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 Title
 Chern Classes of Sheaves of Logarithmic Vector Fields for Free Divisors.
 Creator

Liao, Xia, Aluﬃ, Paolo, Reina, Laura, Klassen, Eric P., Aldrovandi, Ettore, Petersen, Kathleen, Department of Mathematics, Florida State University
 Abstract/Description

The thesis work we present here focuses on solving a conjecture raised by Aluffi about ChernSchwartzMacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(log D)) cap [X]$ in $A_{*}(X)$ for any locally quasihomogeneous free divisor $D$. We prove a stronger...
Show moreThe thesis work we present here focuses on solving a conjecture raised by Aluffi about ChernSchwartzMacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(log D)) cap [X]$ in $A_{*}(X)$ for any locally quasihomogeneous free divisor $D$. We prove a stronger version of this conjecture. We also report on work aimed at studying the Grothedieck class of hypersurfaces of low degree. In this work, we verified the Geometric ChevalleyWarning conjecture in several low dimensional cases.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7467
 Format
 Thesis
 Title
 ChernSchwartzMacpherson Classes of Graph Hypersurfaces and Schubert Varieties.
 Creator

Stryker, Judson P., Aluﬃ, Paolo, Van Engelen, Robert, Aldrovandi, Ettore, Hironaka, Eriko, Van Hoeij, Mark, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of...
Show moreThis dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of certain families of these varieties is proven. The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation. In the second chapter, a series of equivalences of graph hypersurfaces that are useful for reducing the number of cases that must be calculated are developed. A table of CSM classes of all but one graph with 6 or fewer edges are explicitly computed. This table also contains Fulton Chern classes and Milnor classes for the graph hypersurfaces. Using the equivalences and a series of formulas from a paper by Aluffi and Mihalcea, a new series of formulas for the CSM classes of certain families of graph hypersurfaces are deduced. I prove positivity for all graph hypersurfaces corresponding to graphs with first Betti number of 3 or less. Formulas for graphs equivalent to graphs with 6 or fewer edges are developed (as well as cones over graphs with 6 or fewer edges). In the third chapter, CSM classes of Schubert varieties are discussed. It is conjectured by Aluffi and Mihalcea that all Chern classes of Schubert varieties are represented by effective cycles. This is proven in special cases by B. Jones. I examine some positivity results by analyzing and applying combinatorial methods to a formula by Aluffi and Mihalcea. Positivity of what could be considered the ``typical' case for low codimensional coefficients is found. Some other general results for positivity of certain coefficients of Schubert varieties are found. This technique establishes positivity for some known cases very quickly, such as the codimension 1 case as described by Jones, as well as establishing positivity for codimension 2 and families of cases that were previously unknown. An unexpected connection between one family of cases and a second order PDE is also found. Positivity is shown for all cases of codimensions 14 and some higher codimensions are discussed. In both the graph hypersurfaces and Schubert varieties, all calculated ChernSchwartzMacPherson classes were found to be positive.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1531
 Format
 Thesis
 Title
 Closed Form Solutions of Linear Difference Equations.
 Creator

Cha, Yongjae, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this idea is how to find a solved equation Lg to which Linp can be reduced. To solve this problem, we use local data of a difference operator, that is invariant under the transformation.
 Date Issued
 2011
 Identifier
 FSU_migr_etd3960
 Format
 Thesis
 Title
 Factoring Univariate Polynomials over the Rationals.
 Creator

Novocin, Andrew, Van Hoeij, Mark, Van Engelen, Robert, Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is et up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior...
Show moreThis thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is et up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior algorithms in many common classes of polynomials (including irreducibles).
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2515
 Format
 Thesis
 Title
 Finding All Bessel Type Solutions for Linear Differential Equations with Rational Function Coefficients.
 Creator

Yuan, Quan, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if...
Show moreA linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if change of variables is a rational function. In this thesis we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions. This algorithm can be easily extended to a Whittaker/Kummer solver. Combine the two algorithms, we can get a complete algorithm for all 0F1 and 1F1 type solutions. We also use our algorithm to analyze the relation between Bessel functions and Heun functions.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5296
 Format
 Thesis
 Title
 On Elliptic Fibrations and FTheory Compactifications of String Vacua.
 Creator

Fullwood, James, Aluﬃ, Paolo, Reina, Laura, Van Hoeij, Mark, Aldrovandi, Ettore, Hironaka, Eriko, Department of Mathematics, Florida State University
 Abstract/Description

We investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible...
Show moreWe investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible dimensions of a base at once. As such, we construct generating series $f(t)$ corresponding to each invariant such that the coefficient of $t^k$ encodes the invariant of the elliptic fibration over a base of dimension $k$, solely in terms of invariants of the base. From the Ftheory perspective, we highlight aspects of elliptic fibrations other than Weierstrass models, and construct a new orientifold limit of Ftheory associated with $D_5$ fibrations, i.e., elliptic fibrations whose elliptic fiber is realized via a complete intersection of two quadrics in $\mathbb{P}^3$. We verify tadpole relations as predicted by the (conjectural) equivalence between Ftheory and typeIIB, as well as `universal tadpole relations', which are mathematical generalizations of the tadpole relations predicted by the physics of Ftheory. We also simplify formulas for invariants of CalabiYau fourfolds, and suggest that all Hodge numbers of CalabiYau fourfolds depend linearly on $c_1(B)^3$, where $B$ is the base of the fibration.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4848
 Format
 Thesis
 Title
 On Picard 2Stacks and Length 3 Complexes of Abelian Sheaves.
 Creator

Tatar, Ahmet Emin, Aldrovandi, Ettore, Capstick, Simon, Agashe, Amod, Aluﬃ, Paolo, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the...
Show moreIn Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the category of Picard stacks with morphisms being the isomorphism classes. In this dissertation, we generalize his work, following closely his steps in SGA4, to the case of Picard 2stacks. But this generalization requires first a clear description of a Picard 2category as well as of a 2functor between such 2categories that respects Picard structure. Once this has been done, we can talk about category of Picard 2stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2stacks.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1674
 Format
 Thesis
 Title
 Openmath Library for Computing on Riemann Surfaces.
 Creator

Lebedev, Yuri, Seppälä, Mika, Van Engelen, Robert, Van Hoeij, Mark, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis carefully reviews computational methods that will act as a tool in the research of Riemann surfaces. We are interested in representing a Riemann surface from many equivalent points of view. The goal is to define a Riemann surface so it can be freely and unambiguously exchanged between mathematical servers by creating a set of suitable OpenMath CDs.
 Date Issued
 2008
 Identifier
 FSU_migr_etd3208
 Format
 Thesis
 Title
 Periods and Motives: Applications in Mathematical Physics.
 Creator

Li, Dan, Marcolli, Matilde, Reina, Laura, Aluﬃ, Paolo, Agashe, Amod, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebro...
Show moreThe study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebrogeometric approach to the spectral theory of Harper operators in solid state physics. When the parameters are irrational, the compactification of its Bloch variety is an indprovariety, which is a Cantorlike geometric space and it is compatible with the picture of Hofstadter butterfly. On each approximating component the density of states of the electronic model can be expressed in terms of period integrals over Fermi curves, which can be explicitly computed as elliptic integrals or periods of elliptic curves. The above density of states satisfies a PicardFuchs equation, whose solutions are generally given by hypergeometric functions. We use the idea of mirror maps as in mirror symmetry of elliptic curves to derive a qexpansion for the energy level based on the PicardFuchs equation. In addition, formal spectral functions such as the partition function are derived as new period integrals. Secondly, we consider generalized Feynman diagram evaluations of an effective noncommutative field theory of the PonzanoRegge model coupled with matter in loop quantum gravity. We present a parametric representation in a linear kapproximation of the effective field theory derived from a kdeformation of the PonzanoRegge model and define a generalized Kirchhoff polynomial with kcorrection terms. Setting k equal to 1, we verify that the number of points of the corresponding hypersurface of the tetrahedron over finite fields does not fit polynomials with integer coefficients by computer calculations. We then conclude that the hypersurface of the tetrahedron is not polynomially countable, which possibly implies that the hypersurface of the tetrahedron as a motive is not mixed Tate.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5390
 Format
 Thesis
 Title
 Predegree Polynomials of Plane Configurations in Projective Space.
 Creator

Tzigantchev, Dimitre G. (Dimitre Gueorguiev), Aluﬃ, Paolo, Reina, Laura, Aldrovandi, Ettore, Klassen, Eric, Seppälä, Mika, Department of Mathematics, Florida State University
 Abstract/Description

We work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the...
Show moreWe work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the dimension of the orbit. We find the predegrees and other invariants for all surfaces supported on unions of planes. The information is encoded in the socalled adjusted predegree polynomials, which possess nice multiplicative properties allowing us to easily compute the predegree (polynomials) of various special plane configurations. The predegree has both a combinatorial and geometric significance. The results obtained in this thesis would be a necessary step in the solution of the problem of computing predegrees for all surfaces.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd1747
 Format
 Thesis
 Title
 Solutions of Second Order Recurrence Relations.
 Creator

Levy, Giles, Van Hoeij, Mark, Van Engelen, Robert A., Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such...
Show moreThis thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such a transformation exists. The second algorithm `Find Liouvillian' finds a gttransformation to a recurrence relation of the form u(n+2) + b(n)u(n) = 0 for some b(n) in C(n), if such a transformation exists. The third algorithm `Database Solver' takes advantage of a large database of sequences, `The OnLine Encyclopedia of Integer Sequences' maintained by Neil A. J. Sloane at AT&T Labs Research. It employs this database by using the recurrence relations that they satisfy as base equations from which to return a gttransformation, if such a transformation exists.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3099
 Format
 Thesis
 Title
 Solving Linear Differential Equations in Terms of Hypergeometric Functions by ₂Descent.
 Creator

Fang, Tingting, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

Let L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2descent has already...
Show moreLet L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2descent has already been treated before, a complete implementation could not be given because it involved a step for which we do not have a complete implementation. Our key novelty is to give an approach that is fully implementable. We describe and implement the algorithm for order 2, and show by examples that the same also work for higher order. By doing 2descent for L, the number of true singularities drops to at most n/2 + 2 (n is the number of true singularities of L). This provides us ways to solve L in closed form(e.g.in terms of hypergeometric funtions).
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5350
 Format
 Thesis