## 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

Fall 2020 -- current: Fei Hu (postdoc).

Fall 2020 -- current: Viktor Balch Barth (PhD student).

## Conferences/Workshops organised

Spring 2021: Seminar series on (Weil)'s Riemann hypothesis, Standard conjectures (including Hodge's conjecture) and Dynamical systems. Co-organised by Fei Hu and Tuyen Trung Truong. [Seminar's website]

December 2021: Several complex variables meeting Meeting's website

June 2022: Conference on Dynamical systems and systems of equations, at Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy. Co-organised by Cinzia Bisi, Viktor Balch Barth, Fei Hu, Stefano Luzzatto and Tuyen Trung Truong. [Conference's website]

June 2023: Conference on Oka Theory and Complex Geometry, at Fjordane Folkehøgskule, Norway, i.e. the Sophus Lie conference centre. Co-organised by Tuyen Trung Truong and Erlend Fornæss Wold. [Conference's website]

## Talks (seminars, conferences...)

Tuyen Trung Truong presents at the conference Complex analysis, Geometry and Dynamics, in Portoroz, Slovenia, June 2023.

Fei Hu presents at Ottawa/Carleton Number Theory Seminar, Ottawa, Canada, November 2022.

Fei Hu presents at Algebra & Geometry Seminar, University of Toronto, Toronto, Canada, November 2022.

Fei Hu presents at Seminar on Algebraic and Complex Dynamics, Tsinghua University, Beijing, China, November 2022.

Fei Hu presents at Algebra and Algebraic Geometry Seminar, University of Oslo, Norway, October 2022.

Fei Hu presents at Algebraic Geometry Seminar, Fudan University, Shanghai, China, October 2022.

Tuyen Trung Truong presents a talk at the Complex Geometry, Dynamical Systems and Foliation theory conference, CIRM Luminy (France), October 2022.

Viktor Balch Barth presents a seminar talk at University of Ljubljana (Slovenia), October 2022.

Tuyen Trung Truong presents a talk at the Workshop in Complex Geometry, University of Bochum (Germany), September 2022.

Fei Hu presents at Conference of Algebraic Geometry and Number Theory in Nanjing, Nanjing University, China, September 2022.

Tuyen Trung Truong presents a talk at the special session on Complex Analysis and Geometry, the 28th Nordic Congress of Mathematicians, Helsinski (Finland), August 2022.

Tuyen Trung Truong presents a talk at the semi-algebraic geometry seminar University of Dalat (Vietnam), May 2022.

Fei Hu presents at Informal Workshop in Arithmetic and Algebraic Dynamics, Harvard University, Cambridge, US, May 2022.

Fei Hu presents at Algebra/Number Theory Seminar, Brown University, Providence, US, April 2002.

Fei Hu presents at Harvard--MIT Algebraic Geometry Seminar, Cambridge, US, December 2021.

Viktor Balch Barth presents at SCV Conference, December 2021, UiO.

Tuyen Trung Truong presents at Logic seminar, UiO, November 2021.

Fei Hu presents at Algebraic Dynamics Seminar, Harvard University, Cambridge, US, October 2021.

Tuyen Trung Truong presents at the VinAI Research seminar about Deep Learning, August 2021 (online).

Fei Hu presents at Workshop on the Standard Conjectures, (Weil's) Riemann Hypothesis, and Relations to Dynamical Systems, University of Oslo, Norway, June 2021.

Tuyen Trung Truong presents at the SCV conference, Oslo, December 2020.

Tuyen Trung Truong presents at the Algebraic Geometry in East Asia, August 2020, online talk.

Tuyen Trung Truong presents at Recent Developments in Algebraic Geometry, Arithmetic and Dynamics (Part I), Institute for Mathematical Sciences, National University of Singapore, Singapore, June 2021.

Fei Hu presents at Recent Developments in Algebraic Geometry, Arithmetic and Dynamics (Part I), Institute for Mathematical Sciences, National University of Singapore, Singapore, June 2021.

Fei Hu presents at Web-séminaire Arithmétique et Géométrie Algébrique, Strasbourg, France, June 2021.

Fei Hu presents at the Loo-Keng Hua Young Mathematics Forum, Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China, May 2021.

Tuyen Trung Truong presents at SCV conference, University of Oslo, December 2020.

Tuyen Trung Truong presents at the Logics seminar, University of Oslo, November 2020.

## Research visits (university visits, conference attendances,...)

<Participating departments/centres, time frame, theoretical basis, scientific method, etc.>

## Research visitors

<Participating departments/centres, time frame, theoretical basis, scientific method, etc.>

## Publications/Preprints (partially) supported by the project

Preprints:

Fei Hu and Chen Jiang, An upper bound for polynomial log-volume growth of automorphisms of zero entropy, 2022, submitted.

Tuyen Trung Truong, Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations, arXiv:2209.05378. This paper supersedes the 3 previous ones: arXiv:2108.10249, arXiv: 2109.11395, and arXiv:2110.07403. Source code is available at GitHub's link: [Link]

Cinzia Bisi, Jonathan D. Hauenstein and Tuyen Trung Truong, Some interesting birational morphisms of smooth affine quadric 3-folds, arXiv:2208.14327

Fei Hu and Chen Jiang, Polynomial log-volume growth of quasi-unipotent automorphisms of abelian varieties (with an appendix in collaboration with Chen Jiang), arXiv:2208.11120.

Charles Favre, Tuyen Trung Truong and Junyi Xie, Topological entropy of a rational map over a complete metrized field, arXiv:2208.00668

Fei Hu and Tuyen Trung Truong, A dynamical approach to generalized Weil's Riemann hypothesis and semisimplicity, arXiv:2102.04405 (The older version contains also other results which will be developed in another paper later)

Tuyen Trung Truong, Some observations on the properness of Identity plus linear powers: parts 1 and 2, arXiv:2004.03309 and arXiv:2005.02260.

Tuyen Trung Truong, Unconstrained optimisation on Riemannian manifolds, arXiv:2008.11091.

Accepted/Published papers:

Fei Hu and Tuyen Trung Truong, An inequality for polarized endomorphisms, arXiv:2104.12660. Accepted in Archiv der Mathematik.

Tuyen Trung Truong, Tat Dat To, (Tuan Hang) Hang-Tuan Nguyen, Thu Hang Nguyen, Hoang Phuong Nguyen and Maged Helmy, A modification of quasi-Newton's methods helping to avoid saddle points, arXiv:2006.01512. Source code is available on GitHub (Link)

Tuyen Trung Truong and (Tuan Hang) Hang-Tuan Nguyen, Backtracking gradient descent method and some applications in Large scale optimisation. Part 1: Theory, accepted in Minimax Theory and its Applications. This is the more theoretical part of arXiv: 1808.05160, with some additional experiments. Accompanying source codes are available on GitHub: [Link]

Tuyen Trung Truong. Strong sub measures and applications to non-compact dynamical systems, (this is the more dynamical part of arXiv: 1712.02490), accepted in Ergodic Theory and Dynamical Systems. The paper is Open Access. doi.org/10.1017/etds.2020.132

Tuyen Trung Truong, When will a sequence of points in a Riemannian submanifold converge? (Mostly a survey paper, invited submission.) Special issue "Riemannian geometry of submanifolds", journal: Mathematics (MDPI), 2020, 8 (11), 1934. This is open access [Link to the paper] .

Tuyen Trung Truong and (Tuan Hang) Hang-Tuan Nguyen, Backtracking Gradient Descent method and some applications in Large scale optimisation. Part 2: algorithms and experiments. The main part of the paper is based on the more experimental part of arXiv:1808.05160, together with arXiv:2001.02005 and arXiv:2007.03618. Accompanying source codes are available on GitHub: [Link] Published online in Applied Mathematics and Optimization. The paper is Open Access. doi:10.1007/s00245-020-09718-8.

Tuyen Trung Truong, Some new theoretical and computational results around the Jacobian conjecture, Vol. 31, No. 7 (2020) 2050050, 27 pages. A Mathematica file is available on the arXiv version of the paper.

## Source codes

Python source code for Backtracking New Q-Newton's method on GitHub: [Link]

Python source code for New Q-Newton's method on GitHub: (Link)

An implementation of Backtracking Gradient descent for Deep Neural Networks: available on GitHub: [Link]

A Mathematica file for some computations around the Jacobian conjecture is in the appendix here: [Mathematica's source code]

## Financing

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