### 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 |

### Other

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

City | Hyderabad |

Period | 8/19/10 → 8/27/10 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

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

**L1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry.** / Naor, Assaf.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the International Congress of Mathematicians 2010, ICM 2010.*pp. 1549-1575, International Congress of Mathematicians 2010, ICM 2010, Hyderabad, India, 8/19/10.

}

TY - GEN

T1 - L1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

AU - Naor, Assaf

PY - 2010

Y1 - 2010

N2 - 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 L1.

AB - 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 L1.

KW - Bi-Lipschitz embeddings

KW - Heisenberg group

KW - Sparsest Cut Problem

UR - http://www.scopus.com/inward/record.url?scp=84869239127&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869239127&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9814324302

SN - 9789814324304

SP - 1549

EP - 1575

BT - Proceedings of the International Congress of Mathematicians 2010, ICM 2010

ER -