# Surgery, square integrable cohomology, and quantitative geometry

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Chicago (IL) · $150,000

## Abstract

This project addresses two aspects of topology as it is conventionally understood, through the study of particular cases that seem ripe for attack: The first is that it tends to be a qualitative subject, which typically produces statements that certain kinds of objects or deformations exist.  It does not tell us how complicated such an object is, nor how much of some resource (think energy) is necessary to expend in producing the deformation.  The second aspect is that for many problems, topology progresses by reduction to algebra -- and the algebraic problems are themselves extremely difficult.  In some cases, this reduction can better be thought of as a reformulation of the problem in very different terms, but not necessarily easier ones.  The PI will work with and mentor younger researchers on these projects, present his findings at conferences, and work on building bridges to other disciplines.

The central attack envisioned in this project is to use analytic methods that are already known to connect to ring theoretic constructions in some cases (square integrable cohomology, and the Betti numbers defined for them using von Neumann algebras, pioneered by Atiyah) to study problems related to the number of handles necessary for manifold representatives of homology classes and how that contrasts with earlier work of the PI on the number of simplices (or volume).  This will involve, in the case of lattices, representation theory of semisimple Lie groups.  Within pure topolo

## Key facts

- **NSF award ID:** 2505738
- **Awardee organization:** University of Chicago (IL)
- **SAM.gov UEI:** ZUE9HKT2CLC9
- **PI:** Shmuel Weinberger
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **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=2505738

## Citation

> US National Science Foundation, Award 2505738, Surgery, square integrable cohomology, and quantitative geometry. Retrieved via AI Analytics 2026-06-09 from https://api.ai-analytics.org/grant/nsf/2505738. Licensed CC0.

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