# CAREER: Ergodic Geometry Beyond Compactness and Uniform Hyperbolicity

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · George Washington University (DC) · $528,342

## Abstract

This project explores questions in ergodic geometry, an interdisciplinary field connecting ergodic theory and geometry. Ergodic theory, a branch of dynamical systems, investigates systems that evolve over time, often exhibiting unpredictable and chaotic behaviors. Examples of such systems include planetary motion, weather patterns, and stock markets. Geometry, on the other hand, studies the shape and structure of objects. Ergodic geometry combines these perspectives, using tools from dynamical systems to address geometric problems. For instance, the shape of an object can influence the complexity of certain dynamical systems occurring on that object.  This project aims to deepen our understanding of the relationship between an object’s shape and key dynamical quantities associated with it, such as entropic quantities, which serve as indices of complexity. In addition to advancing mathematical knowledge, the PI will establish the GW Experimental Mathematics Lab, creating a collaborative and vertically integrated research environment for students at George Washington University. This lab will foster hands-on learning and mentorship, preparing students to engage in cutting-edge mathematical research.

In ergodic geometry, significant milestones have been achieved when the underlying dynamical systems are compact and uniformly hyperbolic. This project aims to extend these findings to geometric systems that are non-compact, nonuniformly hyperbolic, or both, using thermodynamic f

## Key facts

- **NSF award ID:** 2441413
- **Awardee organization:** George Washington University (DC)
- **SAM.gov UEI:** ECR5E2LU5BL6
- **PI:** Lien-Yung Kao
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** CAREER-Faculty Erly Career Dev, REU SUPP-Res Exp for Ugrd Supp
- **Estimated total:** $528,342
- **Funds obligated:** $194,335
- **Transaction type:** Continuing Grant
- **Period:** 07/01/2025 → 06/30/2030

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2441413

## Citation

> US National Science Foundation, Award 2441413, CAREER: Ergodic Geometry Beyond Compactness and Uniform Hyperbolicity. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2441413. Licensed CC0.

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