# Analysis of noncompact and singular Ricci flows

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Arizona State University (AZ) · $149,911

## Abstract

The Ricci flow is the prototypical example of a geometric heat flow -- a natural process which evolves the geometric structure of a space by a heat-type differential equation, smoothing out bumps and other irregularities much in the same way that the laws governing the diffusion of heat drive the temperatures of all objects in a room, hot or cold, toward the same value over time. Geometric flows arise as models for physical phenomena as diverse as the evolution of grain boundaries in annealing metal and the weathering of stones at the ocean's edge, and through their tendency to "improve" a given space into something more symmetric and homogeneous, they have proven to be remarkably effective tools in efforts to resolve fundamental mathematical questions at the intersection of geometry and topology. This project belongs to these efforts, seeking to better understand the extreme situations where the analogy between the (linear) heat equation and the (nonlinear) Ricci flow begins to break down. The main aims are to study the nature of solutions in singular regions (where the space is becoming irrecoverably curved or pinched), and to extend the analytic theory of the equation to solutions which may become arbitrarily highly curved near spatial infinity. The project also includes an educational component, naturally incorporating the mentorship and research training of graduate students.  


This project has two components. In one direction, the PI will build on his past collabo

## Key facts

- **NSF award ID:** 2506807
- **Awardee organization:** Arizona State University (AZ)
- **SAM.gov UEI:** NTLHJXM55KZ6
- **PI:** Brett Kotschwar
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $149,911
- **Funds obligated:** $149,911
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2506807

## Citation

> US National Science Foundation, Award 2506807, Analysis of noncompact and singular Ricci flows. Retrieved via AI Analytics 2026-06-09 from https://api.ai-analytics.org/grant/nsf/2506807. Licensed CC0.

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