Kaloshin Group

Dynamical Systems, Celestial Mechanics, and Spectral Rigidity

“Can you hear the shape of a drum?” Essentially, this question (and title of a famous paper by M. Kac) asks if the sound of a drum determines its shape—a question that has deep mathematical roots, and for the most part, remains open. Vadim Kaloshin and his group explore how deformations of a drum deform its sound, and if it is possible to change the shape of a drum without changing the sound. In particular, they study the Laplace spectrum of convex, planar domains, and work to show that these eigenvalues determine such domains locally. Similar questions can be posed for Riemannian manifolds, and are also of interest to the group.

Another main focus of the Kaloshin group is stochastic behavior in our solar system. Between the orbits of Mars and Jupiter, there are nearly two million asteroids with diameters greater than one kilometer. When astronomers look at the distribution of these asteroids with respect to semi-major axis or—equivalently—period of this asteroid belt, they see gaps, known as Kirkwood gaps. These gaps are located near low-order resonances with Jupiter, most famously with period ratios of 1:3, 2:5, and 3:7. The 1:3 gap is explained by a well-known mechanism proposed by Wisdom, and is supported by numerical experiments. This mechanism also seems to apply to the 2:5 gap. In this area, the Kaloshin group seeks to achieve two goals: first, to develop a mathematical theory of stochastic behavior at the gaps 1:3, 2:5, 3:7 and second, to explain the shape of the distribution of these gaps.