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 …

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): …

averaging inflections in plane curves

We all know that a generic complex plane curve of degree $$d$$ has $$3d(d-2)$$ inflections. Since Klein it is also known what the (strict) upper bound on the number of real inflections in real plane curves of degree $$d$$ is: $$d(d-2)$$, one third of the number of complex inflections. But what about the average real …

martingales and isochrons

Limit Cycles and Isochrones Consider smooth vector field $$v$$ on a manifold with an exponentially stable limit cycle $$\gamma$$: in other words, any trajectory starting close enough to $$\gamma$$, converges to it exponentially fast. In this situation Guckenheimer proved the existence of isochrons, – a foliation of an open vicinity $$U\supset \gamma$$ of the limit …

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 …