CRM COLLOQUIUM 2026

Susanna Terraccini

Università di Torino

CRM Colloquium 2026

Colloquium

July 14, 2026

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ABSTRACT

Rigidity for Kepler billiards

A point mass moves inside a convex domain $\ \Omega \subset \mathbb{R}^2$

under the gravitational attraction of a fixed positive mass. Upon reaching the boundary, the particle is reflected back into $\ \Omega$

according to the geometric–optics law: the angle of incidence equals the angle of reflection. It is known that this type of dynamical system is integrable (for all energy levels) when $\ \Omega$

is an ellipse and the attracting mass is placed at one of its foci [1] or, more generally, when $\ \Omega$

is composed of arcs of confocal quadrics, again with the attracting mass at a focus [2].

In this talk we will discuss the following rigidity phenomenon: among analytic, centrally symmetric, compact convex domains, the only ones for which the corresponding Kepler billiard is analytically integrable (for sufficiently large energies) are circles with the attracting mass placed at their centres and ellipses, with the attracting mass at a focus. Our approach is based on an explicit construction of a symbolic dynamics for the system.

If time permits, I will discuss some connections between this construction and the presence of a non-rational invariant curve of rotation number $\ 1/2$

for the standard billiard map inside $\ \Omega$

and how to build billiard tables with $\emph{exotic} 1/2$

-invariant curves.

Joint work with Stefano Baranzini, Vivina Barutello, Irene De Blasi

References

$\begin{array}{l} [1]\ \text{S. Baranzini, \emph{Billiards with singular invariant curves}, arXiv:2508.08990 (2025).}\\[4pt] [2]\ \text{S. Baranzini, V. Barutello, I. De Blasi, S. Terracini, \emph{On the Birkhoff conjecture for Kepler billiards}, arXiv:2507.08446 (2025).}\\[4pt] [3]\ \text{V. V. Kozlov, \emph{Some integrable extensions of Jacobi's problem of geodesics on an ellipsoid}, J. Appl. Math. Mech. (1994).}\\[4pt] [4]\ \text{A. Takeuchi, L. Zhao, \emph{Conformal transformations and integrable mechanical billiards}, Adv. Math. (2024).} \end{array}$

SPEAKER

Susanna Terracini

Università di Torino

Terracini was born in South London. She earned a laurea in 1986 in mathematics at the University of Turin, supervised by Fulvia Skof. She completed her Ph.D. at the International School for Advanced Studies in 1990. Her dissertation, Periodic Solutions to Singular Newtonian Systems, was supervised by Ivar Ekeland and Sergio Solimini. She was a researcher at Paris Dauphine University from 1988 to 1989, and became a faculty member at the Polytechnic University of Milan in 1990. In 2001 she became a full professor at the University of Milano-Bicocca, and in 2012 she returned to Turin as a professor.

One of Terracini’s papers on the n-body problem was selected for a featured review in Mathematical Reviews. She was the winner of the 2002 Vinti Prize, a prize of the Italian Mathematical Union for young researchers in mathematical analysis. In 2007 she won the Bruno Finzi Prize of the Istituto Lombardo Accademia di Scienze e Lettere. In 2020 she was awarded the Schauder Medal from the Juliusz P. Schauder Center for Nonlinear Studies at the Nicolaus Copernicus University in Toruń, Poland.

SCHEDULE

Time	Tuesday 14 July 2026
CRM Colloquium
16:00→ 16:10	Welcoming
16:10→ 17:10	On the Birkhoff conjecture for Kepler billiards Susanna Terracini Università di Torino (and member of the SAB)
	A toast of cava

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