research

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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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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