{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $M$ be a closed smooth manifold and let $P$ be an [[elliptic differential operator|MT9P2X85V8]] between vector bundles over $M$.\nThe [[analytic index|44GBFZ16SG]] $\\operatorname{ind} P = \\dim \\ker P - \\dim \\operatorname{coker} P$ is unchanged under continuous deformations of $P$ through elliptic operators.\nFind a formula for $\\operatorname{ind} P$ in terms of topological invariants of $M$ and of the principal symbol of $P$.\n\n## Remarks\n\nGelfand raised the question in an article on elliptic equations published in 1960.\nAtiyah and Singer solved it with their index theorem, announced in 1963: the analytic index equals the [[topological index|QDCD33CS21]].\nEarlier special cases were known, such as the Riemann–Roch theorem for curves and its generalization by Hirzebruch, and the Hirzebruch signature theorem.\n\n## Explanations\n\n### L1 human\n\nThe index of an elliptic operator is a whole number that does not change when the operator is deformed.\nNumbers with this kind of stability are usually topological, so Gelfand asked for a topological formula for it.\nThe answer, the Atiyah–Singer index theorem, became one of the central results of twentieth-century mathematics.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "44GBFZ16SG"
    ],
    "history": {
      "first_stated": "1960",
      "posed_by": "Israel Gelfand"
    },
    "kind": "problem",
    "msc": [
      "58J20",
      "35J99"
    ],
    "name": "Gelfand's problem on the index",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem"
      }
    ],
    "tag": "25G4AP9P4R",
    "uses": [
      "44GBFZ16SG",
      "MT9P2X85V8"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/25G4AP9P4R/",
  "links": [
    {
      "field": "about",
      "target": "44GBFZ16SG"
    },
    {
      "field": "uses",
      "target": "44GBFZ16SG"
    },
    {
      "field": "uses",
      "target": "MT9P2X85V8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/25G4AP9P4R.md",
  "msc": [
    "58J20",
    "35J99"
  ],
  "name": "Gelfand's problem on the index",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/25G/25G4AP9P4R.md",
  "status": null,
  "tag": "25G4AP9P4R",
  "uri": "m2://25G4AP9P4R",
  "url": "https://tamnd.github.io/math2/wiki/gelfands-problem-on-the-index/",
  "writing": {
    "tag": "25G4AP9P4R",
    "kind": "problem",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}