Sage-Days Workshop on Computational Arithmetic Dynamics 2019

Saint Louis University, Saint Louis MO

Investigators

Abstract

This award supports participation in the Sage-Days Workshop on Computational Arithmetic Dynamics held November 17-20, 2019 at Saint Louis University. The use of computers in the sciences over the last several decades has expanded enormously. With the increasing complexity of problems to be solved and increasing capacity of hardware to solve these problems, it is essential that students and researchers have cutting-edge computational tools available to them. This workshop brings together researchers in dynamical systems to further the development of open-source computational tools for use both in research and applications. Arithmetic dynamics is a relatively new area of mathematics at the intersection of number theory, dynamical systems, and arithmetic geometry. At its most fundamental, it concerns the resolution of systems of equations that arise from iterated systems. Its value to other disciplines lies in concrete constructions and computational tools. Applications range from weather prediction, to cellular modelling, to cryptography. Modern computational tools have great potential in these applications. These tools will be implemented in the open-source software Sage and made freely available to the public under the GNU public license. In addition to the development of computational tools, another main goal of the workshop is the training in the use of these computational tools. Participants of this workshop will become proficient in the current tools and be able to train colleagues and students in their use. Finally, these tools will be used to improve the database of dynamical systems. For ages, mathematicians have been compiling tables of computationally intensive information for widespread use. These tools are currently capable of computing an extensive set of properties that would be of use to the research community in an organized searchable form. The database of dynamical systems performs this service for the dynamical systems community. A goal of this workshop is to improve the current set of computational tools for dynamical systems included in the computer algebra system Sage. The tools allow one to quickly and efficiently work with explicit examples to test and prove new theories, to compute examples too complicated to be worked out by hand, and to allow students to explicitly investigate conjectures and specific examples. With the maturity of the current set of tools, we have reached the point where we can compute many properties of dynamical systems. Consequently, the creation of a database to organize the known examples is feasible. A comprehensive efficiently searchable collection of examples will be of considerable use to the research community. The current prototype of the database of dynamical systems will be improved and expanded at this workshop. With the tools freely available, the main barrier to their use is training/education. The organizers will actively recruit women and other under-represented in mathematics groups to this workshop, with a focus on graduate students, post-docs, and early career faculty. More information is available at the workshop web page http://mathstat.slu.edu/~hutzba/sage_days_2019.html This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.