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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.

A note on PL handlebodies, the Hausmann trick and some homology spheres

11. July 202210. December 2022Karim AdiprasitoLeave a comment

There is a surprising lack of intuition for PL manifolds around, which always surprised me. And it turns out you can answer some questions. We stay in the category of PL manifolds throughout. The following question was asked by Gil Kalai to Ed Swartz some 15 years ago. Ed could not answer the question, and popularized here.

Question (Kalai) Are there different (homeomorphism types of) triangulated closed compact homology $(d-1)$ -spheres $H$ with $g_3=0$ (for any $d$ )?

Without going too deep into what $g_3=0$ means, we note quite simply that it is equivalent to saying that $H$ is boundary to a triangulated manifold $D$ that has no interior simplices of dimension $d-3$ .

The answer to this is yes. In fact, there are infinitely many (). For this purpose, we will make the following observation:

Theorem H. Let $d \ge 2$ . Let $M$ be a PL $d$ -manifold. The following are equivalent:

1. $M$ admits a PL handle decomposition into handles of index $\le k$ ,

2. $M$ admits a PL triangulation in which all $(d-k-1)$ -simplices are on $\partial M$ .

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Sofia, Lefschetz via shelling and bonk and the holy trinity (kind of) (part 1)

10. July 202215. October 2024Karim Adiprasito1 Comment

This is born out of an attempt to find equivariant Lefschetz elements, so to have a combinatorial Lefschetz theorem that is a little less generic for some conjectures in geometric topology. Actually this succeeds to give some interesting results, and I will update in a second part. It will take me some days, in the meantime I made this here simple and clear.

Dear X,

so, Covid seems almost over (fingers crossed). I lost three I loved during the time (none of them due to covid, funny enough; two suicides and a heart failure), I had covid twice (fingers crossed for the hat trick) despite three vaccinations. I am in Sofia, at a inaugural conference of the ICMS.

But that’s not what this post is about. This post is about another proof of the g-conjecture. Well, the Lefschetz property for simplicial cycles, really. It is the simplest one yet, but that is not why it is important. It is also the third one (that is essentially different) and combines the ideas of three teams. (I count the original one by me here, and the characteristic two proof by Papadakis and Petrotou; our joint paper is a combination of the ideas from the former and yada yada yada creative counting to make the holy number work out. Deal with it.) If the first is a refined choreography of slashes and parries that is difficult to follow, and the second the equivalent of wooshing around with bloodhound step (using a miraculous formula that comes out of nowhere), then this is the equivalent of bonking the boss with a hammer: we write down a rational function and examine it, observing it has a pole to show it is nontrivial. Unfortunately it seems to be less general than either of the previous proofs, but I will see whether it can be pushed.

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K(π,1) for Artin groups and some interviews

7. August 202110. December 2022Karim AdiprasitoLeave a comment

I forgot to mention: Paolini and Salvetti proved the $K(\pi,1)$ -conjecture for affine Artin groups a while ago, congratulations! It is a very exciting result in combinatorics and topology!

Plus, Toufik Mansour is conducting interviews with renowned combinatorialists, including my friends Gil and Igor. Ah and also me.

Degrees, curious identities, Lefschetz maps and biased pairings (and a cliffhanger)

7. August 202110. December 2022Karim AdiprasitoLeave a comment

This post is long overdue, but being in the middle of nowhere helps finally progressing a little.

In the last post, I discussed a lemma of Kronecker. Let me discuss another version

Lemma (Kronecker, second version) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that im(A) and B ker(A) intersect trivially. Then the generic linear combination of A and B has kernel equal to the intersection of kernels of A and B, in other words, the kernel is as small as possible.

This is a very powerful tool to construct high-rank maps in vector spaces, in particular in the case when X and Y are dual in a Poincaré duality algebra. Then im A and ker A are orthogonal complements, and linear algebra tells us that orthogonal complements intersect trivially if and only if the Poincaré pairing does not degenerate on either space. This is exploited in my proof of the g-conjecture to construct Lefschetz elements, see here, and also in this survey (which I was invited to for winning the EMS prize, which I still cannot quite believe).

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Schmidt and analytic ranks, the Kronecker lemma and Hadamard lectures

11. February 202110. December 2022Karim Adiprasito2 Comments

Just a short post: David Kazhdan, Tammy Ziegler and I have recently (and finally) posted our proof that the analytic rank of a cubic, the logarithm of its bias, and its Schmidt rank (how hard it is to write it as products of lower degree polynomials), are linearly related.

This came out of our discussion of Kronecker’s lemma. One version can be stated as follows:

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Minkowski sums (and how big they must be)

24. January 202110. December 2022Karim AdiprasitoLeave a comment

So, this is the promised actual first post, motivated by and answering a question recently posed to me by Yue Ren (and several before him that I apologize for forgetting). It comes with a problem, and that is, to prove the result in a simpler, more elementary way.

I had written a longer paper (with Raman Sanyal) on a problem concerning Upper Bounds on the complexity of Minkowski sums that dealt with the problem quite thoroughly.

*Yue Ren (who asked me most recently), and Raman Sanyal (coauthor)*

But after we submitted the paper, and it was accepted and published in Publications IHES, someone asked me a question. I thought for a while, and answered that it follows from that and that lemma in the paper. But the question came up again and again, seemingly being quite relevant. On the other hand, writing a new paper on the matter was tedious, as it promised to be 90% repetition of paper number one. Which was an annoying prospect at best.

The problem is this: Consider a bunch (say m of polytopes $(P_i)$ in a euclidean vector space of dimension d. For simplicity, and to be interesting, let us assume that all these polytopes are of positive dimension. Also, let us assume that all these polytopes are in general position with respect to each other.

Question: How many vertices must the Minkowski sum of the family $(P_i)$ have? Surely, it has to be at least the number of vertices as the Minkowski sum of m segments in general position.

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