I forgot to mention: Paolini and Salvetti proved the 
Plus, Toufik Mansour is conducting interviews with renowned combinatorialists, including my friends Gil and Igor. Ah and also me.
K(π,1) for Artin groups and some interviews
Degrees, curious identities, Lefschetz maps and biased pairings (and a cliffhanger)
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).
Continue readingSchmidt and analytic ranks, the Kronecker lemma and Hadamard lectures
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:
Continue readingMinkowski sums (and how big they must be)
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.
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 
Question: How many vertices must the Minkowski sum of the family
First post (the why and what)
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.
For a cup of coffee more 