problems

merge tree monoid

\(\def\Real{\mathbb{R}}\def\tree{\mathcal{T}}\) Consider a path metric space \(M\) and a continuous function \(f:M\to \Real\). For simplicity, all such pairs here will have sublevel sets of \(f\) compact (so that the function is bounded from below). Then one can form the so-called merge tree (see, e.g. here, for motivation, and some history), a new topological space (let’s […]

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domain restrictions and topology

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\) Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \( e_k=(x_k,1)\) and \( x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments ( s_k:=[0,e_k]) is a zonotope ( \Z). Rhombus in this context is the Minkowski sums \( \Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n\) of a pair of the segments, perhaps

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analytic combinatorics in several variables (aimath, san jose 4-9.4.2022)

\(\def\Comp{\mathbb{C}}\def\Proj{\mathbb{P}}\def\Nat{\mathbb{N}}\) This workshop is about complex analytic techniques usable in applications from classical combinatorial problems to asymptotic representation theory and cluster algebras. The scope is approximately what is covered by Pemantle-Wilson(-Melczer) book. I plan to report on what is going on here (reporting is sporadic and idiosyncratic). Lectures are streamed. Day One (all times PDT):

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singularities, biparametric persistence and cubical complexes

\(\def\Real{\mathbb{R}}\def\phd{\mathbf{P}H}\def\CAT{\mathtt{CAT}}\) The goal of this note is to define the biparametric persistence diagrams for smooth generic mappings \(h=(f,g):M\to\Real^2\) for smooth compact manifold (M). Existing approaches to multivariate persistence are mostly centered on the workaround of absence of reasonable algebraic theories for quiver representations for lattices of rank 2 or higher, or similar artificial obstacles. Singularities

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