Plenary lecture

Day 1 (Tuesday 22 April) @ 10:10 – 11:00

Jacob Tsimerman (University of Toronto)

Exceptional Integrability and Primitives of Differential Forms

ranscendence theory is one of the oldest subjects in mathematics, with some of the most basic questions (such as the transcendence of $\pi+e$) still being open. A natural place of study for this field is that of integrals like $\log(2)=\int_1^2 dx/x$. There is a fascinating conjectural theory, much of it described by the Period Conjecture, describing exactly when these “period integrals” ought to have algebraic relations. 

We consider indefinite period integrals such as $log(t)=\int_1^t dx/x$ which we refer to as primitives, and go beyond transcendence. Specifically, we consider the question of when certain primitives can be `expressed’ in terms of others. This study was initiated by Liouville who classified those differential forms whose integrals can be expressed using iterated operations of logarithms, exponents, and algebra.  We answer the question of when an arbitrary primitive can be expressed using finitely many others, both in a theoretical sense and by giving a decision procedure. Even though the question we study is purely complex-theoretic, it turns out to involve finiteness results in number theory. We also explain that the logarithm function is canonical, in that it is the minimal transcendental primitive you can obtain with a logarithmic residue! Everything here is joint with Jonathan Pila.

Biography

Jacob Tsimerman is professor at the University of Toronto specialised in number theory and related areas. He studied at the University of Toronto, graduating in 2006 with a bachelor’s degree in math. He obtained his PhD from Princeton in 2011 under the guidance of Peter Sarnak.

Together with Jonathan Pila, Tsimerman demonstrated the André–Oort conjecture for Siegel modular varieties. Later, he completed the proof of the full André-Oort conjecture for all moduli spaces of abelian varieties by reducing the problem to the averaged Colmez conjecture which was proved by Xinyi Yuan and Shou-Wu Zhang as well as independently by Andreatta, Goren, Howard and Madapusi-Pera.

In 2018, Tsimerman was an invited speaker at the International Congress of Mathematicians. In 2019, he was awarded the Coxeter–James Prize by the Canadian Mathematical Society. He is also one of winners of the 2022 New Horizons in Mathematics Prize, associated with the Breakthrough Prize in Mathematics. He was awarded for “outstanding work in analytic number theory and arithmetic geometry, including breakthroughs on the André-Oort and Griffiths conjectures”. In 2023, Tsimerman received the Ostrowski Prize