---
tag: 25G4AP9P4R
kind: problem
name: Gelfand's problem on the index
msc: [58J20, 35J99]
uses: [44GBFZ16SG, MT9P2X85V8]
about: [44GBFZ16SG]
sources:
  - url: https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem
history:
  posed_by: "Israel Gelfand"
  first_stated: "1960"
---

## Problem

Let $M$ be a closed smooth manifold and let $P$ be an [[elliptic differential operator|MT9P2X85V8]] between vector bundles over $M$.
The [[analytic index|44GBFZ16SG]] $\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P$ is unchanged under continuous deformations of $P$ through elliptic operators.
Find a formula for $\operatorname{ind} P$ in terms of topological invariants of $M$ and of the principal symbol of $P$.

## Remarks

Gelfand raised the question in an article on elliptic equations published in 1960.
Atiyah and Singer solved it with their index theorem, announced in 1963: the analytic index equals the [[topological index|QDCD33CS21]].
Earlier special cases were known, such as the Riemann–Roch theorem for curves and its generalization by Hirzebruch, and the Hirzebruch signature theorem.

## Explanations

### L1 human

The index of an elliptic operator is a whole number that does not change when the operator is deformed.
Numbers with this kind of stability are usually topological, so Gelfand asked for a topological formula for it.
The answer, the Atiyah–Singer index theorem, became one of the central results of twentieth-century mathematics.