ERC grant for soon-to-be IST Austria professor Vadim Kaloshin

In mathematics, rigidity means that any type of object can’t be ‘shaped’ into a different object. Imagine a square. If you pushed two corners, the square would become a rhombus. If you would try the same on a triangle, you would quickly realize that a triangle stays a triangle, no matter how hard you push. Therefore, a triangle is a simple example of a rigid geometrical shape.

Rigidity of shapes and sounds

However, rigidity is not limited to geometrical shapes. This property can be applied to many systems. For example, many scientists study rigidity in billiards or drum sounds. Likewise did Professor Vadim Kaloshin at the University of Maryland, USA. The mathematician is an expert in mathematical physics, deterministic and stochastic dynamics, who will join IST Austria in 2021. In this research, he solved spectral rigidity problems, stating that one can’t deform a nearly circular drum without deforming a sound. In a different vein, Vadim Kaloshin is developing a stochastic technique aimed to explain the formation of the Kirkwood Gaps in the Asteroid belt in astronomy.

Can one hear the shape of a drum?

“The idea: you listen to a drum sound and try to recover its shape. Mathematically, this question is wide open. In my research, I am studying if one can deform the drum without deforming its sound,” explains Professor Kaloshin. He developed mathematical arguments showing that deforming a nearly circular drum inevitably leads to a change of the way it sounds.

At IST Austria, Professor Vadim Kaloshin and his group will develop tools to study rigidity of drum sounds. If their approach is successful, it will open a path to study rigidity and integrability of many dynamical systems. In a different direction, Professor Kaloshin studies different forms of stochastic behavior and the formation of Kirkwood gaps in the Asteroid belt.