We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $${\mathbb {X}}$$ X equipped with a continuous function $$f: {\mathbb {X}}\rightarrow \mathbb {R}$$ f : X → R . We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $$\mathbb {R}$$ R . We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $$({\mathbb {X}}, f)$$ ( X , f ) when it is applied to points randomly sampled from a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of $$({\mathbb {X}}, f)$$ ( X , f ) , a constructible $$\mathbb {R}$$ R -space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $$({\mathbb {X}},f)$$ ( X , f ) to the mapper of a super-level set of a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

No abstract

Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell.

We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al.