Algebraic techniques in geometry: The 10th anniversary

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Abstract

This year we are celebrating the 10th anniversary of a dramatic revolution in combinatorial geometry, fueled by the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz [14, 15], the second of which has (almost completely) solved the celebrated distinct distances problem of Paul Erdős [11], open since 1946. In this talk I will survey, as time permits, some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, and on cycle elimination for lines and triangles in three dimensions.

Original languageEnglish
Title of host publicationISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation
PublisherAssociation for Computing Machinery
Pages1-5
Number of pages5
ISBN (Electronic)9781450355506
DOIs
StatePublished - 11 Jul 2018
Event43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018 - New York, United States
Duration: 16 Jul 201819 Jul 2018

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Conference

Conference43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018
Country/TerritoryUnited States
CityNew York
Period16/07/1819/07/18

Keywords

  • Algebraic Geometry
  • Combinatorial Geometry
  • Distances
  • Incidences
  • Polynomial method

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