# Archive | problems

RSS feed for this section## Protected: Shuffling the sheep

## Protected: Curvilinear Origami

## Fact of the day: Brownian centroids

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

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 \(o(x_1,\ldots,x_n)\)) …

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

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

\)

Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \(e_k=(x_k,1)\) and \(x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments \(s_k:=[0,e_k]\) is a zonotope \(\Z\). *Rhombus* in this context are the Minkowski sums \(\Z(k,l)=s_k\oplus s_l, …

## 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), …

## 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 …

## unimodal category

We say that a continuous function is unimodal if all the upper excursion sets

\(e_f(c):=\{f≥c\}\) are contractible. For a nonnegative function

\(f:\mathbb{R}^d\to\mathbb{R}_+\) with compact support its unimodal category UCat(f) as the least number \(u\) of summands in a decomposition \(f=\sum_{i=1}^u …

## linear search

Linear search problems deals with locating an object hidden on a half line (“long road”) by an absent-minded robot which gather everything it sees (see, e.g. here for more details). A *program* for the robot is a sequence \(0<x_1<x_2<\ldots\) of …