# Probabilistic and Geometric Themes in Combinatorics

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Carnegie Mellon University (PA) · $150,000

## Abstract

This project aims to make progress on several areas at the interface of probability, geometry, and combinatorics. Three specific directions will be investigated that range from more theoretical to more applied. The project also involves a research experience for undergraduates.

The first direction of the project is that of enabling statistical inference in the realm of probability distributions characterized by geometric constraints.  For example, can we efficiently choose a uniformly random partition of a region into geometrically nice pieces?  When can we efficiently choose a uniformly random partition of a graph into, say, connected subgraphs? The second directions concerns the minimum lengths of combinatorial structures like spanning trees and Hamilton cycles among "cities" in geometric space.  Here, some aims push our knowledge in directions that are more closely motivated by applications, while others probe the limits of our current theoretical approaches. Finally, the third direction aims to explore competitive/online analogs of Radon's theorem for high-dimensional point-sets.  This direction places results in geometry and classical machine learning in a new context, while offering new paradigms for inquiry.  One example is the notion of a pseudo-randomized trial, in which controlling curse-of-dimensionality effects can replace randomization in rigorous comparison between groups under a linear effects model.

This award reflects NSF's statutory mission and has b

## Key facts

- **NSF award ID:** 2452213
- **Awardee organization:** Carnegie Mellon University (PA)
- **SAM.gov UEI:** U3NKNFLNQ613
- **PI:** Wesley Pegden
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** REU SUPP-Res Exp for Ugrd Supp
- **Estimated total:** $150,000
- **Funds obligated:** $150,000
- **Transaction type:** Standard Grant
- **Period:** 08/15/2025 → 07/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2452213

## Citation

> US National Science Foundation, Award 2452213, Probabilistic and Geometric Themes in Combinatorics. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2452213. Licensed CC0.

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