# research

## math department and its research

This visual continues the exploration started here. I used the dataset of MathSciNet records of the publications by department members, collected by the IGL participants. Each article reviewed by MSN carries a few MSC codes, roughly classifying them. I used these codes to create a sankey diagram, showing who published in what area. A few …

## 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 …

## Protected: allons enfants de la patrie

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## Protected: kod*fest

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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): …

## derivatives and random matrices

In connection with our IGL project, I remembered a result that might be useful, and is not widely enough known. Let $$z_1\leq \ldots\leq z_N$$ be a sequence of real numbers forming the spectrum of a Hermitian operator $$A$$ acting in $$N$$-dimensional Hermitian space $$U$$:$P_A(z):=\det(zE-A)=\prod_k (z-z_k).$ Proposition: Let $$V$$ be a codimension $$1$$ hyperplane in $$U$$, …

## 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 …