MATH2019RABBY57606 MATH

**Type:** Graduate

**Author(s):**
Fazle Rabby
Mathematics

**Advisor(s):**
Scott Nollet
Mathematics

**Location: **
Session: 1; Basement; Table Number: 12

View PosterAn 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

**Type:** Graduate

**Author(s):**
Douglas Wagner
Mathematics

**Advisor(s):**
José Carrión
Mathematics

**Location: **
Session: 1; 3rd Floor; Table Number: 2

View PosterA 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

**Type:** Graduate

**Author(s):**
Isai Chavarri
Mathematics

**Advisor(s):**
Ken Richardson
Mathematics

View PosterUnderstanding 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

**Type:** Undergraduate

**Author(s):**
Thinh Doan
Mathematics

**Advisor(s):**
Ken Richardson
Mathematics

View PosterUsing 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

**Type:** Undergraduate

**Author(s):**
Jacob Howell
Mathematics

**Advisor(s):**
Eric Hanson
Mathematics

(Poster is private)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.

MATH2018RABBY50952 MATH

**Type:** Graduate

**Author(s):**
Fazle Rabby
Mathematics

**Advisor(s):**
Scott Nollet
Mathematics

View PosterAn 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

**Type:** Graduate

**Author(s):**
Jeremy Smith
Mathematics

**Advisor(s):**
George Gilbert
Mathematics

View PosterAn 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

**Type:** Graduate

**Author(s):**
Nathanael Hellerman
Mathematics

**Advisor(s):**
Efton Park
Mathematics

View PosterThe 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

**Type:** Undergraduate

**Author(s):**
Jake Howell
Mathematics

**Advisor(s):**
Eric Hanson
Mathematics

View PosterPersonality 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

**Type:** Graduate

**Author(s):**
Jeremy Smith
Mathematics

**Advisor(s):**
George Gilbert
Mathematics

View PosterAn algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. It is well-known that there is not always a single algebraic integer that can generate the ring of algebraic integers contained in a field extension of the rational numbers. The index of an algebraic integer is a natural number that measures how far a ring of integers is from having such a "primitive element." We investigate these indices in cubic fields and determine which natural numbers occur as indices in given families.