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.

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Two Riemann surfaces *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GThe Implicit Function Theorem And Free Algebraic Sets, 2016 Washington University in St Louis

#### The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

*Mathematics Faculty Publications*

We prove an implicit function theorem for non-commutative functions. We use this to show that if *p* ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of *p* ( X;Y ) = 0 will commute with X

Aspects Of Non-Commutative Function Theory, 2016 Washington University in St Louis

#### Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy

*Mathematics Faculty Publications*

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.

The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, 2016 Thua Thien Hue College of Education

#### The Log-Exponential Smoothing Technique And Nesterov’S Accelerated Gradient Method For Generalized Sylvester Problems, N. T. An, Daniel J. Giles, Nguyen Mau Nam, R. Blake Rector

*Mathematics and Statistics Faculty Publications and Presentations*

The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.

Topology Of The Affine Springer Fiber In Type A, 2016 University of Massachusetts - Amherst

#### Topology Of The Affine Springer Fiber In Type A, Tobias Wilson

*Doctoral Dissertations May 2014 - current*

We develop algorithms for describing elements of the affine Springer fiber in type

A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular,

it is known that the affine Springer fiber, X, has a paving by affines resulting from

the intersection of Schubert cells with X. Our description of the elements of Xallow

us to understand these affine spaces and write down explicit dimension formulae. We

also explore some closure relations between the affine spaces and begin to describe the

moment map for the both the regular and extended torus action.

Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, 2016 University of Massachusetts Amherst

#### Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin

*Doctoral Dissertations May 2014 - current*

Consider any rational Hodge isometry

$\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$

surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$

is a polynomial in Chern classes of coherent analytic sheaves

over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic

whenever $S_1$ and $S_2$ are algebraic.

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, 2015 Rose-Hulman Institute of Technology

#### Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of *PSL _{2}(q)*. A quasi-platonic action of a group

*G*on a closed Riemann

*S*surface is a conformal action for which

*S/G*is a sphere and

*S->S/G*is branched over

*{0, 1,infinity}*. The unit interval in

*S/G*may be lifted to a dessin d'enfant

*D*, an embedded bipartite graph in

*S.*The dessin forms the edges and vertices of a tiling on

*S*by dihedrally symmetric polygons, generalizing the ...

Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, 2015 Rose-Hulman Institute of Technology

#### Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean Broughton

*S. Allen Broughton*

This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of *PSL _{2}(q)*. A quasi-platonic action of a group

*G*on a closed Riemann

*S*surface is a conformal action for which

*S/G*is a sphere and

*S->S/G*is branched over

*{0, 1,infinity}*. The unit interval in

*S/G*may be lifted to a dessin d'enfant

*D*, an embedded bipartite graph in

*S.*The dessin forms the edges and vertices of a tiling on

*S*by dihedrally symmetric polygons, generalizing the ...

Integrability And Regularity Of Rational Functions, 2015 Washington University in St. Louis

#### Integrability And Regularity Of Rational Functions, Greg Knese

*Mathematics Faculty Publications*

Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting ...

Quasi-Platonic Psl(2,Q)-Actions On Closed Riemann Surfaces, 2015 Rose-Hulman Institute of Technology

#### Quasi-Platonic Psl(2,Q)-Actions On Closed Riemann Surfaces, Sean Broughton

*S. Allen Broughton*

*PSL(2,q)*. A quasi-platonic action of a group

*G*on a closed Riemann

*S*surface is a conformal action for which

*S/G*is a sphere and

*S -> S/G*is

*S/G*may be lifted to a dessin d'enfant

*D*, an embedded bipartite graph in

*S*.The dessin forms the edges and vertices of a tiling on

*S*by dihedrally symmetric polygons, generalizing the ...

Analytic Geometry And Calculus I, 2015 Valdosta State University

#### Analytic Geometry And Calculus I, Shaun V. Ault, Sudhir Goel

*Mathematics Open Textbooks*

This Grants Collection Open Textbook for Analytic Geometry and Calculus I was created under a Round Two 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

A Study Of Green’S Relations On Algebraic Semigroups, 2015 The University of Western Ontario

#### A Study Of Green’S Relations On Algebraic Semigroups, Allen O'Hara

*Electronic Thesis and Dissertation Repository*

The purpose of this work is to enhance the understanding regular algebraic semigroups by considering the structural influence of Green's relations. There will be three chief topics of discussion.

- Green's relations and the Adherence order on reductive monoids
- Renner’s conjecture on regular irreducible semigroups with zero
- a Green’s relation inspired construction of regular algebraic semigroups

Primarily, we will explore the combinatorial and geometric nature of reductive monoids with zero. Such monoids have a decomposition in terms of a Borel subgroup, called the Bruhat decomposition, which produces a finite monoid, **R**, the Renner monoid. We will explore ...

Algorithms To Compute Characteristic Classes, 2015 The University of Western Ontario

#### Algorithms To Compute Characteristic Classes, Martin Helmer

*Electronic Thesis and Dissertation Repository*

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases).

We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and ...

Symbolic Powers Of Edge Ideals, 2015 Dordt College

#### Symbolic Powers Of Edge Ideals, Mike Janssen

*Faculty Work: Comprehensive List*

No abstract provided.

Manipulating The Mass Distribution Of A Golf Putter, 2015 University of Rhode Island

#### Manipulating The Mass Distribution Of A Golf Putter, Paul J. Hessler Jr.

*Senior Honors Projects*

Putting may appear to be the easiest but is actually the most technically challenging part of the game of golf. The ideal putting stroke will remain parallel to its desired trajectory both in the reverse and forward direction when the putter head is within six inches of the ball. Deviation from this concept will cause a cut or sidespin on the ball that will affect the path the ball will travel.

Club design plays a large part in how well a player will be able to achieve a straight back and straight through club head path near impact; specifically the ...

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

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

*Electronic Theses & 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.

A Note On Characters Of Algebraic Groups, 2014 Rose-Hulman Institute of Technology

#### A Note On Characters Of Algebraic Groups, Sean Broughton

*S. Allen Broughton*

It is proven that a nowhere-vanishing polynomial function on a connected algebraic group, over an algebraically closed field, is a multiple of a (one-dimensional) character..

The Homology And Higher Representations Of The Automorphism Group Of A Riemann Surface, 2014 Rose-Hulman Institute of Technology

#### The Homology And Higher Representations Of The Automorphism Group Of A Riemann Surface, Sean Broughton

*S. Allen Broughton*

The representations of the automorphism group of a compact Riemann surface on the first homology group and the spaces of q-differentials are decomposed into irreducibles. As an application it is shown that M24 is not a Hurwitz group.