lefschetz properties

Convex sets can run, but cannot hide. Also, Parseval-Rayleigh identities in characteristic two and Lefschetz properties for monoid algebras.

Karim AdiprasitoLeave a comment

Let me start with some overdue news, finally released after passive aggressive messages and open bullying. And England’s prime minister got usurped by lettuce. I promise I will start making more sense now.

Erdős and Szekeres famously showed that every set of general position points in the plane contains a large subset of points that is in convex position. And quite recently Andrew Suk improved this so much, that by now we have an essentially tight bound on how many points we can find in convex position, at least in the plane. This was improved a little furtherby Holmsen, Mojarrad, Pach and Tardos here.

But some found that all too plain, and pointed out that in higher dimensions, convex sets could have an even harder time hiding, and that there could be substantially larger convex sets in higher dimensional pointclouds. Alas, noone could prove this until recently, when my friend Cosmin provided, in joint work with Dmitrii Zakharov the first substantial improvement in dimension 3. Congratulations Dima and Cosmin (and see you soon, looking forward to a wintery northeast).

Dmitry Zakharov and Cosmin Pohoaţă in New York

Secondly, our (announcement) preprint on the unimodality of the h*-polynomial is finally online. One curious discovery in this research is that there is a Parseval-Raleigh type identity (Lemma 5.2) for the degree map of the Chow ring associated to the monoid.

Seen as a rational function.

In characteristic two.

In dependence on the torus action/the linear system of parameters.

This really remains to be understood, and we have no clue where it comes from. No phenomenon like that seems to be explored, and we suspect some deep connections to residue theory.

Two coauthors (who unlike others had the decency to stand in alphabetical order) and a dog during a recent joint trip to Cortona, Italy: Stavros Argyrios Papadakis and Vasiliki Petrotou

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

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

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

Karim 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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