Quantum Calculus
by oliverknill
7h ago
Since finding the isospectral deformation of the exterior derivative (see “An integrable evolution equation in geometry” from June 1, 2013 and “Isospectral Deformations of the Dirac operator” from June 24, 2013), I tried to find discrete time integrable evolutions of the Dirac operator. Last Sunday, while experimenting in a coffee … The post Kublanovskaya-Francis Transform of Dirac matrix appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
6d ago
A 2-manifold with boundary is a finite simple graph for which every unit sphere is a circular graph with 4 or more nodes or a path graph with 3 more nodes. The boundary is the set of vertices for which the unit sphere is a path graph, the interior is … The post Curvatures for 2-manifolds with boundary appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
1w ago
An abstract delta set (G,D,R) is a finite set G with n elements, a selfadjoint Dirac matrix D=d+d* with and a dimension vector defining a partition and Hilbert spaces called k-forms. The exterior derivative maps to . The Hodge Laplacian is a block diagonal matrix defining the Hodge blocks and … The post Form Curvatures appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
2w ago
This is a presentation from Saturday, July 13, 2024. Curvatures are usually located on the zero dimensional part of space. I look here at curvature located on one or two dimensional parts of space. In the special case of a triangulation of a 2-dimensional surface, where the usual curvature is … The post More Curvatures appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
1M ago
In this presentation, there is a bit of advertisement for finite geometry and delta sets in particular. I also tried to get a bit into the history of finitist ideas in geometry and physics(starting with Riemann). One usually thinks about finite projective spaces when talking about “finite geometries”. I like … The post Geometry of Delta Sets appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
1M ago
I looked at a quadratic cohomology example. For theoretical backgroun, see the ArXiv paper “Fusion inequality for quadratic cohomology”. It is the case when U is a union of two disjoint smallest open sets in a 2-sphere for which I take the Icosahedron, one of the Platonic solids and a … The post Quadratic Cohomology appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
1M ago
While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect. If the simplicial complex is split into a closed set K (a sub-simplicial complex) and an open set U, one can distinguish 5 different cases of interactions. In principle there would … The post Fusion Inequality for Quadratic Cohomology appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
2M ago
Delta sets are very general. They include simplicial complexes, open sets in simplicial complexes, quotients of simplicial complexes, quivers and so multi-graphs or simply hypergraphs, sets of sets. For the later, the geometry is not that interesting in general. As for quivers, the associated delta set is one dimensional only … The post The most general finite geometric structure appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
2M ago
Quivers are graph in which multiple connections and loops are allowed. Since there is a Dirac operator for them, they define a one-dimensional delta set (G,D,r), where G is the union of vertices and edges (loops count as edges) and r is the dimension. For finite simple graphs, there is … The post Delta sets from Quivers appeared first on Quantum Calculus ..read more
Quantum Calculus
by oliverknill
2M ago
If is a finite abstract simplicial complex, a finite set of non-empty sets closed under the operation of taking non-empty subsets, we can ask about what f-vectors can occur if counts the number of sets of cardinality k in G. The case of the complete complex with gives a hint … The post Kruskal-Katona appeared first on Quantum Calculus ..read more

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