Geometric and dynamical properties of rational maps (GrandDrm)

Iterative processes have applications in many aspects of science and life. For example, a numerical optimisation method (such as gradient descent, or Newton's method) proceeds to finding minimum of a function by starting from a random initial guess (initial point), and then uses an iterative update rule to obtain a new point, hence creating a sequence of points which hopefully will converge to a minimum. Numerical optimisation methods have a lot of applications in real life. Study of long term behaviours of such iterative processes (besides that of an orbit as described above, also that of periodic points, of actions on cohomology groups, of actions on measures - which gives rise to entropy) is the subject of Dynamical Systems. When applying an iterative process to an algebraic function, one studies algebraic dynamical systems. This project studies algebraic dynamical systems in 2 settings. One is to construct very nice maps in the field of complex numbers (that of automorphisms) on very basic geometric objects (that of so called rational varieties) and have a very special property (positive entropy and not originating from smaller dimensional ones), which are not many for now. The other is for studying the action on cohomology for maps in positive characteristic, which there has been very little research on so far. This is very difficult but promises breakthrough if successes because it contains as a special case the famous Weil's Riemann hypothesis (corresponding to the study of periodic points and solved by Pierre Deligne in the 1970s). The last topic (algebraic Oka theory) of this project aims to construct new and interesting algebraic varieties to which there are many algebraic maps from affine space. Such algebraic Oka varieties are for the moment very few and not very well understood. The project touches on several fields simultaneously: Algebraic Geometry, Dynamical Systems, Several Complex Variables, Computer Algebra and Number Theory. Results from this project can be also useful for understanding numerical optimisation methods, in particular that for algebraic functions.

Several major unsolved questions in mathematics are formulated in terms of complex and algebraic varieties and maps between them (e.g. the Hodge and Jacobian conjectures). For example, Weil's Riemann hypothesis, the algebraic analog of the famous Riemann hypothesis, can be formulated in terms of periodic points of Frobenius maps. Weil's Riemann hypothesis has been at the centre of the development of algebraic geometry and recognised by various highest distinctions (Fields medal and Abel prizes) for work on it. One major recent result of the PI is to state a vast generalisation of Weil's Riemann hypothesis, inspired by complex dynamics, together with proof of a weaker version of it. This project is in mathematics and studies some very general properties of varieties and maps between them. Topic 1 of the project concerns with the fact that it is difficult to construct interesting rational selfmaps of an interesting complex variety, such as a rational variety, by looking at quotients of Abelian varieties and automorphisms of affine spaces, and by using computer algebra for effective computations of important invariants such as dynamical degrees. In Topic 2, these dynamical degrees enter into fields of positive characteristic to provide the generalisation of Weil's Riemann hypothesis mentioned above. Very recent work by Fei Hu on Abelian varieties gives more significant support to this conjecture. Topic 3 studies a class of complex manifolds which are targets of many maps from Stein manifolds, as well as a related classical conjecture of Forster about embedding of open Riemann surfaces into complex planes. There have been recent works on these two topics, including some by the PI, and the goal is to study these questions for algebraic manifolds. The success of the project, in particular of Topic 2, will bring breakthroughs with further applications in mathematics. The project will help to boost future research capacity through research training.