Application Information
APPLICATIONS FOR SUMRY 2025 ARE NOW OPEN
The SUMRY program is now open to both currently enrolled Yale undergraduates AND non-Yale undergraduate students. We are not able to support non-Yale international students.
SUMRY 2025: June 2, 2025 to July 30, 2025 at Yale University in New Haven, CT
The due date for applications is February 3, 2025. Accepted students should expect to hear back from us by middle to late February. Students will need to accept or decline our invitations by the national REU deadline of March 8, 2025.
Yale students - you need only fill out the application form available on the side bar. You do not need to submit a letter of recommendation.
Non-Yale students - you need to both fill out the application form on the side bar AND apply via mathprograms. The mathprograms application requires a letter of recommendation from a math faculty member, to be uploaded directly via mathprograms.
SUMRY will run the following projects during summer 2025
Vertex domination-critical graphs and well-dominated graphs
Given a graph G with vertex set V(G), a subset D of V(G) is a dominating set of G if every vertex of G is either in D or adjacent to a vertex in D. The minimum cardinality among all dominating sets of G is the domination number of G, denoted g(G). G is said to be well-dominated if every minimal dominating set has cardinality g(G). Additionally, a vertex v in V(G) is said to be critical if g(G-v) < g(G) and G is vertex domination-critical if every vertex in G is critical. The properties of being well-dominated and vertex domination-critical seem completely unrelated in the literature. In fact, finding classes of graphs which are vertex domination-critical is very difficult and only a few are known. All of the examples of vertex domination-critical graphs are in fact well-dominated which begs the question of whether this is always true. In this project, we try to extend the known classes of vertex domination-critical graphs to obtain new examples. We also search for vertex domination-critical graphs which are not well-dominated, or alternatively show that there are none. Moreover, we study the class of well-dominated graphs to determine if they contain potential counterexamples to Vizing’s conjecture regarding the domination number of the Cartesian product of two graphs.
Mentor: Kirsti Kuenzel and Ryan Pellico (Trinity College)
Capillary Water Waves
Fast-moving ripples appear when the wind blows through the surface of the water: they are examples of the capillary water waves aroused by the surface tension. In this project, we will explore the mathematics behind the linearised capillary water wave equations, and find creative ways to stabilise those waves in a 2D aquarium or in a 3D lake, with some numerical simulation.
Discipline: Pure and Applied Analysis, Partial Differential Equations.
Mentors: Ruoyu P. T. Wang (Yale)
Machine Learning in Computational Fluid Dynamics
Computational Fluid Dynamics (CFD), extensively utilized in industries such as aerospace, automotive, and power generation, plays a vital role in modern engineering and scientific research. By leveraging advanced mathematical models and numerical methods, CFD provides profound insights into fluid flow behavior across various applications. Its primary strength lies in its ability to mathematically analyze and optimize fluid flow phenomena, thereby further understanding the mechanisms and thus helping to enhance system performance and safety. Big data and machine learning are catalyzing profound transformations, swiftly revolutionizing the tools and methodologies employed by applied scientists. As continuous developments in CFD methods improve our ability to collect high-quality data, machine learning tools become increasingly viable and promising also in CFD. This proposed project aims to explore how machine learning can be integrated and combined with CFD results/methods to solve complex problems, especially the high-speed turbulent boundary layer in aerodynamics. The objective of this project is to study and significantly improve the state-of-the-art analytical analysis and numerical modeling of complex vortex structures in high-speed flows.
In this project, we will explore how to use PCA more effectively to understand complex flow fields, especially complex vortex structures in high-speed flows, based on existing CFD data and numerical solvers (Navier-Stokes Equations). In addition, we will also explore how to use deep learning, especially those for time series, to explore and predict nonlinear dynamics in complex flow fields and their impact on the evolution of vortex structures.
Mentors: Yonghua Yan, Caixia Chen (Jackson State University) and Sam Panitch (Yale)
Superalgebras and Polynomial Spaces
The special linear Lie algebra sl(2) is a ubiquitous entity of representation theory and often realized as two-by-two matrices with trace zero. Another realization of sl(2) leads to Laplace’s equation, the solutions of which are used in animation. Our project opens with sl(2) and brings in its super analogue, the orthosymplectic Lie superalgebra osp(1|2). Now osp(1|2) contains the matrix realization of sl(2) in the form of three-by-three super matrices and provides a key to entering the world of super representation theory. We will take steps toward new realizations of osp(1|2) by familiarizing ourselves with super linear algebra and superalgebras. We will highlight osp(1|2) as a space of linear maps on polynomials, namely, differentiation and multiplication with respect to the indeterminant. The foundations of the project will rely on solidifying topics in linear algebra (countable bases), ring theory (ideals), and calculus (general Leibniz rule) for grasping osp(1|2) and opening pathways to answer several questions about Lie superalgebras and spaces of polynomials. One particular problem is solving for certain linear maps on the finite-dimensional spaces which decompose the polynomials and polynomial-like spaces based on degree.
Advancing further, the project generalizes in several directions, including the realization of higher-dimensional orthosymplectic Lie superalgebras as polynomials in several indeterminants or the introduction of reduction superalgebras associated with osp(1|2) through hands-on computation. Participants may continue on to provide concrete applications such as the determination of polynomial solutions to Dirac’s equation.
Big picture: Bring forth notions of supermathematics from linear algebra, answer some questions, and pose many more. Summer goal: Solve a physics equation using a superalgebra and/or solve for an infinite family of matrices which govern an infinite-dimensional representation of a superalgebra.
Mentors: Dwight Anderson Williams II (Morgan State University) and Mengwei Hu (Yale)
Graph-based Manifold Learning
With the advent of high dimensional high throughput data in many fields including biomedicine, social science, physics, finance, etc, there is an increasing need to understand the topology (shape, structure) and geometry (intrinsic distance, curvature) of the data. This process starts with the manifold hypothesis, that high dimensional data actually live on or near a much lower dimensional manifold (or surface). One can then study the geometric and topological properties of this manifold to reveal important features of the data. This project will focus on graph based techniques , where the manifold is imputed from high dimensional data via a weighted graph. The graph Laplacian and graph signal processing will allow us to probe manifold features.
Mentor: Smita Krishnaswamy, Ian Adelstein, Rahul Singh, Siddharth Viswanath (Yale)