# Dynamical Data: from Algebra to Topology

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Regents of the University of Michigan - Ann Arbor (MI) · $199,999

## Abstract

Dynamical systems are ubiquitous: they govern the motion of the planets, the weather, the stock market, and the ecosystems. These systems depend on different parameters, and as these parameters change, the corresponding system changes. Sometimes there are special values of the parameters for which the corresponding dynamical system is relatively simple. In complex dynamics, these are known as postcritically finite parameters. The postcritically finite parameters form a thin but sprawling collection that helps the understanding of the global structure of the parameter space. This project aims to extensively study the postcritically finite parameters and record several essential data that uniquely encode them. Broader impacts of the project are through mentoring at all levels, including a Math Corps summer camp for middle school campers and high school mentors from Ypsilanti, Michigan. 

In complex dynamics, one typically studies rational maps on the Riemann sphere from the point of view of iteration. A main principle in the subject asserts that in order to understand the dynamical behavior of a rational map, it is necessary to study the orbits of the critical points of the map. Those rational maps for which all critical points have finite forward orbits are quite special; they are known as postcritically finite rational maps. These maps possess only repelling and superattracting cycles, and their Julia sets are locally connected. Furthermore, (excluding well-understood excep

## Key facts

- **NSF award ID:** 2506489
- **Awardee organization:** Regents of the University of Michigan - Ann Arbor (MI)
- **SAM.gov UEI:** GNJ7BBP73WE9
- **PI:** Sarah C Koch
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $199,999
- **Funds obligated:** $199,999
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2506489

## Citation

> US National Science Foundation, Award 2506489, Dynamical Data: from Algebra to Topology. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2506489. Licensed CC0.

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