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MATH2021DANG27067 MATH

Wound Healing Process Modeling Using Partial Differential Equations and Deep Learning

Type: Undergraduate
Author(s): Hy Dang Mathematics
Advisor(s): Ken Richardson Mathematics
Location: Zoom Room 4, 03:27 PM

The process of successful skin healing from a wound involves different combinations of interactions. Moreover, by clearly understanding this process, we can provide and determine the appropriate amount of medicine to give to patients with varying types of wounds. Thus, this can improve the healing process of patients. In this research, we use the ADI method to solve a partial differential equation that models the wound healing process. Moreover, we try to explore the relationship between parameters in the model for different patients. Wound images are used as our dataset experiment. To segment the image's wound, we implement U-Net, a deep learning-based model, as our model for this segmentation problem. We believe the combination of ADI and Deep Learning helps us understand the process of wound healing.

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MATH2021NAGEL27835 MATH

Analysis of the Settlers of Catan

Type: Undergraduate
Author(s): Lauren Nagel Mathematics
Advisor(s): Drew Tomlin Mathematics
Location: Zoom Room 3, 12:38 PM

Markov chains are stochastic models characterized by the probability of future states depending solely on one's current state. Google's page ranking system, financial phenomena such as stock market crashes, and algorithms to predict a company's projected sales are a glimpse into the array of applications for Markov models. In this research, we analyzed the board game "The Settlers of Catan" using transition matrices. Transition matrices are composed of the current states which represent each row i and the proceeding states across the columns j with the entry (i,j) containing the probability the current state i will transition to the state j. Using these transition matrices, we delved into addressing the question of which starting positions are optimal. Furthermore, we worked on determining optimality in conjunction with a player's gameplay strategy. After building a simulation of the game in python, we tested the results of our theoretical research against the mock run throughs to observe how well our model prevailed under the limitations of time (number of turns before winner is reached).

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MATH2021NGUYEN38212 MATH

Metrics of Positive Scalar Curvature on Riemannian Manifolds

Type: Undergraduate
Author(s): Khoi Nguyen Mathematics
Advisor(s): Ken Richardson Mathematics
Location: Zoom Room 3, 03:03 PM

In Riemannian geometry, the concept of the scalar curvature on a Riemannian manifold is a generalization of the concept of curvature in curve theory or the concept of Gaussian curvature in surface theory. This is a measure of how curved a higher-dimensional manifold is, or another way to put it, how much does the volume of a ball in a curved manifold deviate from that in the usual Euclidean space. In this research, we attempt to dive deeper into the question of what kind of manifold in dimension 3 and above admits a metric of positive scalar curvature (PSC). Given the widespread application of the concept of curvature to the field of higher physics, specifically general relativity and quantum field theory, this question proves essential for furthering the works of mathematicians in classifying manifolds according to their intrinsic geometry. First, we attempt to come up with some useful formulas to calculate various quantities related to scalar curvature. Then, we will look into past results concerning PSC metric on Riemannian manifolds. Finally, we will prove some important theorems relating to this question.

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MATH2020DANG42837 MATH

Modeling Wound Healing Using Deep Learning

Type: Undergraduate
Author(s): Hy Dang Mathematics
Advisor(s): Ken Richardson Mathematics

The process of successful skin healing from a wound involves different combinations of interactions. Moreover, by clearly understanding this process, we can provide and determine the appropriate amount of medicine to give to patients with varying types of wounds. Thus, this can improve the healing process of patients. In this research, we use the ADI method to solve a partial differential equation that models wound healing and also determine the necessary parameters to achieve the stability of the ADI method. The data, which we are using, are pictures of the wounds, and the task is finding the initial conditions, that is exact boundary data from photos. We believe that Deep Learning is an excellent method to deal with this segmentation problem.

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MATH2020WAGNER17549 MATH

Deeper Exploration of the C*-Algebras Arising from Uniformly Recurrent Subgroups and their Relationship with Crossed Products

Type: Graduate
Author(s): Douglas Wagner Mathematics
Advisor(s): José Carrión Mathematics

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 further develop understanding of these C*-algebras using tools from other areas of operator theory. In particular, comparisons with the well-known crossed-product construction have proven useful.

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MATH2019RABBY57606 MATH

Invariants of Triple Conics in Projective Three Space

Type: Graduate
Author(s): Fazle Rabby Mathematics
Advisor(s): Scott Nollet Mathematics
Location: Session: 1; Basement; Table Number: 12

presentation location

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.

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MATH2019WAGNER64069 MATH

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

Type: Graduate
Author(s): Douglas Wagner Mathematics
Advisor(s): José Carrión Mathematics
Location: Session: 1; 3rd Floor; Table Number: 2

presentation location

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.

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MATH2018CHAVARRI16535 MATH

Modeling wound healing using MATLAB

Type: Graduate
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.

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MATH2018DOAN62209 MATH

A Numerical Approximation for Eigenvalues of Hyperbolic Polygons

Type: Undergraduate
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.

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MATH2018HOWELL16095 MATH

Analyzing Differences in Personality Structure by Age Using Topological Data Analysis

Type: Undergraduate
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.

(Presentation is private)

MATH2018RABBY50952 MATH

Double Structures on Conics in Projective Three Space

Type: Graduate
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.

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MATH2018SMITH34045 MATH

Indices of Algebraic Integers in Cubic Fields

Type: Graduate
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.

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MATH2017HELLERMAN41492 MATH

Winding Numbers and Toeplitz Operators

Type: Graduate
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.

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MATH2017HOWELL42763 MATH

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

Type: Undergraduate
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.

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MATH2017SMITH36813 MATH

Indices of Algebraic Integers in Cubic Fields

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

An 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.

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