Automorphisms Of Graph Curves On K3 Surfaces, 2015 Georgia Southern University

#### Automorphisms Of Graph Curves On K3 Surfaces, Joshua C. Ferrerra

*Electronic Theses and Dissertations*

We examine the automorphism group of configurations of rational curves on $K3$ surfaces. We use the properties of finite automorphisms of $\PP^1$ to examine what restrictions a given elliptic fibration imposes on the possible finite order non-symplectic automorphisms of the $K3$ surface. We also examine the fixed loci of these automorphisms, and construct an explicit fibration to demonstrate the process.

Computing Intersection Multiplicity Via Triangular Decomposition, 2014 The University of Western Ontario

#### Computing Intersection Multiplicity Via Triangular Decomposition, Paul Vrbik

*Electronic Thesis and Dissertation Repository*

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by triangular sets) and then, with some generic assumptions, to l many hypersurfaces.

Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at points on curves.

Continuous Dependence Of Solutions Of Equations On Parameters, 2014 Rose-Hulman Institute of Technology

#### Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map *T : C^{n} -> C^{n}*.

Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, 2014 The University of Western Ontario

#### Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, Youlong Yan

*Electronic Thesis and Dissertation Repository*

The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One important device relevant for this study is the notion of tilting sheaf.

This thesis is concerned with the existence of tilting sheaves on some smooth projective varieties. The main technique we use in this thesis is Galois descent theory. We first construct tilting bundles on general Brauer-Severi varieties. Our main result shows the existence of tilting bundles on some Brauer-Severi schemes. As an application, we prove that there are tilting bundles on an arithmetic toric variety whose toric variety ...

Light Pollution Research Through Citizen Science, 2014 CSU Sacramento

#### Light Pollution Research Through Citizen Science, John Kanemoto

*STAR (STEM Teacher and Researcher) Presentations*

Light pollution (LP) can disrupt and/or degrade the health of all living things, as well as, their environments. The goal of my research at the NOAO was to check the accuracy of the citizen science LP reporting systems entitled: Globe at Night (GaN), Dark Sky Meter (DSM), and Loss of the Night (LoN). On the GaN webpage, the darkness of the night sky (DotNS) is reported by selecting a magnitude chart. Each magnitude chart has a different density/number of stars around a specific constellation. The greater number of stars implies a darker night sky. Within the DSM iPhone ...

The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, 2014 University of Nebraska-Lincoln

#### The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, Nora Youngs

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

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode ...

A Kleinian Approach To Fundamental Regions, 2014 Joshua L Hidalgo

#### A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo

*Electronic Theses, Projects, and Dissertations*

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius ...

The Dual Gromov-Hausdorff Propinquity, 2014 University of Denver

#### The Dual Gromov-Hausdorff Propinquity, Frédéric Latrémolière

*Mathematics Preprint Series*

Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*- algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity, which resolves several important issues raised by recent research in noncommutative metric geometry: our new metric makes *-isomorphism a necessary condition for distance zero, is well-adapted to Leibniz seminorms, and — very importantly — is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a new tool for noncommutative metric geometry which offers a solution ...

A Metric On Max-Min Algebra, 2014 University of Oregon

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

*Mathematics*

No abstract provided.

Tropical Convexity Over Max-Min Semiring, 2014 West Chester University of Pennsylvania

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

*Mathematics*

No abstract provided.

Early Investigations In Conformal And Differential Geometry, 2014 University of Arkansas, Fayetteville

#### Early Investigations In Conformal And Differential Geometry, Raymond T. Walter

*Inquiry: The University of Arkansas Undergraduate Research Journal*

The present article introduces fundamental notions of conformal and differential geometry, especially where such notions are useful in mathematical physics applications. Its primary achievement is a nontraditional proof of the classic result of Liouville that the only conformal transformations in Euclidean space of dimension greater than two are Möbius transformations. The proof is nontraditional in the sense that it uses the standard Dirac operator on Euclidean space and is based on a representation of Möbius transformations using 2x2 matrices over a Clifford algebra. Clifford algebras and the Dirac operator are important in other applications of pure mathematics and mathematical physics ...

Function Of Several Variable, 2014 Parkland College

#### Function Of Several Variable, Weiting Li

*A with Honors Projects*

This Parkland A with Honors project discusses the function of several variables, it's limits and partial derivatives.

Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, 2014 Harvey Mudd College

#### Arithmetical Graphs, Riemann-Roch Structure For Lattices, And The Frobenius Number Problem, Jeremy Usatine

*HMC Senior Theses*

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss ...

Geometric Study Of The Category Of Matrix Factorizations, 2013 University of Nebraska-Lincoln

#### Geometric Study Of The Category Of Matrix Factorizations, Xuan Yu

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

We study the geometry of matrix factorizations in this dissertation.

It contains two parts. The first one is a Chern-Weil style

construction for the Chern character of matrix factorizations; this

allows us to reproduce the Chern character in an explicit,

understandable way. Some basic properties of the Chern character are

also proved (via this construction) such as functoriality and that

it determines a ring homomorphism from the Grothendieck group of

matrix factorizations to its Hochschild homology. The second part is

a reconstruction theorem of hypersurface singularities. This is

given by applying a slightly modified version of Balmer's tensor

triangular ...

Results On Containments And Resurgences, With A Focus On Ideals Of Points In The Plane, 2013 University of Nebraska-Lincoln

#### Results On Containments And Resurgences, With A Focus On Ideals Of Points In The Plane, Annika Denkert

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

Let *K* be an algebraically closed field and *I* ⊆ *R*=*K*[**P**^{N}] a nontrivial homogeneous ideal. We can describe ordinary powers *I*^{r} and symbolic powers *I*^{(m)} of *I*. One question that has been of interest over the past couple of years is that of when we have containment of *I*^{(m)} in *I*^{r}. Bocci and Harbourne defined the resurgence of *I* as rho(*I*)=sup_{m,r}{m/r | *I*^{(m)} is not contained in *I*^{r}}. Hence in particular *I*^{(m)} ⊆ *I*^{r} whenever m/r is at least rho(*I*). Results by Macaulay, Ein-Lazarsfeld-Smith ...

On Toric Symmetry Of P1 X P2, 2013 Harvey Mudd College

#### On Toric Symmetry Of P1 X P2, Olivia D. Beckwith

*HMC Senior Theses*

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 ...

Chip Firing Games And Riemann-Roch Properties For Directed Graphs, 2013 Harvey Mudd College

#### Chip Firing Games And Riemann-Roch Properties For Directed Graphs, Joshua Z. Gaslowitz

*HMC Senior Theses*

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

Symbolic Powers Of Ideals In K[PN], 2013 University of Nebraska-Lincoln

#### Symbolic Powers Of Ideals In K[PN], Michael Janssen

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

Let *I* ⊆ *k*[**P**^{N}] be a homogeneous ideal and *k* an algebraically closed field. Of particular interest over the last several years are ideal containments of symbolic powers of *I* in ordinary powers of *I* of the form *I*^{(m)} ⊆ *I*^{r}, and which ratios *m/r* guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if *I* ⊆ *k*[**P**^{N}], where *k* is an algebraically closed field, then the symbolic power *I*^{(Ne)} is contained in the ordinary power *I*^{e}, and thus, whenever *m/r* ≥ *N* we have the containment *I*^{(m)} ⊆ *I*^{r}. Therefore ...

Propeller, 2013 Claremont Colleges

#### Propeller, Joel Kahn

*The STEAM Journal*

This image is based on several different algorithms interconnected within a single program in the language BASIC-256. The fundamental structure involves a tightly wound spiral working outwards from the center of the image. As the spiral is drawn, different values of red, green and blue are modified through separate but related processes, producing the changing appearance. Algebra, trigonometry, geometry, and analytic geometry are all utilized in overlapping ways within the program. As with many works of algorithmic art, small changes in the program can produce dramatic alterations of the visual output, which makes lots of variations possible.

Galois Representations From Non-Torsion Points On Elliptic Curves, 2013 Bard College

#### Galois Representations From Non-Torsion Points On Elliptic Curves, Matthew Phillip Hughes

*Senior Projects Spring 2013*

Working from well-known results regarding *l*-adic Galois representations attached to elliptic curves arising from successive preimages of the identity, we consider a natural deformation. Given a non-zero point P on a curve, we investigate the Galois action on the splitting fields of preimages of P under multiplication-by-*l* maps. We give a group-theoretic structure theorem for the corresponding Galois group, and state a conjecture regarding composita of two such splitting fields.