geometry – For a cup of coffee more

Inspiration in abstract science, metric spaces of metric spaces and stability of numerical algorithms

6. November 202210. December 2022Karim AdiprasitoLeave a comment

Two revelations from yesterday. First, my anonymous friend M (don’t want them to get pestered with inquiries yet) gave me a draft of a novel so delightfully full of cool ideas that I could not reading even as I was driven along a serpentine mountain road. The revelation, apart from the obvious conviction of their genius, was that my stomach could not handle it and I felt like hurling the contents of a fine dinner over the beautiful scenery. Though really I should have known that from experience. Sidepoint: we are watching DEVS nos, which apart from hammering metaphors in with a sledgehammer, is not bad.

Second was that I kept thinking about a friendly interrogation that Stephen Yang of this post conducted on me, asking me to what extent we incorporate the real world experience in abstract science, or whether we are completely removed from it. (I also invited him to Jerusalem almost immediately, looking forward to your visit!)

I had two immediate answers for him: One is indirect, in that nature (in the sense of everything not inside the mind, so maybe the outside world would be a better term), at least for me, acts as cleaning current, flushing out the noise of repetitive thoughts that accumulate in creative thinking by overwhelming the senses.

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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 $d-3$ .