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 …
\(\def\Real{\mathbb{R}}\def\tree{\mathcal{T}}\) 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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There is no excerpt because this is a protected post.
\(\def\Comp{\mathbb{C}}\def\Proj{\mathbb{P}}\def\Nat{\mathbb{N}}\) 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): …
analytic combinatorics in several variables (aimath, san jose 4-9.4.2022) Read More »
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 \(U\), …
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 …
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 …
\(\def\Real{\mathbb{R}}\def\phd{\mathbf{P}H}\def\CAT{\mathtt{CAT}}\) 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 …
singularities, biparametric persistence and cubical complexes Read More »
