# Geometric and dynamical properties of rational maps (GrandDrm)

## About the project

This project touches several both classical and newly formulated questions and topics in mathematics. It aims to achieve a better understanding of open Riemann surfaces (such as open proper subsets of spheres and torus, or many other physical objects which one can see around oneself), specifically on the longstanding question (Forster's conjecture) of whether such open surfaces can be nicely embedded into the complex space of dimension 2. This topic is relevant to recent joint work by the PI on an algebraic analog of Oka's theory. The project also aims to detect whether some special constructed geometric objects (such as finite quotients of Abelian varieties) could be rational varieties, the latter being developed by the masters of geometry in the 19th century. This topic will be used to construct pathological examples of very nice algebraic varieties with transformational groups having very strong properties. The study of transformational groups on rational varieties is used also to construct other pathological examples of selfmaps having less periodic points than expected. One other major question considered in the project is a fairly recent conjecture (based on advancement in dynamical systems in Several Complex Variables) by the PI generalising Weil's Riemann hypothesis. The latter is an algebraic analog of the famous unsolved Riemann hypothesis, and itself has been crucial for the development of Algebraic Geometry and Number Theory since 1960s, and work on its proof has been recognised by Fields medal and Abel's prize to Deligne. The PI has solved a weaker version of the mentioned conjecture in the affirmative, and recently the conjecture has been solved (also in the affirmative) for the special case of Abelian varieties. The knowledge from Dynamical Systems and Geometry aspects of the project can be beneficial to understanding the convergence behaviour of Gradient Descent, a very popular and effective method in optimisation - with many applications.

## Objectives and Potential Impacts

The project encompasses the following fields: Dynamical Systems, Several Complex Variables and Algebraic Geometry. It will use Computer Algebra (both formally such as Groebner basis or numerically such as homotopy method in Bertini) in helping to solve large systems of polynomial systems. It can be benefited from and beneficial to Deep Learning. On the one hand, Deep Learning techniques have been now used in Algebraic Geometry to help with giving a first good indication what should be the correct answer in case a precise answer is difficult to obtain (for example, a polynomial system is too big that formal algorithms cannot terminate in a reasonable time). There is also now growing interest in applying Deep Learning to Automated Proof Checking - which when successful will surely influence the whole mathematics and has reciprocal effect to computations and applications (via Curry - Howard correspondence, which roughly says that checking the correctness of mathematical proofs are is the same as programming verification). On the other hand, Deep Learning uses crucially numerical (= iterative) optimisation methods, which are essentially the same as random dynamical systems, and the behaviour of sequences constructed by such numerical methods could be understood by researching the relevant geometric features of the space and the optimisation method under consideration.

## PhD student and Postdoc recruited

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## Financing

Research Council of Norway, Independent projects - Young research talent. Project number 300814, total budget 9,8 mill NOK.