The Compositio Prize is awarded every three years by the Foundation Compositio Mathematica in recognition of outstanding research published in Compositio Mathematica.
The 2020–2022 prize was awarded to Ziyang Gao for his paper:
Generic rank of Betti map and unlikely intersections, Compositio Mathematica 156 (2020), pp. 2469-2509.
A symposium celebrating this achievement will be held on Friday, November 14, 2025 in Utrecht.
Please register before November 5th by sending an email with your name and affiliation to apeykar@science.ru.nl
Location: Room "Atlas" in the Koningsberger building (first floor) from 13:00 to 18:00 https://www.uu.nl/en/victor-j-koningsberger-building
Program:
13:00 - 14:00 Rafael von Känel (IAS Tsinghua), TBA
14:00 - 14:30h Coffee break
14:30 - 15:30 Philipp Habegger (Basel University), TBA
15:30 - 16:00 Coffee break
16:00 - 16:15 Award ceremony
16:15 - 17:15 Ziyang Gao (UCLA), TBA
17:15h - 18h Reception
Rafael von Känel's talk
Title: Results for non-degenerate Diophantine equations
Abstract: We discuss various results for Diophantine equations satisfying a certain non-degeneracy criterion. For example a uniform bound for the number of solutions of non-degenerate equations (defining a curve of genus >1) is provided by the deep uniformity result of Dimitrov-Gao-Habegger which relies on Ziyang Gao's prize winning paper. After illustrating this uniform bound, we focus on the size/height of the solutions of non-degenerate equations. We present explicit height bounds which establish in particular the effective Mordell conjecture for large classes of (explicit) curves over the rational numbers. In addition, on combining our explicit height bounds with Diophantine approximation techniques, we were able to solve the Fermat problem inside a classical rational surface and to completely determine the set of rational points of certain explicit families of curves of genus >1. We discuss these applications and we also explain the strategy of proof which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. Our results presented in this talk were obtained in joint works with Arno Kret and Shijie Fan.
Philipp Habegger's talk
Title: Specializing Linear Recurrence Sequences at Roots of Unity
Abstract: The Skolem-Mahler-Lech Theorem characterizes the vanishing members of a linear recurrence sequence. In the setting over a number field, no effective proof of this classical theorem is known. In other words, given a linear recurrence sequence, we know no algorithm that is guaranteed to produce precisely the indices where the sequence vanishes.
We consider linear recurrence sequences of rational functions and study when sequence members vanish at a root of unity. Bilu-Luca and Ostafe-Shparlinski proved finiteness results for linear recurrence sequences of order 2. I will report on a new finiteness result for linear recurrence sequences of order 3. We require linear forms in logarithms combined with o-minimal geometry and follow the Pila-Zannier strategy. This calls for a functional transcendence result. Gao's prize-winning paper treated the universal family of abelian varieties. We will require an older result in the multiplicative setting.
This is joint work in progress with Alina Ostafe and David Masser.
Ziyang Gao's talk.
Title: From sparsity of rational points on curves to the generic positivity of Beilinson-Bloch height
Abstract: It is a fundamental question to find rational solutions to a given system of polynomials, and in modern language this translates into finding rational points in algebraic varieties. It is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying rational points on curves. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.
In my talk, I will explain the historical and recent developments of this problem according to the different grades. I will also mention a recent work (joint with Shouwu Zhang) about a generic positivity property and a Northcott property of the Beilison-Bloch height of the Gross-Schoen cycles and the Ceresa cycles.
Organizers: Ariyan Javanpeykar (Nijmegen) and Marta Pieropan (Utrecht)