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MATH2019RABBY57606 MATH

Invariants of Triple Conics in Projective Three Space

Author(s): Fazle Rabby Mathematics
Advisor(s): Scott Nollet Mathematics
Location: Session: 1; Basement; Table Number: 12 An algebraic curve is a one-dimensional set defined by polynomial equations, such as a parabola in the plane (given by y-x^2=0) or the z-axis in the space (given by x=y=0). Let Y be an algebraic curve. Then a multiplicity structure on Y is another curve Z, which as a set has the same points as Y but with a higher and fixed multiplicity at each point. For example, the y-axis in the plane is given by the equation x=0 and if we intersect it with horizontal lines, say with y-b=0, we get the points (0,b) on the y-axis. Now if we take the line given by x^2=0 and intersect it with the horizontal lines as above we get the points (0,b) with multiplicity 2. Hence we call the later curve a double structure on the previous one. Similarly the equation x^3=0 gives a triple structure on the y-axis in the plane and so on. Structures like these might sound naive but they are crucial to understand the behaviors of families of curves. For example, the family of parabolas ty-x^2=0 deforms into the double line x^2=0 as t approaches 0. Although the notion of multiplicity is pretty geometric, we can use tools from abstract algebra to make it rigorous. This makes the subject challenging and yet very interesting at the same time. Classifying the multiplicity structures on a curve is still a wide open field in algebraic geometry. It is now well understood how the double and triple structures on a line look. A natural question then arises how do the double and triple structures look on conics? It turns out that the answers are much more complicated than for lines. In this poster I am going to show some of my research in that direction.

MATH2019WAGNER64069 MATH

The Crossed Product Structure of C*-Algebras Arising from Uniformly Recurrent Subgroups

Author(s): Douglas Wagner Mathematics
Advisor(s): José Carrión Mathematics
Location: Session: 1; 3rd Floor; Table Number: 2 A group is a mathematical construct that represents the symmetries of an object. These symmetries transform the object through what is called a group action. Graphs—Cayley graphs, in particular—provide a rich source of symmetries for forming groups. A graph and its group action can be modeled by a collection of infinite matrices known as a C*-algebra. In a paper in the Journal of Functional Analysis, Gábor Elek used dynamical systems called Uniformly Recurrent Subgroups (URS) to construct a new type of C*-algebra. We relate this C*-algebra to a well-known construction called the crossed-product. This reinterpretation more prominently displays the group action, which proves useful as we further study the C*-algebra’s structure.

MATH2018CHAVARRI16535 MATH

Modeling wound healing using MATLAB

Author(s): Isai Chavarri Mathematics
Advisor(s): Ken Richardson Mathematics

Understanding of the wound healing process can be used to make more tailor-made medicine and to determine the nature of this healing process. In this research we use MATLAB software along with the ADI method to solve a partial differential equation that models wound healing by treating keratin as a diffusion process. A significant hurdle to overcome is finding the appropriate initial conditions, that is to accurately extract boundary data from photos taken with different equipment, lighting, or resolution.

MATH2018DOAN62209 MATH

A Numerical Approximation for Eigenvalues of Hyperbolic Polygons

Author(s): Thinh Doan Mathematics
Advisor(s): Ken Richardson Mathematics

Using the finite element method as a numerical approximation in solving for eigenvalues of the hyperbolic Laplacian, this research investigates the estimates of the first two eigenvalues with Dirichlet and Neumann boundary conditions on bounded domains in the upper half plane. Examples of finite element code using Matlab are presented to illustrate how to obtain these approximations for hyperbolic polygons. These values can further be used to shed light on the Selberg and Fundamental Gap conjectures.

MATH2018HOWELL16095 MATH

Analyzing Differences in Personality Structure by Age Using Topological Data Analysis

Author(s): Jacob Howell Mathematics
Advisor(s): Eric Hanson Mathematics

In the past, Personality Psychologists have commonly applied clustering techniques on questionnaire data to analyze personality structure. The purpose of this research is to determine if techniques from topological data analysis can provide a greater understanding of personality. Specifically, persistent homology was used to determine clusters topologically and analyze the ‘shape’ of the data. Previous work we have done focused on seeing how persistent homology can provide insight on identifying the key (most persistent) clusters in the data. New analysis looks to see how the Big 5 Personality Factors cluster together as a function of age.

(Poster is private)

MATH2018RABBY50952 MATH

Double Structures on Conics in Projective Three Space

Author(s): Fazle Rabby Mathematics
Advisor(s): Scott Nollet Mathematics

An algebraic curve is a one-dimensional set defined by polynomial equations, such as a parabola in the plane (given by y-x^2=0) or the z-axis in the space (given by x=y=0). Let Y be an algebraic curve. Then a multiplicity structure on Y is another curve Z, which as a set has the same points as Y but with a higher and fixed multiplicity at each point. For example, the y-axis in the plane is given by the equation x=0 and if we intersect it with horizontal lines, say with y-b=0, we get the points (0,b) on the y-axis. Now if we take the line given by x^2=0 and intersect it with the horizontal lines as above we get the points (0,b) with multiplicity 2. Hence we call the later curve a double structure on the previous one. Similarly the equation x^3=0 gives a triple structure on the y-axis in the plane and so on. Structures like these might sound naive but they are crucial to understand the behaviors of families of curves. For example, the family of parabolas ty-x^2=0 deforms into the double line x^2=0 as t approaches 0. Although the notion of multiplicity is pretty geometric, we can use tools from abstract algebra to make it rigorous. This makes the subject challenging and yet very interesting at the same time. Classifying the multiplicity structures on a curve is still a wide open field in algebraic geometry. It is now well understood how the double and triple structures on a line look. A natural question then arises how do the double and triple structures look on conics? It turns out that the answers are much more complicated than for lines. In this poster I am going to show some of my research in that direction.

MATH2018SMITH34045 MATH

Indices of Algebraic Integers in Cubic Fields

Author(s): Jeremy Smith Mathematics
Advisor(s): George Gilbert Mathematics

An algebraic integer is a complex number that is a root of a polynomial with integer coefficients and a leading coefficient of 1. This includes numbers like the square root of 2 and the cube root of 10, for example. A field is a set in which we can add, subtract, multiply, and divide (among other details). Consider the set of all algebraic integers in any given field containing the rational numbers. The index of an algebraic integer in this set is a natural number that measures how close the algebraic integer is to generating the set. For instance, the imaginary number i (the square root of -1) is an algebraic integer which generates the set of all complex numbers of the form a + bi where a and b are integers, and so has index 1. The closer the index is to 1, the closer the algebraic integer is to generating the set. We investigate these indices in cubic fields, determining not only which numbers occur as indices in given families, but also that the minimal index is unbounded as one traverses the set of all cubic fields in those families.

MATH2017HELLERMAN41492 MATH

Winding Numbers and Toeplitz Operators

Author(s): Nathanael Hellerman Mathematics
Advisor(s): Efton Park Mathematics

The winding number of a continuous function on the unit circle counts how many times a graph of the function loops around the origin. It is homotopy invariant and has applications to several areas of Mathematics.
Toeplitz operators with continuous symbol are bounded linear operators on the Hardy Space involving multiplication by a continuous function. The index of such a Toeplitz operator is closely connected to the winding number of its symbol.
This connection is examined and then extended for Toeplitz operators with crossed product symbols.

MATH2017HOWELL42763 MATH

Differences in Personality Structure by Age: Analyzing Clusters with Persistent Homology

Author(s): Jake Howell Mathematics
Advisor(s): Eric Hanson Mathematics

Personality psychologists often apply clustering techniques on questionnaire data to model personality structure. Inspired by this work, we apply techniques from topological data analysis (TDA) to understand the structure of this data. The data comes from Cattell’s Sixteen Personality Factor Questionnaire (collected by Bell, Rose, & Damon in 1972). Subjects were 969 adult male volunteers divided into three age groups: 25 to 34, 35 to 54, and 55 to 82. We use persistent homology (a TDA tool) to cluster the data and identify that personality structure is slightly different between the age groups. It is also curious to note that data from the youngest age group appears to have a topological “hole”, which raises questions of the psychological significance. This work suggests that additional research, including applying TDA tools to other questionnaire data sets can provide insights to the study of personality.

MATH2017SMITH36813 MATH