# Archive | research

## Biparametric persistence for smooth filtrations

$$\def\Real{\mathbb{R}} \def\phd{\mathtt{PH}} \def\CAT{\mathtt{CAT}}$$
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 of mappings into the plane

We will rely on the standard facts about generic smooth mappings into two-dimensional manifolds: for such mappings, the set of critical points is a smooth curve $$\Sigma$$ in $$M$$, which is immersed outside of a finite number of pleats: near generic point of $$\Sigma$$, there are local coordinates on $$M$$ in which the mapping is locally given by
$y_1=x_1, y_2=q(x_2,\ldots,x_m)$
(folds), and near isolated points of the curve of critical points, in some coordinates the mapping is given by
$y_1=x_1, y_2=x_2^3+x_1x_2+q(x_3,\ldots,x_m),$
(pleats).…

## Protected: lake wobegon academic publishing

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## Protected: cache choice conundrum

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## Fact of the day: Functions with given merge tree

$$\def\Real{\mathbb{R}}$$

Consider a Morse function on $latex f:\Real^n\to \Real$ with controlled behavior at infinity, – say, $latex f=|x|^2$ near infinity. Assume further that all critical values $latex a_1<a_2<\ldots<a_k$ are distinct and that all indices of critical points are $latex 0$ or $latex 1$. (Condition obviously holds in one variable.)

Clearly, there are many function that satisfy these conditions. In the (again, most transparent) univariate case, the enumeration of topological types of functions with given critical values is the subject of a nice thread of papers by Arnold on “snakes” (see, e.g., here), relating them to enumeration of ramified coverings of Riemannian spheres, up-down sequences and other cute objects.…

## Protected: Curvilinear Origami

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## Fact of the day: Brownian centroids

$latex \def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}$

Consider $latex n$ points in Euclidean space, $latex \xx=\{x_1,\ldots, x_n\}, x_k\in \Real^d, n\leq d+1$.

Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as $latex o(x_1,\ldots,x_n)$) lies at the intersection of the bisectors $latex H_{kl}, 1\leq k\lt l\leq n$, hyperplanes of points equidistant from $latex x_k, x_l$.

Assume now that the points $latex x_k,k=1,\ldots,n$ are independent standard $latex d$-dimensional Brownian motions. What is the law of the centroid?…

## Fact of the day: Condorset domains, tiling permutations and contractibility

$latex \def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}$

Consider a collection of vectors $latex e_1,\ldots,e_n$ in the upper half-plane, such that $latex e_k=(x_k,1)$ and $latex x_1>x_2\gt \ldots \gt x_n$. Minkowski sum of the segments $latex s_k:=[0,e_k]$ is a zonotope $latex \Z$. Rhombus in this context are the Minkowski sums $latex \Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n$ of pairs of the segments. Tilings of $latex \Z$ by $latex n\choose 2$ rhombuses $latex \Z(k,l)$ define also an oriented graph (formed by the edges of the rhombuses, and oriented up in the plane).…

## Topologically constrained models of statistical physics.

$$\def\Real{\mathbb{R}} \def\Int{\mathbb{Z}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}} \def\blob{\mathcal{B}}$$

### The Blob

Consider the following planar “spin model”: the state of the system is a function from $$\Int^2$$ into $$\{0,1\}$$ (on and off states). We interpret the site $$(i,j), i,j\in\Int$$ as the plaque, i.e. the (closed) square given by the inequalities $$s_{ij}:=i-1/2\leq x\leq i+1/2; j-1/2\leq y\leq j+1/2$$.

To any configuration $$\eta$$ we associate the corresponding active domain,
$A_\eta=\bigcup_{(i,j): \eta(i,j)=1} s_{ij}.$

We are interested in the statistical ensembles supported by the finite contractible active domains – let’s refer to such domains as blobs.…

## dimension of the Internet

As reported in the talk on the hyperbolic geometry of maps and networks, the often claimed approximability of the Internet graph (the routing graph as seen by Border Gateway Protocol) by samples from disks of large radius in the hyperbolic plane is somewhat questionable: the local dimensions of the Rips complexes build from the ASN graphs are wild.

Below are links to Javascript based visualizations
(designed and implemented by Yuriy Mileyko)
of the Rips complexes derived from

## hyperbolic geometry of Google maps

$$\def\Real{\mathbb{R}} \def\views{\mathbb{G}} \def\earth{\mathbb{E}} \def\hsp{\mathbb{H}}$$

##### Google Maps and the Space of Views

Let’s consider the geometry hidden behind one of the best user interfaces in mobile apps ever, – the smartphone maps, the ones which you can swipe, flick and pinch. We are not talking here about the incredible engine that generates the maps on the fly as your finger waddles over the screen – just about the interface, which in a very intuitive and remarkably efficient way allows you to move between various maps.…