Senior Theses

directed by

Thomas Garrity

1990-91

Andrew Beveridge, "Steps towards a Morse-Smale Algorithm"

    Our goal is to determine the rank of the homology groups of a real algebraic manifold M. In this exposition, we outline the steps needed in order to determine the homology of the manifold using a Morse-Smale flow. Our method employs a numerical solution to the Morse-Smale gradient flow of the height function. This reduces the problem to solving certain ordinary differential equations. One major assumption, found in section 4.2, must be made in order to solve these equations. We explicitly approximate the local coordinate system at a certain critical point given by the lemma of Morse. We also find an approximation for every orbit connecting a critical point p to a critical point q where the indices of these points satisfy lp - lq = 1. Further assumptions must be made in section 6.5 in order to use our approximations to determine the homology of the manifold.

    Michael Donofrio, "Calculating the Intersection Number of Complementary-Dimensional Cycles"

    Given a manifold and two intersecting cycles of complementary dimension contained in the manifold, all defined by the complete intersections of polynomial equations, we provide an algorithm for calculating the intersection number of the cycles. The algorithm begins by finding as an orientation for the manifold a basis for its tangent space at any point on the manifold. We then perform a similar construction for the tangent spaces of the intersecting cycles at their intersection points. If the intersection point is transverse then we assign to it either a positive or negative orientation. If the point is not transverse, then the orientation of the cycles at that point is not well-defined. We present a method of perturbing one of the cycles so as to insure that the new cycle will intersect the unperturbed cycle transversely around the original non-transverse intersection, thus enabling us to assign an orientation to that intersection.

    Erin Lagesen, "On the Projective Equivalency of Curves and Surfaces"

    This thesis constructs a theoretical method by which to determine (1) if two curves in the complex-projective plane are projectively equivalent and (2) if two generic smooth irreducible surfaces in complex-projective 3-space are projectively equivalent. We will accomplish this by looking for certain points on the curves, respectively surfaces, which possess distinguishing properties which remain invariant under projective change of coordinates. Knowing that such points on one curve, resp. surface, would necessarily map to the same such points of a projectively equivalent curve or surface, we will use these points to set up all of the projective linear transformations that would possibly map one to the other. We will then test the projective transformations on the curve or surfaces to determine whether they are projectively equivalent.

1991-92

Elizabeth Gibbons, "On Computing the Orientability of Algebraic Manifolds"

    This thesis presents an algorithm which determines in simple exponential time if a given manifold embedded in Rn and described by rational polynomials is orientable, and assigns an orientation to the manifold if it is. For manifolds that are complete intersections, the problem is trivial. Given a manifold that is not, the algorithm finds pieces of the manifold that are complete intersections, which we shall call "chunks", which cover the manifold. Determining whether or not it is pssible to consistently orient each of these chunks so that all orientations agree is equivalent to determining whether the manifold is orientable.

    Amy Huston, "On Whitney Stratifications for Semi-Algebraic Sets"

    Sufficient background is given to understand a number of formulations of the Whitney Conditions. Several versions of the Whitney Conditions are presented and discussed. Time bounds are given to a known stratification algorithm. Also given are possible directions for future work in trying to create an efficient algorithm to Whitney stratify a set.

    Cherie Macauley, "The Complexity Bounds of the Multivariate Sign Sequence Algorithm"

    We provide complexity bounds for John Canny's multivariate sign sequence algorithm. This is an algorithm for determining the signs of the polynomials f1(x1, ...., xn), ...., fn(x1, ...., xn) at the isolated roots of the system of polynomial equations P1(x1, ...., xn),..., P1(x1, ...., xn). We find that the majority of the steps of the algorithm can be reduce to determinant calculations and hence, the complexity bounds of computing a determinant.

 

1994-95

David Dela Cruz, "Analysis of Manifolds Using Morse-Smale Homology

    A topological object can be defined as a set of points in a space. A particularly beautiful topological object to examine is the manifold. Given two n-manifolds, it is interesting to know if they are topologically equivalent -- that is, whether we can bend, twist, and stretch one to get the other. The topological invariants of two manifolds must be the same if two manifolds are the same under topology, one such invariant is homology groups. Singular homology theory describes homology groups for manifolds in the abstract. Morse-Smale homology allows us to capture homology groups algebraically as well. The thesis is an exposition of singular homology theory, Morse theory, and Morse- Smale homology, followed by an algorithm that can be used to approximate Morse Smale gradient flows on algebraically-defined manifolds.

    Christopher French, "Computing the Intersection Homology Groups of a Complex Algebraic Variety"

    An algorithm is given to compute the intersection homology groups for a complex algebraic variety. Two previously developed algorithms, the Collin's cad algorithm and Prill's Adjacency algorithm, are presented and used.

    Dimitry Korsunsky, "One Approach to Factoring Multivariate Rational Polynomials Over the Complex Numbers"

    Factoring a given multivariate polynomial is an important task in symbolic computation. Potential uses for an efficient solution to this problem could be found in various branches of applied mathematics, such as computer-aided design and theorem proving. Several algorithms giving different methods for factoring multivariate polynomials had been created over the years (Noether 1922, Davenport and Trager 1981, Christov and Grigoryev 1983 etc.). The theoretical basis for the Bajaj et al [4]. There it is proved that the suggested approach when implemented in parallel will execute in shorter time as compared to earlier solutions. Although a sequential solution which was implemented achieves lower efficiency there is a significant advantage in that it can be used in a large variety of settings. The program had been written using the Mathematica software package.

 

1995-96

Daniel Ebert, "Probabistic Enumerative Geometry: How many inflection points are real"

    Classically, enumerative, or counting, questions have been answered in complex projective space where the answers are precise. In this project, we ask these enumerative questions in real affine space; in real space, the answers may no longer be precise. For instance, chaging the coefficients of an nth degree polynomial gives different numbers of real roots. So, we must answer these enumerative question in real projective space probabilistically, finding the expected, not exact, number. After examining Edelman and Kostlan's expostion and expansion on Kac's formula for the expected number of real roots of a random polynomial, the thesis focuses on trying to find the number of real inflection points of an nth degree polynomial. Interestingly, through Mathematica calculations, the expected number of real inflection points for a quartic and quintic is a little less than twice that of the expected number of real roots for an mth degree polynomial where m is the degree of the resultant of the Hessian curve with the original quartic or quintic polynomial curve.

    Daniel Kim, "Examing Continuity Along Tubular Neighborhoods"

    We use concepts of geometric continuity to develp a relationship between manifolds and tubular neighborhoods. Specifically, we examine a pair of Ck manifolds of dimension r whose intersection is along a Ck manifold of dimension r - 1; if the r- manifolds meet with Gk continuity, then the boundaries of their corresponding tubular neighborhoods intersect with Gk-1 continuity. We also discuss possible ways to extend the scope of this research.

    Alexander Meadows, "The Construction of the Moduli Space of Quadrilaterals up to Similarity"

    A method for constructing a space of quadrilaterals modulo similarity is provided. We begin with several methods of constructing a space of triangles up to simialrity. Then we construct the moduli space of quadrilateral up to similarity using both algebraic and geometric methods.

 

1998-99

Zachary Grossman, "Relations and Syzygies in Classical Invariant Theory for Vector-Valued Bilinear Forms"

    The goal of invariant theory is to describe the algebra of invariants for a vector space under a given group action. We examine the First Fundamental Theorems for the general linear group and for vector-valued linear forms, which describe the invariant rings for their respective vector space. We also examine the Second Fundamental Theorem for the general linear group, which describes the basis relations between the invariants of that group. The main theorem of the paper, the Second Fundamental Theorem for vector-valued bilinear forms, uses the Kunneth formula to describe the basis relations between invariants of vector-valued bilinear forms and to provide a method for finding the resolution of the entire syzygy chain.

 

1999-2000

Tegan Cheslack-Postava, "Questions of Uniqueness for Triangle Sequences in m Dimensions"

    The present paper focuses on the question of uniqueness Ð when does an m-dimensional triangle sequence correspond to a unique point in Rm. The summer 1999 SMALL group under Garrity showed that in two dimensions, there are infinite triangle sequences which correspond to entire line segments. It was also shown that sequences with particular characteristics are guaranteed to identify unique points. We extend both of these results to higher dimensions.

    Adam Schuyler, "A Structural Analysis of the Triangle Iteration"

    Classically, it is known that the continued fraction sequence for a real number a is eventually periodic if and only if a is a quadratic irrational. In response to this, Hermite posed the general question which asks for ways of representing numbers that reflect special algebraic properties . Specifically, he was inquiring about possible generalizations of continued fractions. In this paper we will study the triangle iteration, a two-dimensional analogue of the continued fraction algorithm. We will take a primarily geometric appraoch and look at the probabilities of the occurrences of certain sequences.

 

2002-03

Michael Baiocchi

Mark Rothlisberger

 

Williams Mathematics