Susanna Terracini delivers the CRM Colloquium 2026

Take a billiard table, a convex one, and let a ball loose on it. It travels in straight lines and bounces off the wall at the same angle it arrived. From that, mathematicians have extracted most of the deep ideas in dynamical systems, and the reason is that the only thing you get to change is the shape of the wall. Everything else follows from the geometry of a single curve. Some tables produce motion so orderly you can predict where the ball will be a thousand bounces from now. Others produce trajectories that scatter across the table and never repeat. The question of which shapes produce which behaviour has been open, in one form or another, for a century.

Susanna Terracini (Università di Torino) came to the CRM on 14 July to talk about what happens when you make the problem harder on purpose. “With my research group, we complicated the problem a bit, putting a black hole in the table,” she said in an interview recorded at the centre that morning. Formally, a fixed gravitational mass sitting inside the domain. The ball still bounces off the wall according to the same optical law, but between bounces it no longer travels in a straight line. It falls. A black hole curves light, and the trajectories curve with it, tracing arcs of ellipses, parabolas and hyperbolas instead of segments. The interaction that governs the system is now between the geometry of the boundary and the optics of a curved ray.

These are Kepler billiards, and Terracini’s colloquium [TK: confirm URL], Rigidity for Kepler billiards, presented a result about which ones stay predictable. Among analytic, centrally symmetric, compact convex domains, only two survive: the circle with the mass at its centre, and the ellipse with the mass at a focus. Nothing else is analytically integrable at high energies. The proof runs through an explicit construction of a symbolic dynamics for the system, and it’s joint work with Stefano Baranzini, Vivina Barutello and Irene De Blasi, available as a preprint under the title On the Birkhoff conjecture for Kepler billiards.

Rigidity results have a particular flavour. They don’t tell you what exists. They tell you that the short list you already have is the whole list, and that no amount of cleverness will extend it.

 

The only breach

Terracini works across territory that looks, from outside, like three or four different careers: the N-body problem, spatial pattern formation in PDEs, regularity of free boundaries, spiralling solutions in competition-diffusion systems. Asked what holds it together, she named a single tension.

“Regularity can be seen as something good, an advantage, but it’s also a bit boring,” she said. “Things which are too regular may not be useful when you wish either to catch, but also to exploit, the complexity of the world.”

“Regularity can be seen as something good, an advantage, but it’s also a bit boring.”

What she looks for is the balance point: solutions complicated enough to be interesting and simple enough to be described. And the historical warrant for that strategy comes from Poincaré, who wrote in Les méthodes nouvelles de la mécanique céleste that periodic orbits are the only breach through which we might enter the chaos of the three-body problem.

Terracini’s version of that idea is more concrete, and better. The set of periodic solutions is dense. Which means you can connect any two configurations by stringing together pieces of them, with small error. So the periodic orbits behave like a network of public transport, a subway you can ride from one region of configuration space to another, changing lines as you go.

It also explains why the same person ends up working on planets and on reaction-diffusion equations. The common thread is methodological, she says: build the simple solutions, understand their symmetries, then concatenate them into something complex. Variational methods and the least action principle transfer between fields that otherwise share nothing.

 

What the board is for

Terracini was at the CRM as a member of its Scientific Advisory Board, which met at the centre from 13 to 15 July. The board is the body responsible for defining the centre’s strategic direction and advising on its scientific programme, and its members are appointed by the Governing Board on the Director’s proposal. It’s chaired by Ari Laptev (Imperial College London) and includes Afonso Bandeira (ETH Zürich), Victoria Gould (University of York), Boris Gutkin (École Normale Supérieure), Kathryn Hess (EPFL), Philippe Michel (EPFL), Alessandra Micheletti (Università degli Studi di Milano), Mikhail Sodin (Tel Aviv University), Juan Soler (University of Granada), Katrin Wendland (Trinity College Dublin) and Terracini herself.

She described the board’s job as bringing an international perspective into a place that necessarily operates in a local one. “As a mathematical community, we are one international community,” she said. “So we have a rather broad common view on what mathematics is. On the other hand, each mathematical centre has to cope with its own local government and its own socio-economic framework.”

That gap is where an external board earns its keep. Comparing policies across countries, she noted, tends to be constructive precisely because mathematics institutions aren’t really competing with one another. “We actually wish to build mathematics institutions all over the world. We share a common aim. So we are really here to help.”

Terracini joined the board recently, though she’s been coming to the CRM for years and regularly sends her students and collaborators to its activities. Asked what she’d change, she declined the premise. “I don’t see real weakness that could endanger the future of the centre. I think it will continue to grow.”