{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "XW2BS4VG3P"
    },
    {
      "field": "uses",
      "target": "ZMRFSYVWQT"
    },
    {
      "field": "defines",
      "target": "YX386FVZQR"
    }
  ],
  "body": "\n## Summary\n\nThe Loewner differential equation describes a family of conformal maps that change continuously in time.\nEach map sends the complement of a growing set onto a fixed reference domain.\nThe whole family is driven by a single real or circle-valued function of time, the driving function.\nCharles Loewner introduced it to prove the case $n = 3$ of the [[Bieberbach conjecture|ZMRFSYVWQT]].\nLouis de Branges used it in the proof of the full conjecture.\nWith a Brownian motion as driving function it defines the [[Schramm–Loewner evolution|AQ75M2JTB2]].\n\n## Explanations\n\n### L1 human\n\nPicture a crack growing into the upper half-plane from a point on the real axis.\nAt each time, there is a conformal map that opens the cracked half-plane back into the whole half-plane.\nThese maps change smoothly in time, and the Loewner equation says how.\nThe speed of the map at a point depends only on the value of the map there and on where the tip of the crack is sent.\nFor a straight vertical crack of height $2\\sqrt{t}$ at $0$, the map is $z \\mapsto \\sqrt{z^2 + 4t}$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "30C35",
      "30C55"
    ],
    "name": "Loewner differential equation",
    "same_as": {
      "wikidata": "Q6666620",
      "wikipedia": "Loewner differential equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Loewner_differential_equation"
      }
    ],
    "tag": "51G4QJZ345"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/51G4QJZ345/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/51G4QJZ345.md",
  "msc": [
    "30C35",
    "30C55"
  ],
  "name": "Loewner differential equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/51G/51G4QJZ345.md",
  "status": null,
  "tag": "51G4QJZ345",
  "uri": "m2://51G4QJZ345",
  "url": "https://math2.tamnd.com/wiki/loewner-differential-equation/",
  "writing": {
    "tag": "51G4QJZ345",
    "kind": "concept",
    "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
      }
    ]
  }
}