In the talk, I mention three catchy open problems in differential geometry and how to formulate discrete versions. The first problem is whether a positive curvature 2d manifold has positive Euler characteristic, the second is whether there is a positive curvature metric on and the third is whether the list … The post Discrete Hopf Theme appeared first on Quantum Calculus ..read more

Finite geometric categories: graphs – simplicial complexes -simplicial sets – delta sets Delta sets were originally called semi-simplicial sets by Samuel Eilenberg and Joseph Zilber in 1950. Similarly than semi-rings are more general than rings or semi-groups are more general than groups, also delta sets are more general than simplicial … The post Gauss-Bonnet for Delta Sets appeared first on Quantum Calculus ..read more

The discrete Sard theorem in the simplest case (which I obtained in 2015) that a function from a discrete d-manifold to {-1,1} has level sets that are (d-1) manifolds or empty. (See here for the latest higher generalization to higher codimension.) A simplicial complex is a d-manifold if every unit … The post Sard for delta sets appeared first on Quantum Calculus ..read more

Here are some links to the articles mentioned in the talk: It surprisingly often happens that a big conjecture tumbles at around the same time. In the case of the Arnold conjecture, several approaches, spear headed by Conley-Zehnder, Eliashberg and Floer have reached the goal. But also almost always with … The post Arnold’s Theme appeared first on Quantum Calculus ..read more

The last geometric theorem of Poincare was conceived shortly before the death of Poincare. Poincare had a prostate problem when he was 58 and went to surgery in 1912 which he did not survive. Fortunately his last theorem was sent to an Italian journal two weeks before he died, but … The post Last geometric theorem appeared first on Quantum Calculus ..read more

Finite topological spaces are only interesting if non-Hausdorff because every Hausdorff finite topological space is the discrete topology. The topology from a simplicial complex is a nice example. I still grapple with some of the non-intuitive features. One example which holds in every topology coming from a positive dimensional complex: … The post Finite Topologies appeared first on Quantum Calculus ..read more

It is not quite yet a poem, but here, as promised in the movie, some code to generate both the Betti vector and Wu betti vector of a random submanifold in a given manifold. It is 25 lines without any additional libraries, so not yet quite a poem, but it … The post Wu Betti Conjecture appeared first on Quantum Calculus ..read more

My experiments so far indicate that the Wu cohomology of a d-manifold G can be read off from the usual cohomology. If is the Betti vector of G then (0,\dots,0,b_d,b_{d-1},\dots,b_1,b_0)$ appears to be the Wu Betti vector. So far, this is only a conjecture. In the talk, the case , … The post Wu Cohomology for Manifolds appeared first on Quantum Calculus ..read more

I refresh here a bit the code written in 2018 for Wu cohomology. a nice concrete cohomology. The selling point then had been that it gives different cohomology to the cylinder and Moebius strip. The Euler characteristic, simplicial cohomology as well as the Wu characteristic are the same for both … The post Cylinder And Moebius Strip appeared first on Quantum Calculus ..read more

The video below is an attempt to get back to an older story of Wu characteristic. One of the things which still needs to be explored badly is the Wu cohomology of the complement K of knots H and more generally of the complement K of k-dimensional manifolds H in … The post Back to Wu Characteristic appeared first on Quantum Calculus ..read more