### Abstract

We survey connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest Cut Problem is a striking example of the deep interplay between analysis, geometry, and probability on the one hand, and computational issues in discrete mathematics on the other. We explain how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. As an important illustrative example, we shall examine recently established connections to the the structure of the Heisenberg group, and the incompatibility of its Carnot-Carathéodory geometry with the geometry of the Lebesgue space L_{1}.

Original language | English (US) |
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Title of host publication | Proceedings of the International Congress of Mathematicians 2010, ICM 2010 |

Pages | 1549-1575 |

Number of pages | 27 |

State | Published - 2010 |

Event | International Congress of Mathematicians 2010, ICM 2010 - Hyderabad, India Duration: Aug 19 2010 → Aug 27 2010 |

### Publication series

Name | Proceedings of the International Congress of Mathematicians 2010, ICM 2010 |
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### Other

Other | International Congress of Mathematicians 2010, ICM 2010 |
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Country | India |

City | Hyderabad |

Period | 8/19/10 → 8/27/10 |

### Keywords

- Bi-Lipschitz embeddings
- Heisenberg group
- Sparsest Cut Problem

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

_{1}embeddings of the Heisenberg group and fast estimation of graph isoperimetry. In

*Proceedings of the International Congress of Mathematicians 2010, ICM 2010*(pp. 1549-1575). (Proceedings of the International Congress of Mathematicians 2010, ICM 2010).