Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, 2017 University of Nebraska-Lincoln

#### Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh

*Dissertations, Theses, and Student Research Papers in Mathematics*

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[**P**^{n}], I^{(mn)} ⊆ I^{m} for all m ∈ N. Over the projective plane, we obtain I^{(4)}< ⊆ I^{2}. Huneke asked whether it was the case that I^{(3)} ⊆ I^{2}. Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then ...

Descartes Comes Out Of The Closet, 2017 Vassar College

#### Descartes Comes Out Of The Closet, Nora E. Culik

*Journal of Humanistic Mathematics*

While “Descartes Comes Out of the Closet” is ostensibly about a young woman’s journey to Paris, the descriptive detail borrows language and images from Cartesian coordinate geometry, dualistic philosophy, neuroanatomy (the pineal), and projections of three dimensions onto planes. This mathematical universe is counterpointed in the natural language of the suppressed love story that locates the real in the human. Thus, at the heart of the story is the tension between competing notions of mathematics, i.e., as either an independent realm apart from history or as a culturally produced and historical set of practices. Of course, the central ...

College Algebra, Trigonometry, And Precalculus (Clayton), 2017 Clayton State University

#### College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

*Mathematics Grants Collections*

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

- Linked Syllabus
- Initial Proposal
- Final Report

Counting Rational Points, Integral Points, Fields, And Hypersurfaces, 2017 The Graduate Center, City University of New York

#### Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther

*All Dissertations, Theses, and Capstone Projects*

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5 ...

Klein Four Actions On Graphs And Sets, 2017 Gettysburg College

#### Klein Four Actions On Graphs And Sets, Darren B. Glass

*Math Faculty Publications*

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (ℤ/2ℤ)2-action to the properties of its quotients generalizes to results about sets and graphs that admit (ℤ/2ℤ)2-actions.

Cox Processes For Visual Object Counting, 2017 Portland State University

#### Cox Processes For Visual Object Counting, Yongming Ma

*Student Research Symposium*

We present a model that utilizes Cox processes and CNN classifiers in order to count the number of instances of an object in an image. Poisson processes are well suited to events that occur randomly in space, like the location of objects in an image, as well as to the task of counting. Mixed Poisson processes also offer increased flexibility, however they do not easily scale with image size: they typically require O(n3) computation time and O(n2) storage, where n is the number of pixels. To mitigate this problem, we employ Kronecker algebra which takes advantage of the ...

Integrating Non-Euclidean Geometry Into High School, 2017 Loyola Marymount University

#### Integrating Non-Euclidean Geometry Into High School, John Buda

*Honors Thesis*

The purpose of this project is to provide the framework for integrating the study of non-Euclidean geometry into a high school math class in such a way that both aligns with the Common Core State Standards and makes use of research-based practices to enhance the learning of traditional geometry. Traditionally, Euclidean geometry has been the only strand of geometry taught in high schools, even though mathematicians have developed several other strands. The non-Euclidean geometry that I focus on in this project is what is known as taxicab geometry. With the Common Core Standards for Math Practice pushing students to “model ...

Student-Created Test Sheets, 2017 Bowling Green State University

#### Student-Created Test Sheets, Samuel Laderach

*Honors Projects*

Assessment plays a necessary role in the high school mathematics classroom, and testing is a major part of assessment. Students often struggle with mathematics tests and examinations due to math and test anxiety, a lack of student learning, and insufficient and inefficient student preparation. Practice tests, teacher-created review sheets, and student-created test sheets are ways in which teachers can help increase student performance, while ridding these detrimental factors. Student-created test sheets appear to be the most efficient strategy, and this research study examines the effects of their use in a high school mathematics classroom.

On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, 2017 The University of Western Ontario

#### On Vector-Valued Automorphic Forms On Bounded Symmetric Domains, Nadia Alluhaibi

*Electronic Thesis and Dissertation Repository*

The objective of the study is to investigate the behaviour of the inner products of vector-valued Poincare series, for large weight, associated to submanifolds of a quotient of the complex unit ball and how vector-valued automorphic forms could be constructed via Poincare series. In addition, it provides a proof of that vector-valued Poincare series on an irreducible bounded symmetric domain span the space of vector-valued automorphic forms.

Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, 2017 University High School

#### Polygons, Pillars And Pavilions: Discovering Connections Between Geometry And Architecture, Sean Patrick Madden

*Journal of Catholic Education*

Crowning the second semester of geometry, taught within a Catholic middle school, the author's students explored connections between the geometry of regular polygons and architecture of local buildings. They went on to explore how these principles apply famous buildings around the world such as the monuments of Washington, D.C. and the elliptical piazza of Saint Peter's Basilica at Vatican City within Rome, Italy.

Session A-3: Three-Act Math Tasks, 2017 Illinois Mathematics and Science Academy

#### Session A-3: Three-Act Math Tasks, Lindsey Herlehy

*Professional Learning Day*

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, 2017 Loyola Marymount University

#### Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

*Blake Mellor*

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

Epimorphisms And Boundary Slopes Of 2–Bridge Knots, 2017 Pitzer College

#### Epimorphisms And Boundary Slopes Of 2–Bridge Knots, Jim Hoste, Patrick D. Shanahan

*Patrick Shanahan*

In this article we study a partial ordering on knots in S^{3} where K_{1}≥K_{2} if there is an epimorphism from the knot group of K_{1} onto the knot group of K_{2} which preserves peripheral structure. If K_{1} is a 2–bridge knot and K_{1}≥K_{2}, then it is known that K_{2} must also be 2–bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2–bridge knot K_{p∕q}, produces infinitely many 2–bridge knots K_{p′/q′} with K_{p′∕q′}≥K_{p∕q ...}

Linked Exact Triples Of Triangulated Categories And A Calculus Of T-Structures, 2017 Loyola Marymount University

#### Linked Exact Triples Of Triangulated Categories And A Calculus Of T-Structures, Michael Berg

*Michael Berg*

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our con. structs to· a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason.

A Journey To Fuzzy Rings, 2017 Georgia Southern University

#### A Journey To Fuzzy Rings, Brett T. Ernst

*Electronic Theses & Dissertations*

Enumerative geometry is a very old branch of algebraic geometry. In this thesis, we will describe several classical problems in enumerative geometry and their solutions in order to motivate the introduction of tropical geometry. Finally, fuzzy rings, a powerful algebraic framework for tropical and algebraic geometry is introduced.

Random Tropical Curves, 2017 Harvey Mudd College

#### Random Tropical Curves, Magda L. Hlavacek

*HMC Senior Theses*

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves ...

Tropical Derivation Of Cohomology Ring Of Heavy/Light Hassett Spaces, 2017 Harvey Mudd College

#### Tropical Derivation Of Cohomology Ring Of Heavy/Light Hassett Spaces, Shiyue Li

*HMC Senior Theses*

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for ...

Chow's Theorem, 2017 Colby College

#### Chow's Theorem, Yohannes D. Asega

*Honors Theses*

We present the proof of Chow's theorem as a corollary to J.P.-Serre's GAGA correspondence theorem after introducing the necessary prerequisites. Finally, we discuss consequences of Chow's theorem.

A Categorical Formulation Of Algebraic Geometry, 2017 University of Massachusetts Amherst

#### A Categorical Formulation Of Algebraic Geometry, Bradley Willocks

*Doctoral Dissertations*

We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a ``spec datum" is introduced, as a certain relation between categories, of which one has been given a Grothendieck topology. A ``geometry" is interpreted as a sub-category of $\Omega$, and a formalism is given by which such a subcategory is to be associated to a spec datum, reflecting the standard construction of the category of schemes from the category of rings by affine charts.

Computation Of Real Radical Ideals By Semidefinite Programming And Iterative Methods, 2016 The University of Western Ontario

#### Computation Of Real Radical Ideals By Semidefinite Programming And Iterative Methods, Fei Wang

*Electronic Thesis and Dissertation Repository*

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but ...