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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.


Current Projects

Spectral rigidity for chaotic geodesic flows | Rigidity of planar convex domains | Rational caustics of domains with constant width


Chen J, Kaloshin V, Zhang HK. 2023. Length spectrum rigidity for piecewise analytic Bunimovich billiards. Communications in Mathematical Physics. View

De Simoi J, Kaloshin V, Leguil M. 2023. Marked Length Spectral determination of analytic chaotic billiards with axial symmetries. Inventiones Mathematicae. 233, 829–901. View

Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics. 408(Part A), 108591. View

Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under conjugacy. Regular and Chaotic Dynamics. 27(6), 525–537. View

Bialy M, Fiorebe C, Glutsyuk A, Levi M, Plakhov A, Tabachnikov S. 2022. Open problems on billiards and geometric optics. Arnold Mathematical Journal. 8, 411–422. View

View All Publications

ReX-Link: Vadim Kaloshin


Since 2021 Professor, Institute of Science and Technology Austria (ISTA)
2011 – 2021 The Brin Chair in Mathematics, University of Maryland
2008 – 2011 The Brin Chair in Mathematics, University of Maryland & Distinguished Professor of Mathematics, Penn State University
2007 – 2008 The Brin Chair in Mathematics, University of Maryland
2006 – 2007 Associate Professor, Penn State University
2005 – 2006 Associate Professor at Caltech with tenure
2004 – 2005 AIM Research Fellow and Associate Professor at Caltech with tenure
2002 – 2004 Member IAS, AIM Research Fellow, and Associate Professor at Caltech
2001 – 2002 C.L.E. Moore Instructor MIT and AIM Research Fellow
2001 PhD Princeton University
2000 – 2001 Courant Institute NYU and American Institute of Math Research Fellow

Selected Distinctions

2020 ERC Advanced Grant
2020 Gold medal International Consortium of Chinese Mathematics (ICCM)
2019 Barcelona Prize in Dynamical Systems
2017 Simons Fellowship in mathematics
2004 – 2006 Alfred Sloan Research Fellowship
2001 Prize of the Moscow Mathematical Society
2000 – 2005 American Institute of Mathematics Five-Year Fellowship

Additional Information

Vadim Kaloshin’s website
Mathematics at ISTA
Klaudiusz Czudek’s website

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