# Research

**Databases**

**Multilevel Picard Research**

Full history recursive multilevel Picard approximations (MLP for short) approximate high-dimensional semilinear PDEs without suffering from the curse of dimensionality. For simulations and articles on MLP see Multilevel Picard Research.

**Research interests**

Stochastic analysis

- Calculate the solution of a (partial) stochastic differential equation numerically. Unfortunately, Euler's method fails for many applied stochastic differential equations and we need to find suitable algorithms instead
- Regularity of the semigroup of solutions of (partial) stochastic differential equations
- Perturbations of (partial) stochastic differential equations
- Approximations of nonlinear parabolic partial differential equations

Biomathematics

- What is the influence of a randomly changing environment on the DNA sequence diversity? Gillespie writes in the Preface of his 1991 book The Causes of Molecular Evolution: "If we are to propose that molecular evolution is due to the action of natural selection, we need a mathematical theory to demonstrate that the dynamics of selection are compatible with the observations of molecular variation. It is my conviction that the only viable model of selection is one based on temporal and spatial fluctuations in the environment. The mathematics of selection in a random environment have never been systematically developed or brought to a point where they serve as a model of molecular evolution." To answer this question, we study branching diffusions in random environment and Wright-Fisher diffusions with random selection.
- What is the footprint of selection and, in particular, of selective sweeps on sequence diversity? There are different effects due to hard sweeps, soft sweeps and overlapping sweeps which are still to be understood.
- Can the positive effect of kin selection make up for a negatively selected side-effect of a new mutation?