# Archive | problems

## Topologically constrained models of statistical physics.

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

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