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Monsters and Moonshine : a booklet

I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.

The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu group $M_{12}$. This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’. We discuss Galois’ last letter and the simple groups $L_2(5),L_2(7)$ and $L_2(11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered ‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover what the largest sporadic simple group, the Monster-group, sees of the modular group.

The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $PSL_2(\mathbb{Z})$. We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using Farey codes and use those to give a series of simple groups including as special members $L_2(5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E(8)$ in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s solution of the $E(8)$-observation.

I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!

If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to the “field with one element” and its connections to (noncommutative) geometry and number theory.



I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so far only exists as prep-notes. The links above will always link to the latest versions of these booklets.

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The martial art of giving talks

Last fall, Matilde Marcolli gave a course at CalTech entitled Oral Presentation: The (Martial) Art of Giving Talks. The purpose of this course was to teach students “how to effectively communicate their work in seminars and conferences and how to defend it from criticism from the audience”.

The lecture notes contain basic information on the different types of talks and how to prepare them. But they really shine when it comes to spotting the badasses in the public and how to respond to their interference. She identifies 5 badsass-types : the empreror and the hierophant (see below), the chariot (the one with a literal mind, asking continuously for details), the fool (the one who happens to sit in the talk but doesn’t belong there) and the magician (the quick smartass).

I’ll just quote here the description of, and most effective strategy against, the first two badass-types. Please have a look at the whole paper, it is a good read!

“The Emperor is the typical figure of power and authority in a given field. It refers to those people who have a tendency to think that the whole field is their own private property, and in particular that only what they do in the field is important, that the work of all others is derivative and that in any case they are not being quoted enough. These are typically pathological narcissists, so one needs to take this into account in interacting with them.
The trouble of having The Emperor in your audience is that he (it is rarely she) can very easily disrupt your presentation completely, by continuous interruptions, by running his own commentary while you are trying to stay focused on delivering your talk and by distracting the rest of the audience.
The Emperor is by far one of the most dangerous encounters you can make in the wilderness of the conference rooms.”

Counter-measure : “Keep in mind that the Emperor is a pathological narcissist: part of the reason why he keeps interrupting your talk is because he cannot stand the fact that, during those fifty minutes, the attention of the audience is focused on you and not on him. His continuous interruptions and complaints are a way to try to divert the attention of the audience back to him and away from you. That your talk gets disrupted in the process, he could not care the less.
A good way to try to avoid the worst case scenario is to make sure (if you know in advance you may be having the Emperor in the audience) that you arrange in your talk to make frequent references to him and his work. In this way, he will hopefully feel that his need to be at the center of attention is sufficiently satisfied that he can let you continue with your talk. Effectiveness: high.”

“The Hierophant represents a priestly figure. What this refers to here is the type of character who feels entitled to represent (and defend) a certain “orthodoxy”, a certain school of thought, or a certain group of people within the field.
Typically the hierophants are the minions and lackeys of the Emperor, his entourage and fan club, those who think that the Emperor represents the only and true orthodoxy in the field and that anything that is done in a different way should be opposed and suppressed.
These characters are generally less disruptive than the Emperor himself, as they are really only fighting you on someone else’s behalf. Nonetheless, they can sometime manage to seriously disrupt your presentation.”

Counter-measure : “This is essentially the same advise as in the case of the Emperor. To an objection that substantially is of the form: “This is not the right way to do things because this is not what what we do (= what the Emperor does)”, which is what you expect to hear from the Hierophant, you can reply along lines such as: “There is also another approach to this problem, developed by the Emperor and his school, which is a very interesting approach that gave nice and important results. However, this is not what I am talking about today: I am talking here about a different approach, and I will be focusing only on the specific features of this other approach…”
Something along these lines would recognize “their” work without having to make any concession on their approach being the only game in town.
Effectiveness: high (unless the Emperor is also present and is delegating to his hierophants the task of attacking you: in that case they won’t give up so easily and the effectiveness of this line of defense becomes medium/low).”

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how noncommutative geometry shot itself

I’ve never apologized for prolonged periods of blogsilence and have no intention to start now.

But, sometimes you need to expose the things holding you back before you can turn the page and (hopefully) start afresh.

Long time readers of this blog know I’ve often warned against group-think, personality cults and the making of exaggerate claims as possible threats to the survival of noncommutative geometry (for example in the group think post).

However, I was totally unprepared for this comment left on the noncommutative geometry blog, begin October:

Noncommutative Geometry is a field whose history is unpredictable.
When should I expect the pickaxe? Yours, Leon Trotsky

After sharing this on Google+ someone emailed suggesting I’d better have a look at some ‘semi-secret’ blogs. I did spend the better part of that friday going through more than 3 years worth of blogposts and cried my eyes out.

It is sad to read a message in a bottle and notice that after more than two years the matter is still far from resolved.

I wish you all a healing and liberating 2012!

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Prep-notes dump

Here are the scans of my crude prep-notes for some of the later seminar-talks. These notes still contain mistakes, most of them were corrected during the talks. So, please, read these notes with both mercy are caution!

Hurwitz formula imples ABC : The proof of Smirnov’s argument, but modified so that one doesn’t require an $\epsilon$-term. This is known to be impossible in the number-theory case, but a possible explanation might be that not all of the Smirnov-maps $q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$ are actually covers.

Frobenius lifts and representation rings : Faithfully flat descent allows us to view torsion-free $\mathbb{Z}$-rings with a family of commuting Frobenius lifts (aka $\lambda$-rings) as algebras over the field with one element $\mathbb{F}_1$. We give several examples including the two structures on $\mathbb{Z}[x]$ and Adams operations as Frobenius lifts on representation rings $R(G)$ of finite groups. We give an example that this extra structure may separate groups having the same character table. In general this is not the case, the magic Google search term is ‘Brauer pairs’.

Big Witt vectors and Burnside rings : Because the big Witt vectors functor $W(-)$ is adjoint to the tensor-functor $- \otimes_{\mathbb{F}_1} \mathbb{Z}$ we can view the geometrical object associated to $W(A)$ as the $\mathbb{F}_1$-scheme determined by the arithmetical scheme with coordinate ring $A$. We describe the construction of $\Lambda(A)$ and describe the relation between $W(\mathbb{Z})$ and the (completion of the) Burnside ring of the infinite cyclic group.

Density theorems and the Galois-site of $\mathbb{F}_1$ : We recall standard density theorems (Frobenius, Chebotarev) in number theory and use them in combination with the Kronecker-Weber theorem to prove the result due to James Borger and Bart de Smit on the etale site of $\mathsf{Spec}(\mathbb{F}_1)$.

New geometry coming from $\mathbb{F}_1$ : This is a more speculative talk trying to determine what new features come up when we view an arithmetic scheme over $\mathbb{F}_1$. It touches on the geometric meaning of dual-coalgebras, the Habiro-structure sheaf and Habiro-topology associated to $\mathbb{P}^1_{\mathbb{Z}}$ and tries to extend these notions to more general settings. These scans are unintentionally made mysterious by the fact that the bottom part is blacked out (due to the fact they got really wet and dried horribly). In case you want more info, contact me.

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On aliens and reality

October 21st : Dear Diary,

today’s seminar was fun, though a bit unconventional. The intention was to explain faithfully flat descent, but at the last moment i had the crazy idea to let students discover the main idea themselves (in the easiest of examples) by means of this thought experiment :

“I am an alien, and a very stubborn alien at that. To us, the only existing field is $\mathbb{C}$ and the only rings we accept are $\mathbb{C}$-algebras. We’ve heard rumours that you humans think there is some geometry hidden under $\mathbb{C}$, in particular we’ve heard that you consider something called real manifolds. Can you explain what an algebra over this non-existant field under $\mathbb{C}$ is in a way we can understand?”

The first hurdle was to explain the concept of complex conjugation, as the alien (me) was unwilling to decompose a number $c$ in two ‘ghost components’ $a+bi$. But, i had to concede that i knew about addition and multiplication and knew $1$ and that $-1$ had a square root which they called $i$.

‘Oh, but then you know what $\mathbb{Z}[i]$ is! You just add a number of times $1$’s and $i$’s.’

‘Why are you humans so focussed on counting? We do not count! We can’t! We have neither fingers nor toes!’

Admittedly a fairly drastic intervention, but i had to keep them on the path leading to Galois descent… After a while we agreed on a map, which they called conjugation, sending sums to sums and products to products and taking a root of unity to its inverse.

Next, they asked me to be a bit flexible and allow for ‘generalized’ fields such as consisting of all elements fixed under conjugation! Clearly, the alien refused : ‘We’re not going on the slippery road called generalization, we’ve seen the havock this has caused in human-mathematics.’

It took them a while to realize they would never be able to sell me an $\mathbb{R}$-algebra $A$, but could perhaps try to sell me the complex algebra $B= A \otimes_{\mathbb{R}} \mathbb{C}$.

Alien : ‘But, how do i recognize one of your algebras? Do they have a special property i can check?’

Humans : ‘Yes, they have some map (which we know to be the map $a \otimes c \mapsto a \otimes \overline{c}$, but you cannot see it) sending sums to sums, products to products and extending conjugation on $\mathbb{C}$.’

Alien : ‘But if i take a basis for any of my algebras and apply conjugation to all its coordinates, then all my algebras have this property, not?’

Humans : ‘No, such maps are good for sums, but not always for products. For example, take $\mathbb{C}[x]/(x^2-c)$ for $c$ a complex-number not fixed under conjugation.’

Alien : ‘Point taken. But then, your algebras are just a subclass of my algebras, right?’

Humans : ‘No! An algebra can have several of such additional maps. For example, take $B = \mathbb{C} \times \mathbb{C}$ then there is one sending $(a,b)$ to $(\overline{a},\overline{b})$ and another sending it to $(\overline{b},\overline{a})$. (because we know there are two distinct real algebras $\mathbb{R} \times \mathbb{R}$ and $\mathbb{C}$ of dimension two, tensoring both to $\mathbb{C} \times \mathbb{C}$.)’

By now, the alien and humans agreed on a dictionary : what to humans is the $\mathbb{R}$-algebra $A$ is to the alien the complex algebra $B=A \otimes \mathbb{C}$ together with a map $\gamma_B : B \rightarrow B$ sending sums to sums, products to products and extending conjugation on $\mathbb{C}$ (the extra structure, that is the map $\gamma_B$ is called the ‘descent data’).

Likewise, a human-observed $\mathbb{R}$-algebra morphism $\phi : A \rightarrow A’$ is to the alien the the $\mathbb{C}$-algebra morphism $\Phi = \phi \otimes id_{\mathbb{C}} : B \rightarrow B’$ which commutes with the extra structures, that is, $\Phi \circ \gamma_B = \gamma_{B’} \circ \Phi$.

Phrased differently (the alien didn’t want to hear any of this) : there is an equivalence of categories between the category $\mathbb{R}-\mathsf{algebras}$ of commutative $\mathbb{R}$-algebras and the category $\gamma-\mathsf{algebras}$ consisting of complex commutative algebras $B$ together with a ringmorphism $\gamma_B$ extending complex conjugation and with morphisms $\mathbb{C}$-algebra morphisms compatible with the $\gamma$-structure.

Further, what to humans is the base-extension (or tensor) functor

$- \otimes_{\mathbb{R}} \mathbb{C}~:~\mathbb{R}-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

is (modulo the above equivalence) to the alien merely the forgetful functor

$\mathsf{Forget}~:~\gamma-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$

stripping off the descent-data.

After the break (yes, it took us that long to get here) we used this idea to invent rings living ‘under $\mathbb{Z}$’, or if you want, algebras over the field with one element $\mathbb{F}_1$.

Alien : ‘Ha-ha-ha a field with one element? Surely, you’re joking Mr. Human’

Note to self : Dare to waste more time like this in a seminar. It may very well be the only thing they will still remember next year.

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meanwhile, at angs+

We’ve had three seminar-sessions so far, and the seminar-blog ‘angs+’ contains already 20 posts and counting. As blogging is not a linear activity, I will try to post here at regular intervals to report on the ground we’ve covered in the seminar, providing links to the original angs+ posts.

This year’s goal is to obtain a somewhat definite verdict on the field-with-one-element hype.

In short, the plan is to outline Smirnov’s approach to the ABC-conjecture using geometry over $\mathbb{F}_1$, to describe Borger’s idea for such an $\mathbb{F}_1$-geometry and to test it on elusive objects such as $\mathbb{P}^1_{\mathbb{F}_1} \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant in Smirnov’s paper) and $\mathsf{Spec}(\mathbb{Z}) \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant to the Riemann hypothesis).

We did start with an historic overview, using recently surfaced material such as the Smirnov letters. Next, we did recall some standard material on the geometry of smooth projective curves over finite fields, their genus leading up to the Hurwitz formula relating the genera in a cover of curves.

Using this formula, a version of the classical ABC-conjecture in number theory can be proved quite easily for curves.

By analogy, Smirnov tried to prove the original ABC-conjecture by viewing $\mathsf{Spec}(\mathbb{Z})$ as a ‘curve’ over $\mathbb{F}_1$. Using the connection between the geometric points of the projective line over the finite field $\mathbb{F}_p$ and roots of unity of order coprime to $p$, we identify $\mathbb{P}^1_{\mathbb{F}_1}$ with the set of all roots of unity together with $\{ [0],[\infty] \}$. Next, we describe the schematic points of the ‘curve’ $\mathsf{Spec}(\mathbb{Z})$ and explain why one should take as the degree of the ‘point’ $(p)$ (for a prime number $p$) the non-sensical value $log(p)$.

To me, the fun starts with Smirnov’s proposal to associate to any rational number $q = \tfrac{a}{b} \in \mathbb{Q} – \{ \pm 1 \}$ a cover of curves

$q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$

by mapping primes dividing $a$ to $[0]$, primes dividing $b$ to $[\infty]$, sending the real valuation to $[0]$ or $[\infty]$ depending onw whether or not $b > a$ and finally sending a prime $p$ not involved in $a$ or $b$ to $[n]$ where $n$ is the order of the unit $\overline{a}.\overline{b}^{-1}$ in the finite cyclic group $\mathbb{F}_p^*$. Somewhat surprisingly, it does follow from Zsigmondy’s theorem that this is indeed a finite cover for most values of $q$. A noteworthy exception being the map for $q=2$ (which fails to be a cover at $[6]$) and of which Pieter Belmans did draw this beautiful graph

True believers in $\mathbb{F}_1$ might conclude from this graph that there should only be finitely many Mersenne primes… Further, the full ABC-conjecture would follow from a natural version of the Hurwitz formula for such covers.

(to be continued)

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#cestGrothendieck

In Belgium the hashtag-craze of the moment is #cestjoelle. Joelle Milquet is perceived to be the dark force behind everything, from the crisis in Greece, over DSK, to your mother-in-law coming over this weekend? #cestjoelle.

Sam Leith used the same meme in his book the coincidence engine.

A hurricane assembling a passenger jet out of old bean-cans? #cestGrothendieck

All shops in Alabama out of Chicken & Broccoli Rica-A-Roni? #cestGrothendieck

Frogs raining down on Atlanta? #cestGrothendieck

As this is a work of fiction, Alexandre Grothendieck‘s name is only mentioned in the ‘author’s note’:

“It is customary to announce on this page that all resemblances to characters living or dead are entirely coincidental. It seems only courteous to acknowledge, though, that in preparing the character of Nicolas Banacharski I was inspired by the true-life story of the eminent mathematician Alexandre Grothendieck.”

The name ‘Nicolas Banacharski’ is, of course, chosen on purpose (the old Bourbaki NB-joke even makes an appearance). The character ‘Isla Holderness’ is, of course, Leila Schneps, the ‘Banacharski ring’ is, of course, the Grothendieck circle. But, I’d love to know the name of the IRL-‘Fred Nieman’, who’s described as ‘an operative for the military’.

Sam Leith surely knows all the Grothendieck-trivia which shouldn’t come as a surprise because he wrote in 2004 a piece for the Spectator on the ‘what is a metre?’ incident (see also this n-category cafe post).

The story of ‘the coincidence engine’ is that Grothendieck did a double (or was it triple) bluff when he dropped out of academia in protest of military money accepted by the IHES. He went into hiding only to work for a weapons company and to develop a ‘coincidence bomb’. As more and more unlikely events happen during a car-ride by a young Cambridge postdoc though the US (to propose to his American girlfriend), the true Grothendieck-aficianado (and there are still plenty of them in certain circles) will no doubt begin to believe that the old genius succeeded (once again) and that Ana’s (Grothendieck’s mother) $\infty$-ring is this devilish (pun intended) device…

However,

“There was no coincidence engine. Not in this world. It existed only in Banacharski’s imagination and in the imaginations he touched. But there was a world in which it worked, and this world was no further than a metre from our own. Its effect spilled across, like light through a lampshade.

And with that light there spilled, unappeased and peregrine, fragments of any number of versions of an old mathematician who had become his own ghost. Banacharski was neither quite alive nor quite dead, if you want the truth of it. He was a displaced person again, and nowhere was his home.”

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3 related new math-sites

F_un Mathematics

Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun.

From now on, this site will be hosted at the University of Ghent and maintained by Koen Thas. So, please update your bookmarks and point your RSS-aggregator to the new feed.

Everyone interested in contributing to this blog dedicated to the mathematics of the field with one element should contact Koen by email.

angst

Though I may occasionally (cross)post at F_un mathematics, my own blog-life will center round a new blog to accompany the master-course ‘seminar noncommutative geometry’ I’m running at Antwerp University this semester. Its URL is noncommutative.org and it is called :

Here, angs is short for Antwerp Noncommutative Geometry Seminar and the additions @t resp. + are there to indicate we will experiment a bit trying to find useful interactions between the IRL seminar, its blog and social media such as twitter and Google+.

The seminar (and blog) are scheduled to start in earnest september 30th, but I may post some prep-notes already. This semester the seminar will try to decode Smirnov’s old idea to prove the ABC-conjecture in number theory via geometry over the field with one element and connect it with new ideas such as Borger’s $\mathbb{F}_1$-geometry using $\lambda$-rings and noncommutative ideas proposed by Connes, Consani and Marcolli.

Again, anyone willing to contribute actively is invited to send me an email or to comment on ‘angst’, tweet about it using the hashtag #angs (all such tweets will appear on the frontpage) or share its posts on Google+.

Noncommutative Arithmetic Geometry Media Library

Via the noncommutative geometry blog a new initiative maintained by Alain Connes and Katia Consani was announced : the Noncommutative Arithmetic Geometry Media Library.

This site is dedicated to maintain articles, videos, and news about meetings and activities related to noncommutative arithmetic geometry. The website is still `under construction’ and the plan is to gradually add more videos (also from past conferences and meetings), as well as papers and slides.

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wp-latex’ sweet revenge : wp+MathJax-> ePub

In the early days of math-blogging, one was happy to get LaTeXRender working. Some years later, the majority of math-blogs were using the, more user-friendly, wp-latex plugin to turn LaTeX-code into png-images. Today, everyone uses MathJax which works with modern CSS and web fonts instead of equation images, so equations scale with surrounding text at all zoom levels.

However, MathJax has one downside : it doesn’t parse in ePub-readers. Peter Krautzberger wrote a post Epub and mathematics in which he suggested two methods to turn MathJax into ePub, but after dozens of experiments I still fail to reproduce these.

No doubt, someone will soon come up with a working alternative, but for the impatient here’s a quick but dirty method to turn your MathJax powered wordpress post into ePub :

the tools

  • download and install the ePub export plugin. It automatically creates an ePub file when a post or page is published or updated. The ePubs are stored in the uploads directory (to be found in the wp-contents directory).
  • download and install the wp-latex plugin. MathJax uses the normal \$ tex-delimeters whereas wp-latex requires \$latex, so this plugin doesn’t interfere with the default use of MathJax.
  • download the wp2latex python script. It converts a standard LaTeX file into a format that is ready to be copied into WordPress.

the routine

  • Edit the post you want to convert to ePub. Copy the contents of the post box to a file say post1.tex and save this in the same directory containing the latex2wp.py script.
  • In Terminal go to that directory and type the command ‘python latex2wp.py post1.tex’. It will produce a new file post1 in the same directory.
  • Copy the contents of post1 into the post box of your WordPress-post and press the update button. This time the TeX-commands in your post will be rendered using wp-latex and the ePub export-plugin will have created an ePub-version of it.
  • Locate this newly created ePub file in the relevant wp-contents/uploads/ folder (file has a number.epub name) and, if wanted, change its name into something easier to recognize and copy it somewhere outside the uploads directory. This will be your desired ePub-version of the post.
  • Replace the contents of the post box of your WordPress-post with the contents of the post1.tex file and hit the ‘Update’ button, to restore your original post (powered by MathJax).
  • Email your ePub-file to your iPad and open it with iBooks. Not quite as nice as MathJax-parsed TeX but a lot better than reading unparsed TeX-commands.
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master seminar ncg 2011

Note to students following this year’s ‘seminar noncommutative geometry’ : the seminar starts friday september 30th at 13h in room G 0.16.

However, if you have another good reason to be in Ghent on thursday september 22nd, consider attending the inaugural lecture of Koen Thas at 17h in auditorium Emmy Noether, campus De Sterre, Krijgslaan 281, 9000 Gent.

Koen’s lecture has one of the longest titles i’ve seen : “De lange weg – een verhaal over wiskundige problemen die denkers al eeuwenlang teisteren, zonderlingen die in afgelegen berghutten de existentie van de duivel willen aantonen, en een mythisch object dat niet bestaat, maar waar we toch naar zoeken” (“The long road – a story on mathematical problems torturing scientists for centuries, weirdos trying to prove the existence of the Devil in desolated mountain-huts and the search for a mythical object that doesn’t exist”).

Knowing Koen a bit I’d say it will be on the Riemann hypothesis, Grothendieck’s theory of motives and the field with one element. A sneak preview of our upcoming seminar, quoi?

More information on the event and to register see here.

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