art – For a cup of coffee more

Messing around with OpenAI

2. December 202211. December 2022Karim AdiprasitoLeave a comment

Having a sick day, messing around with the OpenAI chatbot while enjoying soup.

Do you know who Mikhail Gromov is?

Yes, Mikhail Gromov is a Russian mathematician known for his work in geometry and topology. He was awarded the Fields Medal in 2010 for his contributions to the field of geometry, including his work on the structure of manifolds and the theory of groups of polynomial growth. He is currently a professor at the Institut des Hautes Études Scientifiques in Paris, France.

Do you know who Karim Adiprasito is?

Yes, Karim Adiprasito is a mathematician known for his work in algebraic and discrete geometry. He is the co-author of the book “Algebraic and Combinatorial Aspects of Tropical Geometry”, which has been recognized as a groundbreaking work in the field. Adiprasito is currently a professor at the Institute of Mathematics of the Hebrew University of Jerusalem in Israel.

Write an application letter for someone who wants to work with Karim Adiprasito as a postdoctoral researcher

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Subtlety, minimal resolutions, the value of the mystic, FromSoft, David Lynch and root polytopes

1. December 202210. December 2022Karim AdiprasitoLeave a comment

First, let me praise another excellent talk by Ayah Almousa; she presented newest work with my friend Anton Dochterman as well as Ben Smith on triangulations of root polytopes and the applications to minimal resolutions. Recall: A free resolution is a way to write a module M as a Matryoshka doll of free modules, that is, as an exact sequence

$0 \rightarrow F_i \rightarrow F_{i-1} \rightarrow \cdots \rightarrow F_{1} \rightarrow M \rightarrow 0$

where the $F_i$ are free (that is, as simple as you could ever want). And whoever tried to ever studies something hard knows how valuable it is to write something in simpler terms. A minimal free resolution is just that, and a root polytope is a polytope whose vertices correspond to the root directions of a Coxeter group. Ayah surveyed how triangulations of the latter are related to resolutions of monomial ideals, and applied this to prove some new results as well as give more insightful proofs to previously known ones. Bravo, Ayah, Anton and Ben! Here is her talk.

Now to more serious business. Real serious.

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Just an appreciation of art: Stephen Yang add Alef’s Corner

5. November 202224. April 2025Karim Adiprasito1 Comment

Three artist friends I wanted to appreciate here. I lack the words of a smarter (wo)man, so I will let the images and links speak for themselves. First Stephen Yang who makes me impatient for my travel to the east coast. Most of his photographs depict the miracle that is New York (though he also captured Henry Kissinger and Tahrir Square), although the city being the city, it could just be any part of the world and none at the same time.

Second, there is a secret friend of mine who is more shy about his identity. Depicting another friend of mine who is not so secret about his identity.

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: