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Incidence problems in discrete geometry

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Title
Incidence problems in discrete geometry
Name (type = personal)
NamePart (type = family)
Wolf
NamePart (type = given)
Charles
NamePart (type = date)
1989-
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Charles Wolf
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Saraf
NamePart (type = given)
Shubhangi
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Shubhangi Saraf
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Advisory Committee
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chair
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Kopparty
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Swastik
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Swastik Kopparty
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Advisory Committee
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internal member
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Steiger
NamePart (type = given)
William
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William Steiger
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Advisory Committee
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internal member
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NamePart (type = family)
Yehudayoff
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Amir
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Amir Yehudayoff
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Advisory Committee
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outside member
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Rutgers University
Role
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degree grantor
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Graduate School - New Brunswick
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school
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Text
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theses
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2017
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2017-05
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2017
Place
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xx
Language
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eng
Abstract (type = abstract)
Over the past decade, discrete geometry research has flourished with clever uses of algebraic methods. The polynomial method has had a deep impact on a wide collection of results in combinatorics, such as tight asymptotic lower bounds on finite field Kakeya and Nikodym sets, near optimal lower bound for Erdos' distinct distances problem, and improved bounds for cap sets. Spectral methods and rank bounds for matrices have shed new light on improved bounds for point-line incidences, subspace intersections and graph rigidity. This thesis is focused on developing new ways to improve on these techniques, and applying them in a few discrete geometry settings: 1. Techniques such as matrix scaling and rank bounds for design matrices have recently found beautiful applications for understanding configurations of points and lines over the complex numbers. In particular, they give a different proof of Kelly's Theorem, which says that any configuration of points in complex space must either be contained in a plane or have a line passing through exactly 2 of those points, called an ordinary line. We expand on these techniques to prove the first quantitative bounds for the number of ordinary lines in a non planar configuration of points in complex space. 2. In 2008, showed in a breakthrough result that a Kakeya set over finite fields has an asymptotically tight lower bound using the polynomial method. Later in 2008, Saraf and Sudan improved on the polynomial method by interpolating a polynomial that vanishes with high multiplicity on points of the Kakeya set. We further enhance the polynomial method by introducing the notion of ``fractional multiplicity," and use this improvement to obtain a better lower bound for finite field Kakeya sets in 3 dimensions. 3. While studying these 3-dimensional finite field Kakeya sets, we considered the related 3-dimensional finite field Nikodym sets. Previously, the lower bound for a 3-dimensional finite field Nikdoym sets was also obtained using the polynomial method, and had the same lower bound as for a Kakeya set. We achieve a better lower bound for 3-dimensional finite field Nikodym sets, thus separating it from the Kakeya set lower bound.
Subject (authority = RUETD)
Topic
Mathematics
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Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_7937
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (vii, 60 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Subject (authority = ETD-LCSH)
Topic
Discrete geometry
Note (type = statement of responsibility)
by Charles Wolf
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3W098V3
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Wolf
GivenName
Charles
Role
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Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2017-04-04 17:35:38
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Name
Charles Wolf
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Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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2017-04-04T17:33:24
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