2024 Research Projects
Please select your top three project choices by using the drop down boxes provided at the bottom of the page. The projects are listed by number. Do not select the same project more than once.
The group size for each project will typically be 3 or 4 students and/or counselors.
1. Victor Cepeda, Physically-Based Audio Effects
If you're standing outside a room with loud music playing, you'll probably hear just the bass beat. Then, if a window is opened, you'll hear the music more fully, but only from the direction of that window. This is a result of how sound travels through and around different mediums. It can be really complicated to solve, so most games are content to mostly fake it. But what if we didn't? How far can we take can we take physically simulating the effects of sound and what unique audio experiences could it unlock?
Things to explore:
How is audio encoded in a .wav file?
Fourier Transforms / Low Pass Filters
Representations of 3D geometry
Ray Tracing
Different shapes of audio emitters
2. Victor Cepeda, Explorations of Transform Hierarchy
A lot of how game worlds and characters are represented involves a complex hierarchy of positional and rotational offsets(called transforms). Maintaining accurate final "world space" values of all the transforms can be computationally expensive. Imagine a castle on the back of a giant turtle. Every time the turtle moves, every single object in the castle needs to update its position and rotation! Most transform hierarchies I've encountered are the result of haphazard evolution over the course of several games, and I'm curious how far you could optimize things if you built a system from the ground up with performance in mind.
Things to Explore:
What is a transform?
What are homogeneous coordinates?
What is sheer?
What is a quaternion? How is it helpful?
Is a quaternion better or worse than a matrix?
How can multithreading be used?
What is meant by Array of Structures VS Structure of Arrays?
How can Single Instruction Multiple Destination (SIMD) operations be used?
What is the cache?
3. Christine Lee, The topology and combinatorics of the Turaev genus
The Turaev genus is a measure of how far a knot or a link is from being alternating (alternately going over and under as we walk on a strand). It is defined by minimizing the Euler characteristic over all the diagram-surfaces of a link, as is often the story with knot invariants. Unfortunately, this makes the Turaev genus very difficult to compute for any given link. There is also no known algorithm.
In this project, we will study the Turaev genus using the colored Jones polynomial, a ``quantum” link invariant that is a closed relative of the Jones polynomial. The colored Jones polynomial is defined based on diagrams and has had great success being applied to the question of finding crossing numbers, another link invariant that is defined by minimizing the number of crossings over all possible link diagrams.
Our goal is to develop bounds on the Turaev genus by constructing diagrams and manipulating them using Reidemeister moves, and computing the colored Jones polynomial. We will keep an eye out for finding obstructions to a link admitting a certain type of diagrams as well.
Programming skills are particularly welcome for the purpose of having the computer help construct interesting examples to work with!
4. David Snyder, Topology—topic to be added at a later date
5. 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 interesting, as would be answering many questions, such as: Can complements of superminimal prime graphs have induced circles of arbitrary lengths? Is a positive proportion of all minimal prime graphs superminimal?
6. Steven Hoberman, Creating a More Flexible Test for the Population Mean -- a New Approach to Nonparametrics
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. This assumption is less costly in larger samples, but in smaller samples little progress has been made for many years in testing the mean of a distribution when the data may or may not be 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 or skewed. We also explore applications of this distribution to detecting relationships in data sets where there may be multiple observations on the same unit, and in data sets where multiple statistical tests are considered simultaneously.
7. Suho Oh, Weyl groups of type D and Pattern avoidance
Permutations of [n] = {1,...,n} have lot of interesting stuff going on. There is a generalization of this coming from something called Weyl groups (used in physics) of type B and D, each encoded by signed permutations and even-signed permutations respectively. Signed permutations are just permutations with signs (+ or -) attached to each entry, and even-signed permutations are when the number of signs is negative.
There is an interesting operation on Weyl group elements, called the Demazure product coming from 0-Hecke algebra. This is described using simple transpositions. Before, it was not known how to describe this using just the permutations. With the 2022 HSMC students, we were able to describe this mystical operator just using permutations and a fun new elementary operation called hopping (this paper recently accepted to Electronic Journal of Combinatorics!) We also proved that something similar works for type B as well. Our first goal, would be to extend this to type D, the even-signed permutations. After that, we will investigate the connections to pattern avoidance.
8. Ivan Ojeda-Ruiz, Advancing Fairness in Natural Language Processing: Unveiling and Mitigating Bias Beyond Gender
ChatGPT has brought the importance of studying Natural Language Processing (NLP) and its impact on the world. This project in fairness for NLP delves into the diverse dimensions of bias, expanding beyond the conventional focus on gender. We shall examine bias types prevalent in current machine learning systems, emphasizing the harmful consequences of real-world deployment. Formal definitions of fairness will be explored, showcasing the variance in models resulting from different fairness definitions. Metrics employed to measure bias in word embeddings or models will be scrutinized.
While existing research predominantly targets gender bias, this project proposes a shift towards investigating bias due to race, religion, or class. This is a challenging endeavor, given the difficulty in measurement and potential lower prevalence in commonly used corpora. Moreover, a critical gap exists in assuming a priori knowledge of harmful biases in corpora, necessitating novel methodologies to uncover subtle biases. Current metrics, such as the Word Embedding Association Test (WEAT), focus on gender associations without precisely determining the inherent bias, urging the development of new metrics that better reflect human bias in predictions.
9. Ivan Ojeda-Ruiz, Generalizing PageRank Fairness
Search engines like Google are essential to facets of life. This research investigates the generalization of fairness definitions in the context of PageRank algorithms (which are the basis of the Google search engine). Building on existing notions such as 𝜙-fairness, we explore alternative fairness metrics and their impact on the performance and behavior of PageRank algorithms. By extending the scope of fairness definitions, we aim to gain insights into the nuanced aspects of algorithmic fairness in link analysis. The study involves rigorous evaluation of the algorithms under different fairness definitions, providing a comprehensive understanding of how varying fairness criteria influence PageRank outcomes. The findings contribute to the broader discussion on fairness in network analysis algorithms, opening new avenues for refining fairness definitions and enhancing the applicability of PageRank algorithms across diverse scenarios.
10. Ivan Ojeda-Ruiz, Examining the Influence of Fairness Metrics on K-Means Clustering Algorithm
This project aims to thoroughly investigate the integration of fairness metrics into the widely used K-Means clustering algorithm, with a specific focus on metrics such as balance and social fairness cost. As the deployment of machine learning models becomes increasingly pervasive, the importance of addressing bias and ensuring fairness in algorithmic decision-making is paramount. This research will explore the nuanced dynamics between fairness metrics, particularly balance and social fairness cost, and their impact on the clustering outcomes produced by the K-Means algorithm. By conducting a comprehensive analysis, we aim to elucidate how these fairness considerations influence the overall equity of the model. The inclusion of specific fairness metrics like balance and social fairness cost will provide practical insights for researchers and practitioners striving to develop more equitable and unbiased machine learning systems.
11. Young Ju Lee, Development of a Fast Training Algorithm for Logistic Regression.
Supervised Learning is an important category of Machine Learning. It uses labeled datasets to train or learn a function that can predict or recognize patterns. In this project, we shall consider one of the supervised learning model called, the logistic regression. The logistic regression aims at training a function that classifies a multiple patterns. Our focus is to develop a fast training algorithm. Using the framework of subspace correction method for convex minimization, we shall attempt to develop efficient training algorithm for logistic regression. These algorithms will then be used to train a well-known benchmark datasets called, MNIST, a handwritten digits (0 to 9) with their labels.
12. Xiaoxi Shen, Comparing Performances of Convolutional Neural Networks and Recurrent Neural Networks on Genetic Data.
Many diseases have a complex genetic etiology. With advancements in artificial intelligence, convolutional neural networks and recurrent neural networks have been widely used to extract complex relationships between inputs and outputs. However, their applications to genetic data are still limited. In this project, we will use Keras to build some convolutional neural networks and recurrent neural networks to analyze genetic data, and their performances will be compared. No prior knowledge of Keras is needed, and basic Keras will be taught during the math camp.
13. Burcu Cinarci, Derived Lengths and Character Degrees of Solvable Groups
Throughout this project, we deal with finite groups, and G always denotes a finite group. Let ρ be a C-representation of G. The character of G afforded by the representation ρ is the function χ : G −→ C defined by χ(g) = trace(ρ(g)) for all g ∈ G. Note that χ(1) is called the degree of χ and the set of all irreducible character degrees of G is denoted by cd(G). The character theory of finite groups, a cornerstone of modern algebra, delves into the profound relationships between a group’s algebraic structure and its associated complex-valued characters. The structure of a finite group and the irreducible character degrees of this group have certain relationships. For example, G is an abelian group if and only if every irreducible character degree of G is 1. The famous Ito’s theorem states that for any prime p, every character degree in cd(G) is not divisible by p if and only if G has a normal abelian Sylow p-subgroup.
Let G be a solvable group, and let dl(G) denote the derived length of the group G. One famous open problem in character theory is whether the derived length of any solvable group G can be upper-bounded by the cardinality of the set of irreducible character degrees, that is, whether dl(G) ≤ |cd(G)| holds. This problem is known as the Isaacs-Seitz conjecture or the Taketa problem. Despite researchers working on it for more than half a century, the problem remains unsolved to this day. In this project, we will give an introduction to character theory of finite groups and will provide some sufficient conditions for the Taketa problem.
14. Nathan Warshauer, Video Game Development
Explore video game development by creating a simple game of your choice or add new features or cosmetics to an existing gaming platform. Immerse yourself in the gaming industry’s development technologies and begin your journey in the field of video game development. Brief introduction to game development followed by a project of your choice.
15. I choose not to participate in a research project
