# Ancient Ricci Flow and Ricci Solitons

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of California-Irvine (CA) · $199,999

## Abstract

This project explores a powerful mathematical process called Ricci flow, which smooths out geometric shapes over time--like heat evening out the temperature on a surface. Ricci flow has led to major breakthroughs, including Perelman's resolution of the century-old Poincaré Conjecture, and continues to transform our understanding of the geometry and topology of space. A major challenge lies in understanding the "singularities" that inevitably form during the flow, where the shape becomes infinitely curved, and hold vital clues about the hidden structure of space. A central idea in Perelman's proof is the use of surgeries to continue Ricci flows through singularities. The PI aims to extend this surgery construction in higher dimensions, with the goal of uncovering new geometric and topological applications. The research will be complemented by mentoring graduate students and organizing workshops and conferences.

The research project is split into two parts: The first project focuses on classifying ancient Ricci flows that are asymptotic to cylinders, which serve as potential singularity models. These asymptotically cylindrical flows include the classical rotationally symmetric examples such as the Bryant soliton and Perelman's ovals, as well as new examples recently constructed by the PI, known as flying wings. The flying wings are asymptotic to cylinders with more than one R-factor and break the rotational symmetry. The PI aims to develop a general method to estimate the

## Key facts

- **NSF award ID:** 2506832
- **Awardee organization:** University of California-Irvine (CA)
- **SAM.gov UEI:** MJC5FCYQTPE6
- **PI:** Yi Lai
- **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=2506832

## Citation

> US National Science Foundation, Award 2506832, Ancient Ricci Flow and Ricci Solitons. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2506832. Licensed CC0.

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