# CAREER: Termination of Flips and Cluster Geometry

> **NSF 01002728DB NSF RESEARCH & RELATED ACTIVIT** · University of California-Los Angeles (CA) · $500,000

## Abstract

This project investigates problems in algebraic geometry. Algebraic geometry is the field of mathematics that studies the geometric shape of objects defined by polynomial equations. Often, the shape of these objects, called algebraic varieties, reflects the complexity of the defining equations and vice-versa. For instance, from the defining equations, we can understand if the algebraic variety has singularities, i.e., whether the geometric object has sharp points at which the curvature changes abruptly. These points are known as algebraic singularities. This project aims to develop new tools to understand algebraic singularities and apply these techniques to understand algebraic varieties of positive curvature.  As part of the broader impacts of this project, the PI will run online research seminars for graduate students related to the proposed research. The PI will initiate a summer reading program in algebraic geometry for undergraduates. The educational component of this project includes training in algebraic geometry for graduate students. Two summer research schools will be hosted at UCLA by the PI and experts in the field to train the new generation of mathematicians in birational geometry.

Fano varieties are considered one of the three building blocks of projective varieties. There has been a lot of progress towards understanding the geometry of Fano varieties in the last few decades. Nevertheless, the understanding of log terminal singularities (the local analog of

## Key facts

- **NSF award ID:** 2443425
- **Awardee organization:** University of California-Los Angeles (CA)
- **SAM.gov UEI:** RN64EPNH8JC6
- **PI:** Joaquín Moraga
- **Primary program:** 01002728DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** CAREER-Faculty Erly Career Dev
- **Estimated total:** $500,000
- **Funds obligated:** $87,074
- **Transaction type:** Continuing Grant
- **Period:** 07/01/2025 → 06/30/2030

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2443425

## Citation

> US National Science Foundation, Award 2443425, CAREER: Termination of Flips and Cluster Geometry. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2443425. Licensed CC0.

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