Prague and overdue congratulations

Karim AdiprasitoLeave a comment

I visited Prague last week to distract myself after my cat Misha died way too early (he was only 6). And it was quite an amazing visit, in every way. Something about it made me forget my worries (can you guess what?)

More importantly, I met once again my very first postdoc Zuzka (Patakova) as well as my good friend Martin (Tancer). As is long-establish precedent among postdocs, Zuzka had to share some of her blueberry cake with me. But we also made some amazing advances, proving that realizing simplicial 4-polytopes is at least as hard as existential theory of the reals as well as investigating cool questions about spaces with prescribed geodesics.

Zuzka and a fully intact blueberry pie

Let me close with some overdue congratulations: My former postdoc Gaku Liu (now a professor in University of Washington) extended some joint work of him, Michael Temkin and me to prove an old conjecture in polytope theory: Every sufficiently large dilation of a lattice polytope admits a unimodular triangulation.

Misha Temkin
My former postdoc Gaku

Oh and my coauthor Vasso has succesfully defended her thesis. She will be joining my team in the fall to work on my new ERC project. Congratulations Gaku and Vasso!

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

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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

Karim 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.

Coauthors Tammy and David

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)

Karim 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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First post (the why and what)

Karim AdiprasitoLeave a comment

The idea of this blog, that I was long too lazy to put into action, was as a collection of smaller observations that are not exactly paper worthy. That, and the obvious procrastination from doing actual work.

The final kick was the pandemic, and a question that Yue Ren asked me recently. It occured to me that I had answered the question before, several times, to several people, but had not really recorded the answer. That will be the topic of the actual first post.

So if you look for smaller musings and nonsense, between combinatorics, algebra and geometry. Then that somewhere is here, for this result and others like it. Sometimes also other things, who knows.