Research
General Information
Each project will consist of a small research team consisting of typically 2-4 undergraduates, a graduate mentor, and a faculty mentor. The graduate mentor and undergraduates will meet on a weekly basis, with full team meetings every few weeks as determined by the faculty mentor.
Undergraduates who have been accepted into a project must sign up for the 3-credit "Purdue Experimental Math Lab" course (currently listed under MA 490) and must pledge that they are able to dedicate 10 hours of effort per week to the project. For information on how to apply, click on the tab 'Join PXML' at the top of the screen.
Spring 2025 Projects
Statistics and simulations in the Kardar-Parisi-Zhang class
Faculty Mentor: Prof. Christopher Janjigian
Level: Advanced
Skills required: Experience coding in Python or MATLAB is preferred.
Meeting times: Availability Friday afternoons is requested.
Description: Kardar-Parisi-Zhang statistics are conjectured to describe the fluctuations of many roughly growing interfaces, ranging from the edge of forest fires to the boundary of bacterial colonies. This project will do simulations of simple models which lie in the class to explore some of these statistical properties. There are a variety of projects available depending on student background.
Some potential options include the following:
-I would like to eventually write up lecture notes or a book introducing some of the models in the KPZ class. For that project, it would be useful to have simulations to illustrate many basic properties of models in the KPZ class. These projects are fairly straightforward coding exercises but offer some room for creativity in terms of finding good ways to present ideas visually.
-The “multi-type stationary distributions" in the model of last-passage percolation can be represented as a sequence of queues run in tandem. In this queuing system, the “dual service time” plays an important role in many geometric arguments, but is not well-understood. The goal of this project would be to deliver some exact simulations of these dual service times. This project will involve working with Poisson processes and working with a useful technique for exact simulation known as coupling from the past. This is more challenging, but I have some example code written in Matlab that can serve as a guide.
-Many growth models in this class start to look approximately statistically stationary (meaning that certain statistics stop changing much) after being run for a relatively short time. Previous students have done simulations to illustrate this phenomenon. There are natural follow-up questions about the rate of convergence to stationarity, as well as the statistics of the convergence times. There is some existing Python code that students can look at here to get a sense of what to do.
For all of the above projects, I would like to explore with students how we can use large language models like GPT-o1 to improve and streamline the process of generating simulations. Students will be given access to a node on the Gilbreth cluster to work on these projects.
Fundamental Representations and Stable Conjugacy Classes in Finite and p-adic Groups
Faculty Mentor: Prof. Daniel Johnstone
Level: Intermediate/Advanced
Skills required: Taking or having taken MA 450 or MA 453 is strongly preferred but not required. General programming acumen may be helpful but is not required.
Meeting times: Availability MWF in the afternoon is requested.
Description: We can easily determine what field extension the roots of a quadratic polynomial lie in simply by considering the polynomial’s discriminant; this, equivalently, allows us to determine what type of maximal torus a regular semisimple element of the group GL_2 lies in. Moving beyond the case of the general linear group GL_n (or even for the general linear group for n>2), however, this classification problem becomes quite rich, requiring a deep understanding of Galois theory and Representation theory. The goal of this project is to study this phenomenon over a number of classical and exceptional Lie groups over finite and p-adic fields, notably for the groups SP_4 and G_2.
Numerical Methods for Some Eigenvalue Problems
Faculty Mentor: Prof. Nicholas McCleerey
Level: Intermediate
Skills required: No specific courses are required, but familiarity with Linear Algebra, ODEs, and Computational Methods is necessary. Familiarity with PDEs and analysis is a plus, but not necessary.
Meeting times: Ability to meet on Wednesdays, Thursdays, or Fridays is preferred.
Description: A common problem in partial differential equations (PDEs) is the eigenvalue problem. The original example is to consider a vibrating string, when two endpoints are fixed (e.g. on a violin). The eigenvalues correspond to the natural frequencies at which the string ``prefers" to vibrate, and can be computed by solving a second-order ODE with fixed boundary values.
When you move to higher dimensions, the ODE is replaced with a PDE. For simple PDE, we can compute the eigenvalues over the unit ball by hand. For not-so-simple ones however, this is not tractable, and we need to resort to numerical methods to make progress.
The project I would like to consider is to attempt to compute the first eigenvalue of the Monge-Ampere (MA) equation on the unit ball. The MA equation is a fundamental non-linear PDE appearing in convex and complex geometry; despite this, it seems to be that no eigenvalues are known for this operator, on any domain.
On the unit ball, symmetry reduces the MA equation to a non-linear ODE. Unfortunately, this ODE appears to be too non-linear for basic solvers to evaluate directly. We believe that an iterative approach, based off of the linearization of the PDE, and not the ODE, should be able to produce a solution. If successful, there are a number of possible other directions which could be pursed, e.g. computing the second eigenvalue (if it even exists), or computing first eigenvalues for related geometric PDE.
Visualizing the computational complexity of knots and links
Faculty Mentor: Prof. Eric Samperton
Level: Intermediate
Skills required: Completion of a course in Linear algebra, at least at level of MA 265, ideally at the MA 353 level. Intermediate programming experience, ideally in Python.
Meeting times: Availability before 2:30 on MWF is requested.
Description: Have you ever been annoyed trying to untie headphone cables after stuffing them in your pocket? (Maybe not since most headphones are wireless these days! But almost everyone has at least been annoyed by having to remove a bad knot from shoe laces.) It turns out that you're not wrong to be annoyed, since there are various situations where we can give mathematical proofs that problems related to untying things are "hard" according to computational complexity theory. This project will be an introduction to such ideas.
Specifically, we will attempt to build a computer program that will allow us to visualize the NP-hardness of the trivial sublink problem. This problem asks the following: given a link in the 3-sphere with many components and an integer k, decide if there exists a k-component sublink that is trivial. The NP-hardness of this problem can be proved easily by reducing from the independent set problem in graph theory. We will try to build a computer program that implements this reduction, and is user friendly. By the end of the semester, I hope to have a program that allows a user to input a graph and then outputs a picture of a link who trivial sublinks are identified with the independent sets of the graph. If we finish this sooner than expected, we will try to do similar things for other problems.
This project will require some previous experience with computer programming (preferably Python).
Exploring Quantum Graph Invariants
Faculty Mentor: Prof. Thomas Sinclair
Level: Intermediate
Skills required: Completion of a course in proof-based linear algebra, MA 353 or equivalent. Experience programming either in Python or MATLAB is strongly preferred.
Meeting times: Availability to meet Tuesday/Thursday before 2:00pm is preferred.
Description: The Lovasz theta number of a graph is a famous computable graph invariant which can be used to estimate two important, but hard to compute, graph invariants, the chromatic number and the clique number. (Duan, Severini, and Winter, 2013) introduced a quantum Lovasz theta number for "quantum graphs", subspaces of the n-by-n complex matrices which contain the identity and are closed under conjugate transposition. A related quantum graph invariant "lambda" was introduced in (Araiza, Griffin, and Sinclair, 2024).
In this project we will investigate whether these two quantum graph invariants agree. The lambda invariant can be used to define a "relative Lovasz theta number" for an inclusion of graphs. We will also aim to explore the properties of this relative graph invariant.
Optimizing Matrix Factorizations for Large-Scale Data Analysis
Faculty Mentor: Prof. Mahesh Sunkula
Level: Intermediate
Skills required: Completion of a course in linear algebra at the 200-level or above. Completion of at least one proof-based mathematics course. Experience programming in Python.
Meeting times: Open to meeting based on availability of participants.
Description: Matrix factorizations, including LU, QR, and SVD, play a crucial role in linear algebra and data science, providing the foundation for solving linear systems, dimensionality reduction, and various data analysis tasks. However, these traditional methods face significant challenges when applied to large, sparse, or structured matrices frequently encountered in machine learning, social network analysis, and genomics. This project aims to investigate the limitations of existing factorization techniques while focusing on developing or optimizing algorithms that deliver better performance on large-scale datasets. Students will work on implementing these algorithms, testing their effectiveness on real-world datasets, and evaluating their computational efficiency and accuracy. Time permitting, this project will also involve creating new methods or improving existing ones, specifically designed for different types of matrices, such as low-rank or sparse matrices.
Exploring computational geometry
Faculty Mentor: Prof. Shuyi Weng
Level: Beginner
Skills required: Completion of MA 265 or equivalent. Programming experience, though not language specific.
Meeting times: Availability Tuesday/Thursday after 10:30 is preferred.
Description: Although geometry is as old as mathematics itself, computational geometry was only really cherished and developed in the late 20th century, as “computer” gradually transitioned from a type of job to a type of machine much more capable at the job. The field has since expanded greatly, and the new connections to areas of mathematics, computer science, and engineering areas seem only to be accelerating. Nevertheless, computational geometry still harbors some of the most difficult unsolved questions in mathematics.
The project will begin with experimental mathematics, including an introduction to computing and computational geometry overall. Students will spend time reading about the theory of computational geometry, implementing algorithms to verify stated results, and exploring computational tools to provide evidence for new conjectures.