math – For a cup of coffee more

Jewish Poker

16. December 202216. December 2022Karim AdiprasitoLeave a comment

For quite a while the two of us sat at our table, wordlessly stirring our coffee. Ervinke was bared. All right, he said. Let’s play poker.

No, I answered. I hate cards. I always lose.

Who’s talking about cards? thus Ervinke. I was thinking of Jewish poker.

He then briefly explained the rules of the game. Jewish poker is played without cards, in your head, as befits the People of the Book.

You think of a number, I also think of a number, Ervinke said. Whoever thinks of a higher number wins. This sounds easy, but it has a hundred pitfalls. Nu!

All right, I agreed. Let’s try.

We plunked down five piasters each, and, leaning back in our chairs began to think of numbers. After a while Ervinke signaled that he had one. I said I was ready.

All right, thus Ervinke. Let’s hear your number.

Eleven, I said.

Twelve, Ervinke said, and took the money.

I could have’ kicked myself, because originally I had thought of Fourteen, and only at the last moment had I climbed down to Eleven, I really don’t know why. Listen. I turned to Ervinke. What would have happened had I said Fourteen?

What a question! I’d have lost. Now, that is just the charm of poker: you never know how things will turn out. But if your nerves cannot stand a little gambling, perhaps we had better call it off.

…

Jewish Poker by Ephraim Kimshon

*What just happened?* by Hendrick ter Brugghen

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Angles, gherkins, polytopes and a sociological experiment

6. December 202210. December 2022Karim AdiprasitoLeave a comment

Let me start by admitting that this blog is a social experiment to see how long people take to ask me “what the fuck?” (I am kidding)

(seriously though, I am always a bit of chaos. Nothing out of the ordinary, though admittedly a bit sick and depressed at the moment. Thank you and all the love for caring y’all. Love you all 😉 )

Now, on to the other stuff: At some point in my life, I was doing my PhD and being an all around useless student (I spent most of my office computer hours watching Game of Thrones and Gossip Girl; this is not to say I did not work, but I usually cannot sit in an office chair and work. I paced around outside. Honestly I feel most time was wasted because I was anxious about seeming to work and ending up watching series rather than going out and thinking my own way), my advisor PhD Günter Ziegler and I looked at a gherkin and said: this is going to make a fine math paper. Anyway, tonight I was visited by three ghosts, and they told me a tale.

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A fact and a consequence

18. November 202210. December 2022Karim AdiprasitoLeave a comment

I am currently visiting my friend Paco Santos in Santander, Cantabria, and things are off to a rocky start. As I enter his office, he challenges me to a duel.

“I hold a theorem” he says. And after thinking for a few seconds, he adds: “I am also holding a corollary about symmetries”

I am stumped. It is early, and I lack the mental fortitude of morning coffee. I have to think…

The answer: the vertices of the dodecahedron can be partitioned into 5 regular tetrahedra (these are the green diagonals).

Now, you can use this fact to compute the group of symmetries of the dodecahedron! Well, clearly you can take the vertices of one of the tetrahedra to itself. That is the alternating group $A_4$ . But you can also take any tetrahedron to any other tetrahedron, leading to conjugate copies of the same group. Those are 5 copies. Hence, what you obtain finally is the alternating group $A_5$ .

Kustin-Miller unprojection, degree maps and post-nuclear relationships of the future

11. November 202218. March 2023Karim Adiprasito1 Comment

Dear Diary,

yesterday I heard an amazing talk by Eva Philippe and I will get back to that, as well as related questions I am thinking about with Sergey Avvakumov (who is on the market btw., though I do not expect trouble there cause he is doing great things) and Mark Berezovik (who I got to Jerusalem to finish his masters in exile. Study in exile. Interesting that we do that again).

Anyway, dear diary, it reminded me of something many years in the future.

You see, in 35 years I am going to sit my second grandchild on my lap. The bright one. She was always the bright one. So many questions.

And so, after helping my first, my grandson, build a gauntlet for the mutated squirrel warriors, I will sit her down. Spit the iodine tablet into the bucket and sigh, thinking about the best advice. She is starting to get interested in relationships, and she is asking me.

“Grandpa, I want to ask about pairings”

she begins, and my bioengineered squidheart sinks. Not that one.

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Kabelsalat, Schoolification, science education, flattening PL maps and Ronly Honly Bing

3. November 202210. December 2022Karim AdiprasitoLeave a comment

First bullet point: There is an interesting analysis of the secret deals that led to the war in Ukraine at the moment by the New York Times, as well as an account of the Trump administration involvement.

This post is brought to you by the letter A for anger issues, and the failed attempt to find an HDMI cable. Why is it that usually there are enough of them to choke two species into extinction, rid Paris of her rat-issues and still have enough for the Praelatura Sanctae Crucis et Operis Dei to flagellate themselves biweekly, but now I cannot find a single one.

Second: Something is rotten in the state of Denmark. Ok, I am being overdramatic; though if I was talking about the level of paranoia this country sometimes presents when faced with youth whose skin is not piggy-pink, that could well be appropriate.

I am also not referring to royal family affairs and the queen’s dental care.

Also, the issue is not really restricted to Denmark. I just really wanted to quote Marcellus.

What I am referring to is an increasing effort to make university a school. And as someone who hated school, let me say:

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The Medici bitch and quantifications in commutative algebra

31. October 202210. December 2022Karim AdiprasitoLeave a comment

As forces and probabilities greater than I decide my location in the next 48 hours, let me briefly profess my love for Samantha Morton as Serpent Queen. It is beautiful to watch a sly smile creep across her lips as schemes come to fruition, and perhaps more importantly, how hardships are quickly incorporated into yet another scheme.

*Isn’t it nice to scheme in France. Muhahaha*

Indeed, isn’t it most interesting when things fail? When we invariably hit our nose, the rock rolls down the hill again. For we can scheme how to roll it up next time. Blood in the streets and all that.

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Art, friends and aesthetically pleasing counterexamples

16. October 202224. November 2025Karim AdiprasitoLeave a comment

A while ago, Janos Pach told me a question of Peter Maga originating in topological graph theory: Given an arrangement of curves in the plane, can they be realized as geodesics? Due to my Nikolai Mnev‘s universality theorem (and earlier work by Ringel), it is badly impossible to linearize them (stretch them to line segments), even if we assume that any two curves intersect at most once and at that point transversally, that is, if they form a system of pseudo-segments. However, Janos “just” wanted each curve to be a shortest path.

This had been proven if the segments extend to infinity by Herbert Busemann (who, as I found out, was also an accomplished artist) and so Janos asked whether it was true in general; this of course would be much more reasonable than the realization along lines (note that the space of all metric spaces with given geodesics would be a disk, rather than the arbitrarily nasty deformation space that Kolya proved whe have when looking for arrangements of real lines.)

So, now we want each line to be the shortest, no two points along it should have shortcut somewhere else. A reasonable request; having dabbled in city planning (read: I played Cities: Skylines like, twice) I imagine it must be the worst nightmare of every real estate developer if the carefully named street is not actually used because a detour using other streets is shorter. (Edit: Conferred with a friend who does spatial planning for a German metropolis. It is not something that keeps him up at night. As with all of us, it is depression and anxieties that keep us up and night. And the decision to have coffee in the evening.)

Jokes aside, this can have real applications, as knowing which routes are shortest (geometer speak: knowing the geodesics and shortest paths) makes planning a route considerably easier (CS speak: faster to compute).

Alas, it turns out to be wrong, and Janos and I found that there are examples where the carefully named and arranged streets are not the shortest. And the aesthetically pleasing example you see here:

It is not hard to see that we cannot give a length metric to this graph such that each of the colored lines is a shortest path; here is the calculation:

And as streets should not have negative length, we are out of luck.

Disclaimer: updates to the post were sponsored by an anonymous professor at Rényi Institute

Anisotropy in arbitrary characteristic, Lefschetz beyond positivity, moment curves and lattice polytopes (part 2)

28. July 202218. March 2023Karim Adiprasito1 Comment

Just a quick post that kills some questions from the previous post, and a report on just an overall wonderful REU project summer.

So let me start by the following: myself, and then Papadakis and Petrotou, and then us three jointly, proved the Lefschetz property for triangulated spheres (characteristic two in the case of Papadakis and Petrotou). This may be an unfamiliar word for someone not from algebraic geometry, but essentially, it is a property stating that Poincaré duality in certain manifolds coming from algebraic geometry (for instance, smooth projective varieties) is realized in a concrete way. Now you may think combinatorial Hodge theory à la Rota conjecture I proved with June and Eric, or positivity of Kazhdan-Lustig polynomials established by Elias and Williamson, this is much better. Because while those arguments relied on a known combinatorial trick by McMullen/de Cataldo-Migliorini, this one used an entirely new idea. First, let me state the theorem, without getting too technical:

We proved, given a triangulated sphere $\Sigma$ of dimension $d-1$ , the face ring (in arbitrary characteristic) permits an Artinian reduction and contains a linear element $\ell$ so that

$A^k(\Sigma) \xrightarrow{\ell^{d-2k}} A^{d-k}(\Sigma)$

is an isomorphism.

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Scalar curvature and angles of polyhedra: test your intuition

11. July 20229. December 2022Karim AdiprasitoLeave a comment

I was listening to a marvelous talk by Jean-Pierre Bourguignon today, about generalizations and (geo)metric meaning of scalar curvature, and Misha Gromovs thoughts on the subject. Essentially, just like Alexandrov and Gromov and many others gave us metric understanding of sectional curvature, and Sturm and Lott-Villani and many more gave us metric ways of thinking about Ricci curvature, Misha is at it again to do the same with the weakest notion of curvature.

Jean-Pierre Bourguignon (I will take a less formal picture of him tomorrow)

One of the goals would be to understand polyhedral spaces from the viewpoint of scalar curvature. In this context, Misha asked me to give a reference for the following statement: Given fixed $d$ and any $\epsilon>0$ , there is a finite number of combinatorially distinct $d$ -polytopes all whose dihedral angles are all smaller than $\pi-\epsilon.$ I will write more about this and scalar curvature later, but for now I am stealing a format from Gil’s blog and ask: Did I have a chance to find such a reference, or do you find examples that make such a statement hopeless? And what $\epsilon>0$ can you choose safely?

While I am at it, let me add another question for your intuition: Can you deform a polytope such that all dihedral angles do not get bigger? What deformations can you find? More on this subject, and some partial answers to the questions above, later.