Fermat's Last Theorem, 2013 University of Redlands

#### Fermat's Last Theorem, Rozalyn N. Miner

*Undergraduate Honors Theses*

Throughout history, math has been transformed, theorems have been proven, and great people have become known through their discoveries. One of these great people was Pierre Fermat, who studied law at the University of Orleans. Through his work as a government official and the office he held there, he was able to change his name to Pierre de Fermat. There is controversy as to when Fermat was actually born. Most historians say that he was born in 1601, but through research Klaus Barner, a professor at University of Kassel, Germany, found that Fermat was most likely born in 1607. This ...

Geometry Curriculum For High School Students During A Summer School Program, 2012 The College at Brockport

#### Geometry Curriculum For High School Students During A Summer School Program, Kyle E. Kucsmas

*Education and Human Development Master's Theses*

This geometry curriculum project was designed to be used during a high school summer school credit recovery program. The National Council of Teachers of Mathematics (NCTM), New York State (NYS) Learning Standards for Mathematics, Science, and Technology (MST), and the most recent addition of the Common Core Learning Standards (CCSL) harbor the foundations for each lesson. The curriculum presented will provide teachers with a condensed standards based instrument that can be utilized during a twenty-two day summer school geometry program which contains mathematical content, appropriate rigor, and opportunities for student growth. The curriculum is engaging, obtainable, cyclical, and diverse enough ...

On Hilbert Modular Threefolds Of Discriminant 49, 2012 University of Massachusetts - Amherst

#### On Hilbert Modular Threefolds Of Discriminant 49, Lev Borisov, Paul Gunnells

*Paul Gunnells*

Let K be the totally real cubic field of discriminant 49 , let \fancyscriptO be its ring of integers, and let p⊂\fancyscriptO be the prime over 7 . Let Γ(p)⊂Γ=SL2(\fancyscriptO) be the principal congruence subgroup of level p . This paper investigates the geometry of the Hilbert modular threefold attached to Γ(p) and some related varieties. In particular, we discover an octic in P3 with 84 isolated singular points of type A2 .

The Secant Conjecture In The Real Schubert Calculus, 2012 Sam Houston State University

#### The Secant Conjecture In The Real Schubert Calculus, Luis D. García-Puente, Nickolas Hein, Christopher Hillar, Abraham Martín Del Campo, James Ruffo, Frank Sottile, Zach Teitler

*Zach Teitler*

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

The Secant Conjecture In The Real Schubert Calculus, 2012 Sam Houston State University

#### The Secant Conjecture In The Real Schubert Calculus, Luis D. García-Puente, Nickolas Hein, Christopher Hillar, Abraham Martín Del Campo, James Ruffo, Frank Sottile, Zach Teitler

*Mathematics Faculty Publications and Presentations*

We formulate the Secant Conjecture, which is a generalization of the Shapiro Conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

Weierstrass Points On Families Of Graphs, 2012 University of Connecticut

#### Weierstrass Points On Families Of Graphs, William D. Lindsay Jr.

*Master's Theses*

No abstract provided.

Communal Partitions Of Integers, 2012 Gettysburg College

#### Communal Partitions Of Integers, Darren B. Glass

*Math Faculty Publications*

There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analagous question counting the number of k-tuples of nonnegative integers none of which is more than 1/(k−1) of the sum of all the integers. We give an explicit function for the generating function which counts these k-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question.

The Zeta Function Of Generalized Markoff Equations Over Finite Fields, 2012 University of Nevada, Las Vegas

#### The Zeta Function Of Generalized Markoff Equations Over Finite Fields, Juan Mariscal

*UNLV Theses, Dissertations, Professional Papers, and Capstones*

The purpose of this paper is to derive the Hasse-Weil zeta function of a special class of Algebraic varieties based on a generalization of the Markoff equation. We count the number of solutions to generalized Markoff equations over finite fields first by using the group structure of the set of automorphisms that generate solutions and in other cases by applying a slicing method from the two-dimensional cases. This enables us to derive a generating function for the number of solutions over the degree k extensions of a fixed finite field giving us the local zeta function. We then create an ...

An Investigation Into Three Dimensional Probabilistic Polyforms, 2012 Olivet Nazarene University

#### An Investigation Into Three Dimensional Probabilistic Polyforms, Danielle Marie Vander Schaaf

*Honors Program Projects*

Polyforms are created by taking squares, equilateral triangles, and regular hexagons and placing them side by side to generate larger shapes. This project addressed three-dimensional polyforms and focused on cubes. I investigated the probabilities of certain shape outcomes to discover what these probabilities could tell us about the polyforms’ characteristics and vice versa. From my findings, I was able to derive a formula for the probability of two different polyform patterns which add to a third formula found prior to my research. In addition, I found the probability that 8 cubes randomly attached together one by one would form a ...

Elliptic Curves Of High Rank, 2012 Macalester College

#### Elliptic Curves Of High Rank, Cecylia Bocovich

*Mathematics, Statistics, and Computer Science Honors Projects*

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y^{2} = x^{3} + ax + b, where a and b are coefficients that satisfy the property 4a^{3} + 27b^{2} = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic to Z^{r} ⊕ E ...

A Multiplicative "Conic", 2012 Stephen F Austin State University

#### A Multiplicative "Conic", Tiffany Lundy

*Undergraduate Research Conference*

Early in their mathematical career, students learn about how ellipses and hyperbolas are formed, their properties, and their applications. In particular, an ellipse is the set of all points in the plane whose combined distance from two fixed locations (foci) is constant. A hyperbola is formed in much the same way, except instead of combining (adding) the distance from the foci, the difference is used. This research begins by examining a new locus of points. The locus of points if formed by taking all points whose distance from two fixed locations when multiplied in constant. The difference discernible types of ...

Non-Genera Of Curves With Automorphisms In Characteristic P, 2012 Gettysburg College

#### Non-Genera Of Curves With Automorphisms In Characteristic P, Darren B. Glass

*Math Faculty Publications*

We consider which integers *g *and *r* can occur respectively as the genus and *p*-rank of a curve defined over a field of odd characteristics *p* which admits an automorphism of degree *p. *

Road Trips In Geodesic Metric Spaces And Groups With Quadratic Isoperimetric Inequalities, 2011 Eastern Kentucky University

#### Road Trips In Geodesic Metric Spaces And Groups With Quadratic Isoperimetric Inequalities, Rachel Bishop-Ross, Jon Corson

*Rachel E. Bishop-Ross*

We introduce a property of geodesic metric spaces, called the road trip property, that generalizes hyperbolic and convex metric spaces. This property is shown to be invariant under quasi-isometry. Thus, it leads to a geometric property of finitely generated groups, also called the road trip property. The main result is that groups with the road trip property are finitely presented and satisfy a quadratic isoperimetric inequality. Examples of groups with the road trip property include hyperbolic, semihyperbolic, automatic and CAT(0) groups. DOI: 10.1142/S0218196712500506

Mixed Discriminants, 2011 University of Massachusetts Amherst

#### Mixed Discriminants, Eduardo Cattani, Maria Angelica Cueto, Alicia Dickenstein, Sandra Di Rocco, Bernd Strumfels

*Eduardo Cattani*

No abstract provided.

Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, 2011 Universitat Duisburg-Essen, Germany

#### Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann

*Mathematics Faculty Publications*

Warfield modules are direct summands of simply presented **Z**_{p }- modules, or, alternatively, are **Z**_{p }- modules possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by theor Ilm-Kaplansky and Warfield invariants. Taking a model theoretic point of view and using infinitary languages we give here a complete theoretic characterization of a large class of** Z**_{p }- modules having a nice decomposition basis. As a corollary, we obtain the classical classification of countable Warfield modules. This generalizes results by Barwise and Eklof.

Microrna Expression Data Reveals A Signature Of Kidney Damage Following Ischemia Reperfusion Injury, 2011 Tufts University

#### Microrna Expression Data Reveals A Signature Of Kidney Damage Following Ischemia Reperfusion Injury, Michael D. Shapiro, Jessamyn Bagley, Jeff Latz, Jonathan G. Godwin, Xupeng Ge, Stefan G. Tullius, John Iacomini

*GSBS Student Publications*

Ischemia reperfusion injury (IRI) is a leading cause of acute kidney injury, a common problem worldwide associated with significant morbidity and mortality. We have recently examined the role of microRNAs (miRs) in renal IRI using expression profiling. Here we conducted mathematical analyses to determine if differential expression of miRs can be used to define a biomarker of renal IRI. Principal component analysis (PCA) was combined with spherical geometry to determine whether samples that underwent renal injury as a result of IRI can be distinguished from controls based on alterations in miR expression using our data set consisting of time series ...

Counting Reducible Composites Of Polynomials, 2011 University of Tennessee, Knoxville

#### Counting Reducible Composites Of Polynomials, Jacob Andrew Ogle

*Doctoral Dissertations*

This research answers some open questions about the number of reducible translates of a fixed non-constant polynomial over a field. The natural hypothesis to consider is that the base field is algebraically closed in the function field. Since two possible choices for the base field arise, this naturally yields two different hypotheses. In this work, we explicitly relate the two hypotheses arising from this choice. Using the theory of derivations, and specifically an explicit construction of a derivation with a well-understood ring of constants, we can relate the ranks of the two relative-unit-groups involved, both of which are free Abelian ...

Moduli Problems In Derived Noncommutative Geometry, 2011 University of Pennsylvania

#### Moduli Problems In Derived Noncommutative Geometry, Pranav Pandit

*Publicly Accessible Penn Dissertations*

We study moduli spaces of boundary conditions in 2D topological field theories. To a compactly generated linear infinity-category X, we associate a moduli functor M_X parametrizing compact objects in X. The Barr-Beck-Lurie monadicity theorem allows us to establish the descent properties of M_X, and show that M_X is a derived stack. The Artin-Lurie representability criterion makes manifest the relation between finiteness conditions on X, and the geometricity of M_X. If X is fully dualizable (smooth and proper), then M_X is geometric, recovering a result of Toën-Vaquie from a new perspective. Properness of X does not imply geometricity in general: perfect ...

Homological Projective Duality For Gr(3,6), 2011 University of Pennsylvania

#### Homological Projective Duality For Gr(3,6), Dragos Deliu

*Publicly Accessible Penn Dissertations*

Homological Projective Duality is a homological extension of the classical no-

tion of projective duality. Constructing the homological projective dual of a variety

allows one to describe semiorthogonal decompositions on the bounded derived cat-

egory of coherent sheaves for all the complete linear sections of the initial variety.

This gives a powerful method to construct decompositions for a big class of varieties,

however examples for which this duality is understood are very few.

In this thesis we investigate the case of Gr(3, 6) with respect to the Plucker

embedding.

The Derived Category And The Singularity Category, 2011 University of Pennsylvania

#### The Derived Category And The Singularity Category, Mehmet U. Isik

*Publicly Accessible Penn Dissertations*

We prove an equivalence between the derived category of a variety and the equivari- ant/graded singularity category of a corresponding singular variety. The equivalence also holds at the dg level.