Abstract
We have selected problems that may not yet be well known, but have the potential to push the research in interesting directions. In particular, we state problems that do not require specific knowledge outside the standard circle of ideas in discrete geometry. Despite the relatively simple statements, these problems are related to current research and their solutions are likely to require new ideas and approaches. We have chosen problems from different fields to make this short paper attractive to a wide range of specialists.The article is published in the author’s wording.
Highlights
X is the d-dimensional unit cube with opposite faces identified, and f : X → R is a random continuous function on this d-torus. Another natural choice would be to pick n random points in X and to define f (x) equal to the Euclidean distance of x from the nearest of these random points
Applying the homology functor to the increasing sequence of superlevel sets, we get a tower of homology groups, connected by homomorphisms induced by inclusion
Assuming coefficients in a field, all homology groups are vector spaces, and all homomorphisms are linear maps, which we write as fi,j : Vi → Vj
Summary
We are interested in the typical shape of a persistence diagram, assuming f is chosen at random from some population. To this end, we construct the empirical intensity plot of f , which is the function p : R2 → R whose integral over every region R ⊆ R2 is the expected number of points in R. One can always hope that a more general approach gives the necessary structure to gain insight into this question In this context, we note the “pointy hat” shape of the 0-dimensional plot, with its tip leaning to the left.
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