**Many effects in nature are determined by pure randomness. Máté Gerencsér is investigating the best way to incorporate random phenomena into differential equations and has now been awarded an ERC Starting Grant for his research.**

The ERC Starting Grants of the European Research Council are among the most highly endowed and prestigious grants in the European research landscape. One such grant has now been awarded to mathematician Máté Gerencsér from TU Wien. Gerencsér had only recently obtained a START Prize by the Austrian Research Fund FWF, and now his research project is getting an upgrade: with his ERC grant, endowed with approximately 1.5 million Euros, Máté Gerencsér will now push ahead with his research and set up a research group.

## How to describe the world

Differential equations are probably the most important mathematical tool for the natural sciences today. They are equations whose solution is not merely a number, but usually a function. "For example, the shape of a wave, a temperature curve that depends on place and time, the turbulences in air or water," Máté Gerencsér explains.The laws of nature that we know today are described in the form of differential equations - this applies to Newton’s laws of motion as well as to the equations of electrodynamics or the Schrödinger equation - a partial differential equation that describes the behaviour of quantum particles. These are deterministic equations: If we know the state of a physical system at a certain point in time, the state of the system at any later time is also determined.

## Incorporating randomness into the equation

The equations Máté Gerencsér is working with, however, are somewhat more complicated: "In our research group, we work with stochastic differential equations - that is, differential equations that also have a certain amount of randomness built into them, such as a certain kind of noise," he explains.For example, you can use ordinary differential equations to calculate how fast a hot rod cools down in a cool environment. But what if there is also a randomly fluctuating heat source that heats up the rod? The Schrödinger equation can be used to calculate the motion of an electron. But what if the electron is disturbed completely randomly by other particles in the process?

In order to be able to take such randomness into account in the world of differential equations, one has to develop completely new mathematical tools. For example, it can happen that certain terms of these differential equations become infinitely large - this is called a singularity. Conventional mathematical methods fail in this situation. However, there are so-called renormalisation methods with which one can still determine reliable results even in this case. "There have been some stunning breakthroughs in this field in recent years," says Máté Gerencsér. The goal is to further develop these ideas and answer important open questions.