Honors Research Projects Options

2025 Research Projects

Please read each project choice and select your top three choices below.

 1. Geonhee Cho, The Geometry of Quantum Computing

Quantum computing can be studied through the lens of geometry, where the space of quantum states is naturally equipped with an information-geometric structure. Information geometry is a broad field that studies statistical models using differential geometry. This project will explore these geometric foundations, referencing ‘The Geometry of Quantum Computing’ by E. Ercolessi, R. Fioresi, and T. Weber.

The project will focus on investigating the relationship between entanglement entropy and quantum state geometry through concrete two-qubit quantum circuits. We aim to understand how geometric structures influence quantum information processing and develop an intuition for quantum state evolution in this framework.

We will begin by introducing fundamental concepts, including qubits, density operators, and quantum logic gates, which will serve as the foundation for our study. Next, we will examine key ideas from information geometry, such as the Fisher matrix, and explore their quantum counterparts, leading to the Quantum Geometric Tensor, a natural Kähler metric on the space of qubits. This project is particularly suitable for undergraduate students interested in learning differential geometry at the graduate level in the future.

Prerequisites and Research Approach

Participants should have a background in linear algebra, spectral theory over complex numbers, and abstract algebra, particularly group theory, group actions, and quotient groups. The project will explore the intersection of differential geometry, information geometry, and quantum mechanics, providing a mathematical approach to quantum computing. For computational tools for project, we will use Mathematica.

The research will be conducted online.

2. Burcu Cinarci, Prime Centralizer Graph of Groups

Let G be a finite group. For any element  g in G, the centralizer of g is denoted by 

C_G(g) = {h in G : gh=hg}

We associate a graph PC_G with a group G (called the non-prime centralizer graph of G) as follows: let P be the set of elements of G in which their orders are prime power and take P as the vertex set of PC_G; and two vertices x and y are adjacent if 

C_G(x) is not equal to C_G(y)

where C_G(x) is the centralizer of x in G.  In this project, we study the graph theoretical properties of PC_G and its induced sub-graph on P_G, where P_G is a prime graph which is a famous graph that many prominent researchers work on it.

 Here we try to find the clique number of this graph and the influence of this graph on the prime graph.

The research will be conducted online.

3. Anton Dochtermann, Powers of paths and cycles

A graph consists of a set of nodes along with a collection of edges that connects specified pairs of nodes. One can associate an algebraic object to these combinatorial objects, an "ideal" generated by the edges that appear. In this context it is of interest to understand how properties change when one takes powers. This makes sense algebraically (just multiply!) but the challenge is to interpret the operation on the underlying graphs. What does it mean to take the square of a path? Or of a cycle? In this project we will explore these (and related) questions. Students should be familiar with basic graph theory, and coding skills are strongly encouraged.

4. Anton Dochtermann, Groups and graphs

Graphs are discrete objects that model connectivity in a network, and groups are algebraic objects with a binary operation that is `invertible'.  What happens when we combine the two? A natural instance of this comes from the automorphism group of a graph, where self maps of a graph can be `adjacent'. Inspired by this, as well as fancy notions from topology like Lie groups and G-bundles, in this project we will study sets that can be given both a graph and group structure in a compatible way.  Students should be familiar with graphs and maybe some basic group theory (although this is not required), and coding skills are encouraged.

5. Anton Dochtermann, Cycle systems for matroids

A "matroid" is a combinatorial object that abstract and unifies the notion of independence from several branches of mathematics, including linear algebra and graph theory. In this context one is interested in counting the number of independent sets in a matroid of each size, information encoded in its "h-vector" (related to June Huh's breakthrough work that won him the Fields Medal in 2022). An intriguing conjecture of Stanley says that such sequences can be recovered as the face vector of a multicomplex. In previous work we showed that Stanley's conjecture holds for any matroid that admits a "cycle system", an intersection property that mimics chip-firing on a graph. Determining which matroids admit cycle systems is an open question, and in this project we search for other classes of matroids with this property. No prior knowledge of matroids is necessary, but this project will require good coding skills.

6. Chris Guzelian, Mathematics and Economics

Economics is a social science that often invokes mathematics. Students will have the opportunity to work on a project examining whether price inflation causes environmental harm. The project requires gathering of data and statistical computation involving fractal Hurst exponents, an improved form of statistical analysis for time series.

7. Chris Guzelian, Mathematics and Law

Law is influenced by, and based upon, principles of mathematics and statistics. But law often does not correctly reflect that mathematical knowledge. Students have the opportunity to collaborate with a law professor and attorney conversant in mathematics to select a legal topic of interest, identify and explain the mathematics relevant to that area of law, and propose improvements to current laws.

8. Hamilton Hardison, Artificial Intelligence for Mathematically Pedagogical Purposes

Artificial intelligence (AI) presents new affordances and challenges in mathematics teaching and learning. The landscape is ever changing, and (as with any tool), power resides in how it is used. In this project, we will explore how to use AI to transformatively advance mathematics classrooms to the next level. Different from most projects, our aim will not be to reduce time associated with teaching mathematics; instead, we aim to create student-engineered AI-generated examples and imagery that might support other students' conceptual understanding. All are welcome. 

9. Steven Hoberman, Creating a More Flexible Test for the Population Mean – Extensions to Bivariate and Longitudinal data

Statistics is the discipline where conclusions are reached in the presence of uncertainty using quantitative information. In this context, it is typical to assume the data are normally distributed. The most common test we use for the mean in this setting is the t-test. In this project, we study how a novel probability distribution can be useful in making inferences about the mean when the data may be either symmetric (like normal data) or skewed (different from normal). We will consider for the first time two-sample tests based on this novel distribution. We also explore applications of this distribution to testing for a change in the mean over time where there may be multiple observations on the same unit, and in data sets where multiple statistical tests are considered simultaneously.

10. Wade Hindes, Prime Factors in Dynamical Orbits

Let S = {f_1,…,f_s} be a set of polynomials with integer coefficients, let M_S denote the semigroup of all finite compositions of elements of S, and let b in Z be any basepoint. Then we are interested in studying the prime factors appearing in the orbit of b:  

P_S(b) := { primes p : p| f(b) for some f in M_S }

In particular, we hope to prove that P_S(b) is an infinite and "sparse" set of primes (i.e., Dirichlet density zero). Such a set of primes is useful for cryptography. To begin, we will consider sets of the form 

S = {x^2+c_1,…,x^2+c_s}

for some c_1,…,c_s in Z.

11. Thomas Keller, Prime graphs of finite groups

The subject of this project are prime graphs of finite groups with a focus on groups that are very close to solvable groups. There are several open questions on these graphs which are purely combinatorial in nature. In the realm of solvable groups, an important notion here is the one of minimal prime graphs. A simple graph is called a minimal prime graph if it is connected, has at least two vertices, and its complement is triangle-free and has chromatic number 3, and removing any edge results in a graph whose complement has a triangle or chromatic number 4.     

An interesting notion in this context is the one of superminimal prime graphs which in a sense have as few edges as possible. Finding new families of such graphs would be nteresting, as would be answering many questions, such as: Can complements of superminimal prime graphs have induced circles of arbitrary lengths?       

Another notion that comes up in this context is a certain new product of graphs, and it would be interesting to study properties of this product and see how it interacts with other graph products.

12. Young Ju Lee, Fast Training Algorithm for Logistic regression and Support Vector Machine

Logistic regression and Support Vector machine are valuable tools for predicting the likelihood of an event. They help determine the probabilities between two or more classes. A very simple example is an email spam folder, for which, the Logistic regression can classify emails as spam or regular, and thus it can direct them to their respective inboxes. In recent years, the Logistic regression and SVM have become invaluable tools in the increasingly popular field of Machine Learning. In particular, for collecting and analyzing data, it is easier to implement, interpret, and train than other machine learning models, such as neural networks. Therefore, logistic regression and SVM can be considered to be a foundation of machine learning and it has been used in many areas, which include social sciences, such as Credit scoring, Medicine, Text editing, Hotel Booking, gaming, the acceptance rate for the school, just to cite a few. Logistic regression sets a basis for modern deep learning algorithms as well. On the other hand, it is well-known that logistic regression is difficult to solve. The most well-known solver is known as scikit learn, which is based on the Newton methods. However, since Newton method relies on the computation of the Hessian of the objective functional of the Logistic regression, it is very expensive and slow, in general. Thus, the timely research should be devoted to the development of a fast and efficient solver. In the summer project, we will propose to develop a fast solution technique for logistic regression based on conjugate gradient methods

13. Suho Oh, Attempting Simon's conjecture for 3d case 

 If one looks at an orange or a pizza, its slices are formed nicely in a way that you can remove one slice at a time, so that each slice fits in a pretty way with the current chunk. In math, these kind of objects are called shellable simplicial complex. Now there is a long standing conjecture, that says whenever you have something shellable, you can add an additional piece always so that it is still shellable. We are going to attempt this for three dimensional objects (either prove it or find a counterexample!). For proving, we are going to try various modifications of this conjecture for the 3-d case. For disproving it, we are going to test some recently developed machine learning techniques to try to search for potentially big counterexamples.

14. Ivan Ojeda-Ruiz, Enhancing Image Segmentation with Normalized Cuts and Inverse Filters

This project explores the application of inverse filters to enhance edge detection in image segmentation using the normalized cut algorithm. Image segmentation, a fundamental task in computer vision, involves partitioning an image into meaningful regions, and the normalized cut method is a powerful technique for achieving this by minimizing dissimilarities between segments while maximizing similarities within them. However, its effectiveness relies heavily on accurate edge detection, which can be challenging in noisy or complex images. Inverse filters, known for their ability to sharpen edges and enhance fine details, are integrated into the preprocessing stage to improve edge detection. By applying these filters, the project aims to boost the performance of the normalized cut algorithm, particularly in scenarios with low contrast or high noise. Through experimentation on diverse image datasets, this project evaluates the impact of inverse filters on segmentation accuracy and computational efficiency, offering insights into how advanced preprocessing techniques can enhance traditional computer vision algorithms. This interdisciplinary approach bridges signal processing and computer vision, demonstrating the potential for innovative solutions in image analysis.

* Python, Matlab, or some level of Linear Algebra is preferred but not required.

15. Ivan Ojeda-Ruiz, Exploring Fairness in Simple Clustering Algorithms

This project introduces the fundamental concepts of fairness in data analysis through the lens of simple clustering algorithms. While clustering is a powerful tool for grouping similar data points, it can unintentionally reinforce biases present in the data. This project aims to make the topic of fairness in AI accessible and engaging by focusing on the k-means algorithm, a widely used and easy-to-understand clustering method. Students will explore how fairness can be incorporated into clustering by examining basic fairness metrics and their impact on the results. Through hands-on activities, they will learn to identify potential biases in datasets and apply simple fairness constraints to the k-means algorithm.

* Python, Matlab, or some level of Linear Algebra is preferred but not required.

16. Mohammad Zarrin, Study of the non-power set graph of a group

The research will be conducted online.

17. I choose not to participate in a research project