Oral reading accuracy is an important measurement to assess language proficiency in educational settings. We aim to quantify misclassification rates when reading scores are assessed by humans versus an AI scoring system using speech recognition. The project breaks down misclassification rates into two components: True Positive, where human or AI systems correctly identify the correct words, and True Negative, where they correctly identify the incorrect words. For each reader, and conditional on the true score, the number of true positives and true negatives is assumed to follow binomial distributions. We propose two solutions to estimate misclassification rates, the first model assumes true scores are available, while the second model explores binomial and Poisson distribution structures for true scores with the minimization of normed Euclidean distance when the true scores are not observed. The models are evaluated through a study of elementary school students using ten passages of different lengths and difficulty.

We show that a discrete two-dimensional logistic predator-prey dynamical system with two parameters undergoes a Neimark-Sacker bifurcation under certain conditions. Our evidence includes numerical computations of orbits and bifurcation diagrams.

A Latin square is a nxn square that contains n different symbols, often numbers, and are arranged such that each symbol appears exactly once in each row and column. In this project, we look at the probability of a random arrangement of symbols being a Latin square. I start with n number of n symbols, for example a 3x3 square will contain the numbers 1,1,1,2,2,2,3,3,3 in a random assortment. Using counting methods and statistical estimation through Python, we discover the proportion of total squares that are Latin squares.

Geodesics are significant objects and a major topic in differential geometry. They are "straight" curves on surfaces that can locally represent the shortest path between two points. In this research, we employ the genetic algorithm, an optimization method in classical Artificial Intelligence, to construct a geodesic net on closed surfaces. A geodesic net is a network that connects multiple points with the shortest curves while ensuring that each point is ``balanced'' and stretched equally by its neighbors through those curves.

In this project we examine 2-dimensional cell-complexes and group actions on those cell complexes. We determine topological invariants of the group actions on these complexes using homology, cohomology, and the Euler characteristic.

In the field of Riemannian geometry, the condition on the Riemannian metric so that a manifold has positive scalar curvature (PSC) is important for a number of reasons. Many famous researchers have contributed gradually to this area of geometry, and in this project, we study more about PSC metrics on such manifolds. Specifically, we refine and provide some details to the proof of Gromov and Lawson that the connected sum of 2 n-dimensional manifolds will admit a PSC metric, provided each of the manifolds has a metric with the same condition. We then derive some useful formulas related to the Riemann curvature tensor, the Ricci tensor, and the scalar curvature in many different scenarios. We compute the quantities for a manifold equipped with an orthonormal frame and its dual coframe, namely the connection one-form and the curvature two-form. Then, we observe the change in the structure functions, defined as a function that determines the Lie derivative of the orthonormal frame, under a nearly conformal change of the said frame. The aim of these calculations is that, by expressing the scalar curvature of a manifold M entirely in terms of the structure functions, we can determine a condition on the conformal factor so that when dividing the tangent bundle of M into two sub-bundles, then the scalar curvature restricted to one sub-bundle will “dominate” that of the other one so that if we know the scalar curvature of the former sub-bundle is positive, we can be assured that the scalar curvature of M as a whole is also positive.

This research project focuses on the spreading of random curves in the differential geometry field which arises in statistical mechanics . It is known from the work of Einstein that random walks are connected to Brownian motion and diffusion. We will examine random curves that are not merely continuous but that are smooth and have prescribed bounds on curvature. We examine the distribution of a finite number of endpoints of such random curves. Using Python, we obtain 2-D histograms, graphs, and charts to research the spreading of random curves. A central goal in statistical mechanics is to describe the large-scale behavior of systems with the distribution of randomly generated data; we compare the distributions of curve endpoints to the Gaussian (normal) distribution.

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.

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).

In the field of Riemannian geometry, the condition on the Riemannian metric so that a manifold has positive scalar curvature (PSC) is important for a number of reasons. Many famous researchers have contributed gradually to this area of geometry, and in this project, we study more about PSC metrics on such manifolds. Specifically, we refine and provide some details to the proof of Gromov and Lawson that the connected sum of 2 n-dimensional manifolds will admit a PSC metric, provided each of the manifolds has a metric with the same condition. We then derive some useful formulas related to the Riemann curvature tensor, the Ricci tensor, and the scalar curvature in many different scenarios. We compute the quantities for a manifold equipped with an orthonormal frame and its dual coframe, namely the connection one-form and the curvature two-form. Then, we observe the change in the structure functions, defined as a function that determines the Lie derivative of the orthonormal frame, under a nearly conformal change of the said frame. The aim of these calculations is that, by expressing the scalar curvature of a manifold M entirely in terms of the structure functions, we can determine a condition on the conformal factor so that when dividing the tangent bundle of M into two sub-bundles, then the scalar curvature restricted to one sub-bundle will “dominate” that of the other one, so that if we know the scalar curvature of the former sub-bundle is positive, we c

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.