Institute of Science and Technology Austria (ISTA) organizes a summer school on dynamical systems. The summer school will consist of 3 mini-courses of 3-4 hours each, accessible to PhD students and young researchers, with a few slots of contributed talks and a poster session.

The summer school takes place at the Federal Institute for Adult Education (bifeb), a beautiful location on Lake Wolfgang in Austria, near Salzburg. The organizing institute ISTA is a young international institute dedicated to basic research and graduate education in the natural, mathematical, and computer sciences. ISTA is located in Klosterneuburg on the outskirts of Vienna.

Photo from Summer School 2025

Lecturers

Alfonso Sorrentino (University of Rome Tor Vergata):

Title: Spectral rigidity questions in billiard-like dynamics

Abstract: Billiard systems have long fascinated researchers across multiple mathematical disciplines. Their intuitive physical interpretation and conceptual simplicity stand in contrast to their rich and varied dynamical behavior. Moreover, while it is evident that the geometry dictates the motion, a more subtle and compelling question is to what extent the geometry can be reconstructed from the dynamics. This problem lies at the heart of numerous rigidity phenomena and longstanding conjectures.
One of the most iconic questions in this framework, with a long and rich history at the intersection of dynamical systems and geometry, is: “Can one hear the shape of a billiard?” That is, can we recover the shape of the domain from the action of its periodic orbits?
In these lectures, we shall address this question and describe several rigidity results in the context of four interesting billiard-like models: Birkhoff billiards, outer billiards, symplectic billiards, and outer-length billiards.

Kostiantyn Drach (Universitat de Barcelona):

Title: Length spectral rigidity: the case of expanding circle maps

Abstract: In this mini-course, we will study length spectral rigidity in the seemingly simple dynamical setting of smooth expanding maps of the circle. Such maps have exponentially many periodic orbits of each given period, and to every periodic orbit one can associate its length, defined as the logarithm of its multiplier. The resulting collection of lengths is called the length spectrum of the map.
We will begin by discussing some statistical properties of this spectrum and explaining how one can construct expanding circle maps whose length spectra satisfy some prescribed structure. We will then turn to the rigidity problem. A simple computation shows that the length spectrum is invariant under smooth conjugacies. Therefore, it is natural to ask whether the converse holds: does the length spectrum determine the map up to smooth conjugacy? We will discuss this question both in the marked (easier) and in the unmarked (harder) settings.
Finally, we will place the problem in a broader context by explaining how it fits into a wider family of length spectral rigidity questions arising in geometry and dynamics.

Anna Florio (Université Paris Dauphine):

Title: Reeb flows in dimension 3 and Birkhoff sections

Abstract: We will introduce Reeb flows in dimension 3, defined on contact manifolds via the associated Reeb vector field. We will present canonical examples (Hopf flow, geodesic flow, non-singular Beltrami fields) ; we will discuss their connection to open book and broken book decompositions. Through Birkhoff sections, we will obtain dynamical consequences, such as positive topological entropy and the dichotomy of “2 or infinity” periodic orbits. We will then focus on right-handed Reeb flows on the 3-sphere, highlighting some obstructions connected to periodic orbits.

Organizing Committee

Vadim Kaloshin (ISTA)
Miguel Garrido (ISTA)
Mathieu Helfter (ISTA)
Illya Koval (ISTA)
Yunzhe Li (ISTA)
Alessandra Nardi (ISTA)
Daniel Tsodikovich (ISTA)