# Nonlinear Geometric Flows: Singularity analysis, ancient solutions and regularity

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Columbia University (NY) · $199,056

## Abstract

Some of the most important problems in mathematics and physics are related to the understanding of singularities. These are anomalies in the behavior of a physical quantity where the norm that is used to measure such quantities breaks up. It may be related to the understanding of turbulence, of black holes, the accumulation of cancer cells in human bodies, or the behavior of neurons in the brain. These phenomena are often described mathematically via a differential equation which involves time and space. Studying the qualitative behavior of the solutions of such equations becomes crucial for understanding the related physical problem and is also essential for computing. The main goal of this project is to study the singular behavior of partial differential equations that is related to physical problems, as discussed above, and also to see how the fundamental shapes of spheres, cones, and cylinders, appear in the singularity formation of these equations. The goal of the studies is to enhance our knowledge of the behavior of solutions near singularities. The project provides research training opportunities for graduate students and postdoctoral scholars.

The Ricci flow is a geometric equation that describes the intrinsic change in shape according to its Ricci curvature, a notion of curved space that played a fundamental role in the theory of relativity. G. Perelman, in his seminal 2002 work on the Ricci flow and the resolution of the 100 years old Poincare Conjecture, showed

## Key facts

- **NSF award ID:** 2454018
- **Awardee organization:** Columbia University (NY)
- **SAM.gov UEI:** F4N1QNPB95M4
- **PI:** Panagiota Daskalopoulos
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $199,056
- **Funds obligated:** $199,056
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2454018

## Citation

> US National Science Foundation, Award 2454018, Nonlinear Geometric Flows: Singularity analysis, ancient solutions and regularity. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2454018. Licensed CC0.

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