# Logic seminar: Abhiram Natarajan

Dates: | 6 November 2024 |
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Times: | 15:15 - 16:30 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: Polynomial and Pfaffian Partitioning of Semi-Pfaffian Sets with Applications

Abstract: The polynomial partitioning theorem of Guth and Katz, in addition to helping nearly resolve Erdos' distinct distances problem, has resulted in a wide swath of improvements not just in incidence geometry, but in areas such as harmonic analysis as well. The polynomial partitioning theorem only applies to a finite set of semi-algebraic sets, and thus is not applicable when the sets involved are more general than semi-algebraic sets. Incidence geometry questions, where the involved sets are sets definable in arbitrary o-minimal structures, has not progressed as much as the case where the sets involved are semi-algebraic, and one reason for this is the absence of a polynomial partitioning theorem that works in the definable setting.

We generalize the polynomial partitioning theorem of Guth and Katz to a finite set of semi-Pfaffian sets. Additionally, we also obtain a partitioning theorem for semi-Pfaffian sets where instead of using a polynomial to partition the space, we are able to partition the space using a Pfaffian set instead. The latter method has an important advantage.

Finally, we are able to derive at least one immediate application of our new theorem.

Joint work with Martin Lotz and Nicolai Vorobjov.