# Images from our research

The following images arise from research carried out by the faculty, organized by subfield.

If you are looking for a list of recent undergraduate research projects, visit this page instead.

## Chaos theory

How can we calculate the probability density function that determines the statistical properties of chaotic orbits or trajectories? The animation above illustrates the chaotic trajectory taken by a population that starts at 20% capacity when the logistic equation of its population is $4x(1-x)$.

- *Jiu Ding*

## Computational algebra

How can we efficiently choose a monomial ordering during a dynamic algorithm to compute a Gröbner basis? The animation above illustrates how the growth of the basis increases dramatically the number of orderings we can choose.

## Graph theory

The graph above is a counterexample to a conjecture on the size of the neighborhoods of independent sets.

## Partial differential equations

The ground state and contour of the problem $\Delta u + u^3=0$, $u>0$ on $D$, $u=0$ on $\partial D$, with $D$ being a nonconcentric annular domain.