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Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces./
Author:	Huang, Yifeng.
Description:	1 online resource (142 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
Contained By:	Dissertations Abstracts International84-01B.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29274978click for full text (PQDT)
ISBN:	9798438780922

Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces.
Huang, Yifeng.

Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces. - 1 online resource (142 pages)

Source: Dissertations Abstracts International, Volume: 84-01, Section: B.

Thesis (Ph.D.)--University of Michigan, 2022.

Includes bibliographical references

We investigate three topics that are motivated by the study of polynomial equations in noncommutative rings. These topics have distinct flavors, ranging from number theory, combinatorics to topology.As the first topic, we study a noncommutative analogue of a classical theorem in number theory that the unit equation x+y=1, where both x and y belong to a given finitely generated subgroup of the multiplicative group of nonzero complex numbers, has only finitely many solutions. We show that if x and y are nonzero quaternions expressable as certain products, then the unit equation x+y=1 on the (noncommutative) quaternion algebra only has finitely many solutions. We also give a natural application to the study of iterations of self-maps on abelian varieties whose endormorphism rings lie inside the quaternion algebra.As the second topic, we count the numbers of solutions of several equations on the ring of n by n matrices over a finite field. We investigate the combinatorial behaviors of these counts by giving generating functions. Each of these counts can be viewed as the point count (over a finite field) of a space that parametrizes finite-dimensional modules over a certain algebra that arises from algebraic geometry. The connection between the count and the underlying geometry is also discussed. In Chapter III, we count pairs of mutually annihilating matrices AB=BA=0 over a finite field; the underlying geometry is a nodal singularity on an algebraic curve. In Chapter IV, we count pairs of matrices satisfying AB=ζBA, where ζ is a root of unity in a finite field; the underlying geometry is the quantum plane.As the third topic, we focus on the configuration space, which is the space that parametrizes unordered tuples of distinct points on a base space. We give several results that state that certain combinatorial behaviors of some geometric invariants (namely, Betti numbers and mixed Hodge numbers) of configuration spaces are analogous to the well-known behavior of the point counts of configuration spaces over finite fields. In Chapter V, we give a rational generating function, which is essentially a zeta function, that encodes Betti and mixed Hodge numbers of configuration spaces of a punctured elliptic curve over ℂ. In Chapter VI, we describe the effect of puncturing a point from the base space on the Betti and mixed Hodge numbers of the configuration spaces, under a certain assumption.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798438780922Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Unit equationIndex Terms--Genre/Form:

542853
Electronic books.

Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces.
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245 1 0 $a Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces.
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300 $a 1 online resource (142 pages)
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500 $a Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
500 $a Advisor: Zieve, Michael.
502 $a Thesis (Ph.D.)--University of Michigan, 2022.
504 $a Includes bibliographical references
520 $a We investigate three topics that are motivated by the study of polynomial equations in noncommutative rings. These topics have distinct flavors, ranging from number theory, combinatorics to topology.As the first topic, we study a noncommutative analogue of a classical theorem in number theory that the unit equation x+y=1, where both x and y belong to a given finitely generated subgroup of the multiplicative group of nonzero complex numbers, has only finitely many solutions. We show that if x and y are nonzero quaternions expressable as certain products, then the unit equation x+y=1 on the (noncommutative) quaternion algebra only has finitely many solutions. We also give a natural application to the study of iterations of self-maps on abelian varieties whose endormorphism rings lie inside the quaternion algebra.As the second topic, we count the numbers of solutions of several equations on the ring of n by n matrices over a finite field. We investigate the combinatorial behaviors of these counts by giving generating functions. Each of these counts can be viewed as the point count (over a finite field) of a space that parametrizes finite-dimensional modules over a certain algebra that arises from algebraic geometry. The connection between the count and the underlying geometry is also discussed. In Chapter III, we count pairs of mutually annihilating matrices AB=BA=0 over a finite field; the underlying geometry is a nodal singularity on an algebraic curve. In Chapter IV, we count pairs of matrices satisfying AB=ζBA, where ζ is a root of unity in a finite field; the underlying geometry is the quantum plane.As the third topic, we focus on the configuration space, which is the space that parametrizes unordered tuples of distinct points on a base space. We give several results that state that certain combinatorial behaviors of some geometric invariants (namely, Betti numbers and mixed Hodge numbers) of configuration spaces are analogous to the well-known behavior of the point counts of configuration spaces over finite fields. In Chapter V, we give a rational generating function, which is essentially a zeta function, that encodes Betti and mixed Hodge numbers of configuration spaces of a punctured elliptic curve over ℂ. In Chapter VI, we describe the effect of puncturing a point from the base space on the Betti and mixed Hodge numbers of the configuration spaces, under a certain assumption.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2023
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Unit equation
653 $a Configuration spaces
653 $a Matrix enumeration over finite fields
653 $a Noncommutative algebra
653 $a Moduli spaces
653 $a Mixed Hodge theory
655 7 $a Electronic books. $2 lcsh $3 542853
690 $a 0405
690 $a 0642
710 2 $a ProQuest Information and Learning Co. $3 783688
710 2 $a University of Michigan. $b Mathematics. $3 3185260
773 0 $t Dissertations Abstracts International $g 84-01B.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29274978 $z click for full text (PQDT)