Classification Of Book Representations Of K6, 2017 Merrimack College

#### Classification Of Book Representations Of K6, Dana Rowland

*Mathematics Faculty Publications*

A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and ...

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.

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.

The Jones Polynomial Of An Almost Alternating Link, 2017 Vassar College

#### The Jones Polynomial Of An Almost Alternating Link, Adam M. Lowrance, Dean Spyropoulos

*Faculty Research and Reports*

No abstract provided.

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 ...

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.

Characterization Of Rectifying And Sphere Curves In R^3, 2017 Andrews University

#### Characterization Of Rectifying And Sphere Curves In R^3, Yun Oh, Julie Logan

*Faculty Publications*

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extracondition on the constant curvature, Y.L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s ...

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 Journey To Fuzzy Rings, 2017 Georgia Southern University

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

*Electronic Theses and 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.

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 ...

On The Perfect Reconstruction Of The Structure Of Dynamic Networks, 2016 University of Dayton

#### On The Perfect Reconstruction Of The Structure Of Dynamic Networks, Alan Veliz-Cuba

*Annual Symposium on Biomathematics and Ecology: Education and Research*

No abstract provided.

Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, 2016 Washington University in St Louis

#### Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney

*Mathematics Faculty Publications*

We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results. We apply our results to NC domains consisting of unit balls of rectangular matrices.

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), 2016 The Graduate Center, City University of New York

#### On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

*All Dissertations, Theses, and Capstone Projects*

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight ...

K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, 2016 The University of Western Ontario

#### K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev

*Electronic Thesis and Dissertation Repository*

We give a definition of a root stack and describe its most basic properties. Then we recall the necessary background (Abhyankar’s lemma, Chevalley-Shephard-Todd theorem, Luna’s etale slice theorem) and prove that under some conditions a quotient stack is a root stack. Then we compute G-theory and K-theory of a root stack. These results are used to formulate the theorem on equivariant algebraic K-theory of schemes.

Tropical Convexity Over Max-Min Semiring, 2016 West Chester University of Pennsylvania

#### Tropical Convexity Over Max-Min Semiring, Viorel Nitica, Sergei Sergeev

*Viorel Nitica*

No abstract provided.

A Metric On Max-Min Algebra, 2016 University of Oregon

#### A Metric On Max-Min Algebra, Jonathan Eskeldson, Miriam Jaffe, Viorel Nitica

*Viorel Nitica*

No abstract provided.

On Hyperplanes And Semispaces In Max-Min Convex Geometry, 2016 West Chester University of Pennsylvania

#### On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

*Viorel Nitica*

No abstract provided.

A Metric On Max-Min Algebra, 2016 University of Oregon

#### A Metric On Max-Min Algebra, Jonathan Eskeldson, Miriam Jaffe, Viorel Nitica

*Viorel Nitica*

No abstract provided.

On Hyperplanes And Semispaces In Max-Min Convex Geometry, 2016 West Chester University of Pennsylvania

#### On Hyperplanes And Semispaces In Max-Min Convex Geometry, Viorel Nitica, Sergeĭ Sergeev

*Viorel Nitica*

No abstract provided.