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Cohomology of Configuration Spaces o...

O'Connor, Benjamin Garrett.

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Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane./
Author:	O'Connor, Benjamin Garrett.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
Description:	51 p.
Notes:	Source: Dissertations Abstracts International, Volume: 81-10, Section: B.
Contained By:	Dissertations Abstracts International81-10B.
Subject:	Theoretical mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=27736392
ISBN:	9781658494892

Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane.
O'Connor, Benjamin Garrett.

Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 51 p.

Source: Dissertations Abstracts International, Volume: 81-10, Section: B.

Thesis (Ph.D.)--The University of Chicago, 2020.

This item must not be sold to any third party vendors.

This work discusses the topology of configurations of noncollinear points in the projective plane.As with classical configuration spaces, we can view configurations of noncollinear points in several distinct but compatible ways by leveraging their nature as varieties defined over the integers. For such varieties, a suite of comparison theorems provide a bridge between the singular cohomology and the etale cohomology. Further, the Grothendieck--Lefschetz trace formula establishes a relationship between the etale cohomology and the number of points on the variety over a finite field. In general, this relationship can be mixed by the action of the Frobenius automorphism, but when this action is known to be sufficiently tame, the original topological question can be reduced to a combinatorial problem of counting points.This work consists primarily of three components. We first extract sufficient topological structure to establish control of the Frobenius automorphism, showing that - at least up to configurations of six points - the topology may be understood in terms of a projective linear group and complements of hyperplane arrangements. We then elaborate on the reduction via the Grothendieck--Lefschetz trace formula to a set of point counting problems. We conclude by enumerating through the solutions to these point counting problems.

ISBN: 9781658494892Subjects--Topical Terms:

3173530
Theoretical mathematics.
Subjects--Index Terms:

Algebraic geometry

Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane.
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100 1 $a O'Connor, Benjamin Garrett. $3 3550456
245 1 0 $a Cohomology of Configuration Spaces of Noncollinear Points in the Projective Plane.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2020
300 $a 51 p.
500 $a Source: Dissertations Abstracts International, Volume: 81-10, Section: B.
500 $a Advisor: Farb, Benson.
502 $a Thesis (Ph.D.)--The University of Chicago, 2020.
506 $a This item must not be sold to any third party vendors.
506 $a This item must not be added to any third party search indexes.
520 $a This work discusses the topology of configurations of noncollinear points in the projective plane.As with classical configuration spaces, we can view configurations of noncollinear points in several distinct but compatible ways by leveraging their nature as varieties defined over the integers. For such varieties, a suite of comparison theorems provide a bridge between the singular cohomology and the etale cohomology. Further, the Grothendieck--Lefschetz trace formula establishes a relationship between the etale cohomology and the number of points on the variety over a finite field. In general, this relationship can be mixed by the action of the Frobenius automorphism, but when this action is known to be sufficiently tame, the original topological question can be reduced to a combinatorial problem of counting points.This work consists primarily of three components. We first extract sufficient topological structure to establish control of the Frobenius automorphism, showing that - at least up to configurations of six points - the topology may be understood in terms of a projective linear group and complements of hyperplane arrangements. We then elaborate on the reduction via the Grothendieck--Lefschetz trace formula to a set of point counting problems. We conclude by enumerating through the solutions to these point counting problems.
590 $a School code: 0330.
650 4 $a Theoretical mathematics. $3 3173530
653 $a Algebraic geometry
653 $a Algebraic topology
653 $a Configuration space
653 $a Noncollinear
653 $a Projective plane
690 $a 0642
710 2 $a The University of Chicago. $b Mathematics. $3 2049825
773 0 $t Dissertations Abstracts International $g 81-10B.
790 $a 0330
791 $a Ph.D.
792 $a 2020
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=27736392