# research

## averaging inflections in plane curves

$$\def\Nat{\mathbb{N}} \def\Real{\mathbb{R}}\def\Int{\mathbb{N}} \def\Rat{\mathbb{Q}}\def\Comp{\mathbb{C}}\def\ex{\mathbb{E}}\def\im{\mathtt{img}}\def\iid{{\em{iid}}}\def\tM{\tilde{M}}\def\bn{\mathbf{n}}\def\bz{\mathbf{z}}\def\bw{\mathbf{w}}\def\bk{\mathtt{k}}\def\bn{\mathbf{n}}\def\bz{\mathbf{z}}\def\bw{\mathbf{w}}\def\br{\mathbf{r}}\def\bsi{\mathbf{\sigma}}\def\Proj{\mathbb{P}}\def\pv{\Proj{V}}\def\val{{\vec{\alpha}}}\def\bz{\mathbf{z}}\def\pinv{\pi^{-1}}\def\kom{\rho_K(p)}\def\pt{\tilde{p}}\def\pb{\bar{p}}\def\so{\mathtt{SO}}\def\kk{\mathbf{k}}\def\flags{\mathbf{F}}\def\var{\mathbb{V}}$$ We all know how many inflections a generic complex plane curve of degree $$d$$ has: $$3d(d-2)$$. Since Klein’s work it is known that the (strict) upper bound on the number of inflections in real plane curves of degree $$d$$ situation is $$d(d-2)$$. But what about the average situation? Of course, one would …

## martingals 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 …