarch is also the head of the Church of England), so Edward chose love over duty and fled to France to await the finalization of his beloved’s divorce. They were married in a private, civil ceremony, which the Royal Family boycotted.”

But, what does this wedding have to do with Bourbaki?

For starters, remember that the wedding-card-canular was concocted in the spring of 1939 in Cambridge, England. So, if Weil and his Anglo-American associates needed a common wedding-example, the Edward-Wallis case surely would spring to mind. One might even wonder about the transposed symmetry : a Royal (Betti, whose father is from the Royal Poldavian Academy), marrying an American (Stanislas Pondiczery).

Even Andre Weil must have watched this wedding with interest (perhaps even sympathy). He too had to wait a considerable amount of time for Eveline’s divorce (see this post) to finalize, so that they could marry on october 30th 1937, just a few months after Edward & Wallis.

But, there’s more. The royal wedding took place at the Chateau de Cande, just south of Tours (the A on the google-map below). Now, remember that the 2nd Bourbaki congress was held at the Chevalley family-property in Chancay (see the Escorial post) a bit to the north-east of Tours (the marker on the map). As this conference took place only a month after the Royal Wedding (from 10th till 20th of July 1937), the event surely must have been the talk of the town.

Early on, we concluded that the Bourbaki-Petard wedding took place at 12 o’clock (‘a l’heure habituelle’). So did the Edward-Wallis wedding. More precisely, the civil ceremony began at 11.47 and the local mayor had to come to the castle for the occasion, and, afterwards the couple went into the music-room, which was converted into an Anglican chapel for the day, at precisely 12 o’clock.

The emphasis on the musical organ in the Bourbaki wedding-invitation allowed us to identify the identity of ‘Monsieur Modulo’ to be Olivier Messiaen as well as that of the wedding church. Now, the Chateau de Cande also houses an impressive organ, the Skinner opus 718 organ.

For the wedding ceremony, Edward and Wallis hired the services of one of the most renowned French organists at the time : Marcel Dupre who was since 1906 Widor’s assistent, and, from 1934 resident organist in the Saint-Sulpice church in Paris. Perhaps more telling for our story is that Dupre was, apart from Paul Dukas, the most influential teacher of Olivier Messiaen.

On June 3rd, 1937 Dupre performed the following pieces. During the civil ceremony, an extract from the 29e Bach cantate, canon in re-minor by Schumann and the prelude of the fugue in do-minor of himself. When the couple entered the music room he played the march of the Judas Macchabee oratorium of Handel and the cortege by himself. During the religious ceremony he performed his own choral, adagium in mi-minor by Cesar Franck, the traditional ‘Oh Perfect Love’, the Jesus-choral by Bach and the toccata of the 5th symphony of Widor. Compare this level of detail to the minimal musical hint given in the Bourbaki wedding-invitation

“Assistent Simplexe de la Grassmannienne (lemmas chantees par la Scholia Cartanorum)”

This is one of the easier riddles to solve. The ‘simplicial assistent of the Grassmannian’ is of course Hermann Schubert (Schubert cell-decomposition of Grassmannians). But, the composer Franz Schubert only left us one organ-composition : the Fugue in E-minor.

I have tried hard to get hold of a copy of the official invitation for the Edward-Wallis wedding, but failed miserably. There must be quite a few of them still out there, of the 300 invited people only 16 showed up… You can watch a video newsreel film of the wedding.

As Claude Chevalley’s father had an impressive diplomatic career behind him and lived in the neighborhood, he might have been invited, and, perhaps the (unused) invitation was lying around at the time of the second Bourbaki-congress in Chancay,just one month after the Edward-Wallis wedding…

Sunday january 2nd around 18hr NeB-stats went crazy.

Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day.

Now, the dust has settled, so let’s learn from the experience.

A Reddit-mention is to a blog what doping is to a sporter.

You get an immediate boost in the most competitive of all blog-stats, the number of unique vistors (blue graph), but is doesn’t result in a long-term effect, and, it may even be harmful to more essential blog-stats, such as the average time visitors spend on your site (yellow graph).

For NeB the unique vistors/day fluctuate normally around 300, but peaked to 1295 and 1733 on the ‘Reddit-days’. In contrast, the avg. time on site is normally around 3 minutes, but dropped the same days to 44 and 30 seconds!

Whereas some of the Reddits spend enough time to read the post and comment on it, the vast majority zap from one link to the next. Having monitored the Reddit/math page for two weeks, I’m convinced that post only made it because it was visually pretty good. The average Reddit/math-er is a viewer more than a reader…

So, should I go for shorter, snappier, more visual posts?

Let’s compare Reddits to those coming from the three sites giving NeB most referrals : Google search, MathOverflow and Wikipedia.

This is the traffic coming from Reddit/math, as always the blue graph are the unique visitors, the yellow graph their average time on site, blue-scales to the left, yellow-scales to the right.

Here’s the same graph for Google search. The unique visitors/day fluctuate around 50 and their average time on site about 2 minutes.

The math-related search terms most used were this month : ‘functor of point approach’, ‘profinite integers’ and ‘bost-connes sytem’.

More rewarding to me are referrals from MathOverflow.

The number of visitors depends on whether the MathO-questions made it to the front-page (for example, the 80 visits on december 15, came from the What are dessins d’enfants?-topic getting an extra comment that very day, and having two references to NeB-posts : The best rejected proposal ever and Klein’s dessins d’enfant and the buckyball), but even older MathO-topics give a few referrals a day, and these people sure take their time reading the posts (+ 5 minutes).

Other MathO-topics giving referrals this month were Most intricate and most beautiful structures in mathematics (linking to Looking for F-un), What should be learned in a first serious schemes course? (linking to Mumford’s treasure map (btw. one of the most visited NeB-posts ever)), How much of scheme theory can you visualize? (linking again to Mumford’s treasure map) and Approaches to Riemann hypothesis using methods outside number theory (linking to the Bost-Connes series).

Finally, there’s Wikipedia

giving 5 to 10 referrals a day, with a pretty good time-on-site average (around 4 minutes, peaking to 12 minutes). It is rewarding to see NeB-posts referred to in as diverse Wikipedia-topics as ‘Fifteen puzzle’, ‘Field with one element’, ‘Evariste Galois’, ‘ADE classification’, ‘Monster group’, ‘Arithmetic topology’, ‘Dessin d’enfant’, ‘Groupoid’, ‘Belyi’s theorem’, ‘Modular group’, ‘Cubic surface’, ‘Esquisse d’un programme’, ‘N-puzzle’, ‘Shabat polynomial’ and ‘Mathieu group’.

What lesson should be learned from all this data? Should I go for shorter, snappier and more visual posts, or should I focus on the small group of visitors taking their time reading through a longer post, and don’t care about the appallingly high bounce rate the others cause?

Exactly 7 years ago I wrote my first post. This blog wasn’t called NeB yet and it used pMachine, a then free blogging tool (later transformed into expression engine), rather than WordPress.

Over the years NeB survived three hardware-upgrades of ‘the Matrix’ (the webserver hosting it), more themes than I care to remember, and a couple of dramatic closure announcements…

But then we’re still here, soldiering on, still uncertain whether there’s a point to it, but grateful for tiny tokens of appreciation.

Such as this morning’s story: Chandan deemed it necessary to correct two spelling mistakes in a 2 year old Fun-math post on Weil and the Riemann hypothesis (also reposted on Neb here). Often there’s a story behind such sudden comments, and a quick check of MathOverflow revealed this answer and the comments following it.

I thank Ed Dean for linking to the Fun-post, Chandan for correcting the misspellings and Georges for the kind words. I agree with Georges that a cut&copy of a blogpost-quoted text does not require a link to that post (though it is always much appreciated). It is rewarding to see such old posts getting a second chance…

Above the Google Analytics graph of the visitors coming here via a mobile device (at most 5 on a good day…). Anticipating much more iPads around after tonights presents-session I’ve made NeB more accessible for iPods, iPhones, iPads and other mobile devices.

The first time you get here via your Mac-device of choice you’ll be given the option of saving NeB as an App. It has its own icon (lowest row middle, also the favicon of NeB) and flashy start-up screen.

Of course, the whole point trying to make NeB more readable for Mobile users you get an overview of the latest posts together with links to categories and tags and the number of comments. Sliding through you can read the post, optimized for the device.

I do hope you will use the two buttons at the end of each post, the first to share or save it and the second to leave a comment.

I wish you all a lot of mathematical (and other) fun in 2011 :: lieven.

No christmas- or new-years family party without heated discussions. Often on quite silly topics.

For example, which late 19th-century bookcharacter turned out to be most influential in the 20th century? Dracula, from the 1897 novel by Irish author Bram Stoker or Sir Arthur Conan Doyle’s Sherlock Holmes who made his first appearance in 1887?

Well, this year you can spice up such futile discussions by going over to Google Labs Books Ngram Viewer, specify the time period of interest to you and the relevant search terms and in no time it spits back a graph comparing the number of books mentioning these terms.

Here’s the 20th-century graph for ‘Dracula’ (blue), compared to ‘Sherlock Holmes’ (red).

The verdict being that Sherlock was the more popular of the two for the better part of the century, but in the end the vampire bit the detective. Such graphs lead to lots of new questions, such as : why was Holmes so popular in the early 30ties? and in WW2? why did Dracula become popular in the late 90ties? etc. etc.

Clearly, once you’ve used Books Ngram it’s a dangerous time-waster. Below, the graphs in the time-frame 1980-2008 for Alain Connes (blue), noncommutative geometry (red), Hopf algebras (green) and quantum groups (yellow).

It illustrates the simultaneous rise and fall of both quantum groups and Hopf algebras, whereas the noncommutative geometry-graph follows that of Alain Connes with a delay of about 2 years. I’m sure you’ll find a good use for this splendid tool…

you get emails like this one :

From: Alyssa jasmine
Subject: Interested in writing article for your blog
Date: 28 Sep 2010 13:41:40 GMT+02:00
To: lieven lebruyn

Hi

This is Alyssa

I went through your site while surfing in google, am very much impressed with your site unique informations, and We are pleased to inform you that we do write articles for such unique sites without cost.

Article are sure for its unusual quality to invite traffic to your website. We provide relevant articles based on the topics suggested for your site

We do provide a unique article for your service. No duplication or copying of the article is done. we write contents exclusively for your site on demand. We also give Copy rights for articles to your site on security base.

In return we expect a small link connecting to our website from your webpage(2 links per post/article).

Thanks,
Alyssa

Via Tanya Khovanova I learned yesterday of the 50 best math blogs for math-majors list by OnlineDegree.net. Tanya’s blog got in 2nd (congrats!) and most of the blogs I sort of follow made it to the list : the n-category cafe (5), not even wrong (6), Gowers (12), Tao (13), good math bad math (14), rigorous trivialities (18), the secret blogging seminar (20), arcadian functor (28) (btw. Kea’s new blog is now at arcadian pseudofunctor), etc., etc. . Sincere congrats to you all!

NeverEndingBooks didn’t make it to the list, and I can live with that. For reasons only relevant to myself, posting has slowed down over the last year and the most recent post dates back from february!

More puzzling to me was the fact that F-un mathematics got in place 26! OnlineDegree had this to say about F-un Math : “Any students studying math must bookmark this blog, which provides readers with a broad selection of undergraduate and graduate concerns, quotes, research, webcasts, and much, much more.” Well, personally I wouldn’t bother to bookmark this site as prospects for upcoming posts are virtually inexistent…

As I am privy to both sites’ admin-pages, let me explain my confusion by comparing their monthly hits. Here’s the full F-un history

After a flurry of activity in the fall of 2008, both posting and attendance rates dropped, and presently the site gets roughly 50 hits-a-day. Compare this to the (partial) NeB history

The whopping 45000 visits in january 2008 were (i think) deserved at the time as there was then a new post almost every other day. On the other hand, the green bars to the right are a mystery to me. It appears one is rewarded for not posting at all…

The only explanation I can offer is that perhaps more and more people are recovering from the late 2008-depression and do again enjoy reading blog-posts. Google then helps blogs having a larger archive (500 NeB-posts compared to about 20 genuine Fun-posts) to attract a larger audience, even though the blog is dormant.

But this still doesn’t explain why FunMath made it to the top 50-list and NeB did not. Perhaps the fault is entirely mine and a consequence of a bad choice of blog-title. ‘NeverEndingBooks’ does not ring like a math-blog, does it?

Still, I’m not going to change the title into something more math-related. NeverEndingBooks will be around for some time (unless my hard-disk breaks down). On the other hand, I plan to start something entirely new and learn from the mistakes I made over the past 6 years. Regulars of this blog will have a pretty good idea of the intended launch date, not?

Until then, my online activity will be limited to tweets.

Early 1936, Andre Weil and Evelyne Gillet made a pre-honeymooning trip to Spain and visited El Escorial. Weil was so taken by the place that he planned the next Bourbaki-conference to be held in a nearby college. However, the Bourbakis never made it to to Spain that summer as the Spanish Civil War broke out July 17th, a few weeks before the intended conference. Can we GEO-tag the exact location of Bourbaki’s “Escorial”?

As explained in the bumpy-road-post, Andre Weil and Evelyne Gillet became involved sometime in 1935.
Early 1936, they made a pre-honeymooning trip to Spain and visited El Escorial. Weil was so taken by the place that he planned the next Bourbaki-conference to be held in a nearby college.

However, the Bourbakis never made it to to Spain that summer as the Spanish Civil War broke out July 17th, a few weeks before the intended conference. Still, the second Bourbaki-meeting remains often referred to as the ‘Escorial conference’. Can we GEO-tag the exact location of Bourbaki’s “Escorial”?

Claude Chevalley came up with a Plan-B and suggested they would use his parents’ place in Chançay as their venue. Chevalley’s father was a French diplomat and his house sure did possess a matching ‘grandeur’ as can be seen from the famous picture below, taken at the (second) Chançay meeting in 1937 (Weil to the left, Chevalley to the right and Weil’s sister Simonne standing).

Thanks to the Bourbaki archives we know that the meeting took place from september 16th to 28th, that each of them had to pay 16 francs for full pension and had to bring along their own sheets and towels.

But where exactly is this beautiful house? Jacques Borowczyk has written a nice paper Bourbaki et la touraine in which he describes the Bourbaki congresses of 1936 and 1937 at the Chevalley-house in Chançay and further those held in 1956, 1957 and 1959 in ‘hôtel de la Brèche’ in Amboise.

Borowczyk places the Chevalley house in the little hamlet of Chançay, called “La Massoterie”. The village files assert that in 1931 three people were living at La Massoterie : father Abel Chevalley, who took residence there after his retirement in 1931, his wife Marguerite and their son Claude. But, at the time of the Bourbaki congres in 1936, Marguerite remained the only permanent inhabitant. Sadly,
Abel Chevalley, who together with Marguerite compiled the The concise Oxford French dictionary, died in 1934.

Usually when you know the name of the hamlet, of the village and add just to be certain ‘France’, Google Maps takes you there within metres. So, this was going to be a quick post, for a change… Well, much to my surprise, typing ‘La Massoterie, Chançay, France’ only produced the answer “We could not understand the location La Massoterie, Chançay, France”.

Did I spell it wrong? Or, did the name change over times? No, Googling for it the first hit gives you the map of a 10km walk around Chançay passing through la Massoterie!

Now what? Fortunately Borowczyk included in his paper an old map, from Napoleonic times, showing the exact location of La Massoterie (just above the flash-sign), facing the castle of Volmer. If you compare it with the picture below from present day Chançay (via Google earth) it is surprising how many of the landmarks have survived the changes over two centuries.

It is now easy to pinpoint the exact location and zoom into the Chavalley-house, and, you’re in for a small surprise : the place is called La Massotterie with 2 t’s…

Probably, Googles database is more reliable than the information provided by the village of Chançay, or the paper by Borowczyk as it is the same spelling as on the old Napoleonic map. Anyway, feel free to have a peek at Bourbaki’s Escorial yourself!

Next time you visit your math-library, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s Riemann-Roch algebra and Donald Knutson’s lambda-rings and the representation theory of the symmetric group.

I wouldn’t be surprised if one or more of these books are borrowed out, probably all of them to the same person. I’m afraid I’m that person in Antwerp…

Lately, there’s been a renewed interest in $\lambda$-rings and the endo-functor W assigning to a commutative algebra its ring of big Witt vectors, following Borger’s new proposal for a geometry over the absolute point.

However, as Hendrik Lenstra writes in his 2002 course-notes on the subject Construction of the ring of Witt vectors : “The literature on the functor W is in a somewhat unsatisfactory state: nobody seems to have any interest in Witt vectors beyond applying them for a purpose, and they are often treated in appendices to papers devoting to something else; also, the construction usually depends on a set of implicit or unintelligible formulae. Apparently, anybody who wishes to understand Witt vectors needs to construct them personally. That is what is now happening to myself.”

Before doing a series on Borger’s paper, we’d better run through Lenstra’s elegant construction in a couple of posts. Let A be a commutative ring and consider the multiplicative group of all ‘one-power series’ over it $\Lambda(A)=1+t A[[t]]$. Our aim is to define a commutative ring structure on $\Lambda(A)$ taking as its ADDITION the MULTIPLICATION of power series.

That is, if $u(t),v(t) \in \Lambda(A)$, then we define our addition $u(t) \oplus v(t) = u(t) \times v(t)$. This may be slightly confusing as the ZERO-element in $\Lambda(A),\oplus$ will then turn be the constant power series 1…

We are now going to define a multiplication $\otimes$ on $\Lambda(A)$ which is distributively with respect to $\oplus$ and turns $\Lambda(A)$ into a commutative ring with ONE-element the series $~(1-t)^{-1}=1+t+t^2+t^3+\ldots$.

We will do this inductively, so consider $\Lambda_n(A)$ the (classes of) one-power series truncated at term n, that is, the kernel of the natural augmentation map between the multiplicative group-units $~A[t]/(t^{n+1})^* \rightarrow A^*$.
Again, taking multiplication in $A[t]/(t^{n+1})$ as a new addition rule $\oplus$, we see that $~(\Lambda_n(A),\oplus)$ is an Abelian group, whence a $\mathbb{Z}$-module.

For all elements $a \in A$ we have a scaling operator $\phi_a$ (sending $t \rightarrow at$) which is an A-ring endomorphism of $A[t]/(t^{n+1})$, in particular multiplicative wrt. $\times$. But then, $\phi_a$ is an additive endomorphism of $~(\Lambda_n(A),\oplus)$, so is an element of the endomorphism-RING $End_{\mathbb{Z}}(\Lambda_n(A))$. Because composition (being the multiplication in this endomorphism ring) of scaling operators is clearly commutative ($\phi_a \circ \phi_b = \phi_{ab}$) we can define a commutative RING $E$ being the subring of $End_{\mathbb{Z}}(\Lambda_n(A))$ generated by the operators $\phi_a$.

The action turns $~(\Lambda_n(A),\oplus)$ into an E-module and we define an E-module morphism $E \rightarrow \Lambda_n(A)$ by $\phi_a \mapsto \phi_a((1-t)^{-1}) = (1-at)^{-a}$.

All of this looks pretty harmless, but the upshot is that we have now equipped the image of this E-module morphism, say $L_n(A)$ (which is the additive subgroup of $~(\Lambda_n(A),\oplus)$ generated by the elements $~(1-at)^{-1}$) with a commutative multiplication $\otimes$ induced by the rule $~(1-at)^{-1} \otimes (1-bt)^{-1} = (1-abt)^{-1}$.

Explicitly, $L_n(A)$ is the set of one-truncated polynomials $u(t)$ with coefficients in $A$ such that one can find elements $a_1,\ldots,a_k \in A$ such that $u(t) \equiv (1-a_1t)^{-1} \times \ldots \times (1-a_k)^{-1}~mod~t^{n+1}$. We multiply $u(t)$ with another such truncated one-polynomial $v(t)$ (taking elements $b_1,b_2,\ldots,b_l \in A$) via

$u(t) \otimes v(t) = ((1-a_1t)^{-1} \oplus \ldots \oplus (1-a_k)^{-1}) \otimes ((1-b_1t)^{-1} \oplus \ldots \oplus (1-b_l)^{-1})$

and using distributivity and the multiplication rule this gives the element $\prod_{i,j} (1-a_ib_jt)^{-1}~mod~t^{n+1} \in L_n(A)$.
Being a ring-qutient of $E$ we have that $~(L_n(A),\oplus,\otimes)$ is a commutative ring, and, from the construction it is clear that $L_n$ behaves functorially.

For rings $A$ such that $L_n(A)=\Lambda_n(A)$ we are done, but in general $L_n(A)$ may be strictly smaller. The idea is to use functoriality and do the relevant calculations in a larger ring $A \subset B$ where we can multiply the two truncated one-polynomials and observe that the resulting truncated polynomial still has all its coefficients in $A$.

Here’s how we would do this over $\mathbb{Z}$ : take two irreducible one-polynomials u(t) and v(t) of degrees r resp. s smaller or equal to n. Then over the complex numbers we have
$u(t)=(1-\alpha_1t) \ldots (1-\alpha_rt)$ and $v(t)=(1-\beta_1) \ldots (1-\beta_st)$. Then, over the field $K=\mathbb{Q}(\alpha_1,\ldots,\alpha_r,\beta_1,\ldots,\beta_s)$ we have that $u(t),v(t) \in L_n(K)$ and hence we can compute their product $u(t) \otimes v(t)$ as before to be $\prod_{i,j}(1-\alpha_i\beta_jt)^{-1}~mod~t^{n+1}$. But then, all coefficients of this truncated K-polynomial are invariant under all permutations of the roots $\alpha_i$ and the roots $\beta_j$ and so is invariant under all elements of the Galois group. But then, these coefficients are algebraic numbers in $\mathbb{Q}$ whence integers. That is, $u(t) \otimes v(t) \in \Lambda_n(\mathbb{Z})$. It should already be clear from this that the rings $\Lambda_n(\mathbb{Z})$ contain a lot of arithmetic information!

For a general commutative ring $A$ we will copy this argument by considering a free overring $A^{(\infty)}$ (with 1 as one of the base elements) by formally adjoining roots. At level 1, consider $M_0$ to be the set of all non-constant one-polynomials over $A$ and consider the ring

$A^{(1)} = \bigotimes_{f \in M_0} A[X]/(f) = A[X_f, f \in M_0]/(f(X_f) , f \in M_0)$

The idea being that every one-polynomial $f \in M_0$ now has one root, namely $\alpha_f = \overline{X_f}$ in $A^{(1)}$. Further, $A^{(1)}$ is a free A-module with basis elements all $\alpha_f^i$ with $0 \leq i < deg(f)$.

Good! We now have at least one root, but we can continue this process. At level 2, $M_1$ will be the set of all non-constant one-polynomials over $A^{(1)}$ and we use them to construct the free overring $A^{(2)}$ (which now has the property that every $f \in M_0$ has at least two roots in $A^{(2)}$). And, again, we repeat this process and obtain in succession the rings $A^{(3)},A^{(4)},\ldots$. Finally, we define $A^{(\infty)} = \underset{\rightarrow}{lim}~A^{(i)}$ having the property that every one-polynomial over A splits entirely in linear factors over $A^{(\infty)}$.

But then, for all $u(t),v(t) \in \Lambda_n(A)$ we can compute $u(t) \otimes v(t) \in \Lambda_n(A^{(\infty)})$. Remains to show that the resulting truncated one-polynomial has all its entries in A. The ring $A^{(\infty)} \otimes_A A^{(\infty)}$ contains two copies of $A^{(\infty)}$ namely $A^{(\infty)} \otimes 1$ and $1 \otimes A^{(\infty)}$ and the intersection of these two rings in exactly $A$ (here we use the freeness property and the additional fact that 1 is one of the base elements). But then, by functoriality of $L_n$, the element
$u(t) \otimes v(t) \in L_n(A^{(\infty)} \otimes_A A^{(\infty)})$ lies in the intersection $\Lambda_n(A^{(\infty)} \otimes 1) \cap \Lambda_n(1 \otimes A^{(\infty)})=\Lambda_n(A)$. Done!

Hence, we have endo-functors $\Lambda_n$ in the category of all commutative rings, for every number n. Reviewing the construction of $L_n$ one observes that there are natural transformations $L_{n+1} \rightarrow L_n$ and therefore also natural transformations $\Lambda_{n+1} \rightarrow \Lambda_n$. Taking the inverse limits $\Lambda(A) = \underset{\leftarrow}{lim} \Lambda_n(A)$ we therefore have the ‘one-power series’ endo-functor
$\Lambda~:~\mathbf{comm} \rightarrow \mathbf{comm}$
which is ‘almost’ the functor W of big Witt vectors. Next time we’ll take you through the identification using ‘ghost variables’ and how the functor $\Lambda$ can be used to define the category of $\lambda$-rings.