Well-edge-dominated graphs
A set F of edges in a graph G is an edge dominating set if every edge of G that is not in F shares a vertex with at least one edge in F, and F is a matching if no pair of edges in F share a common vertex. The minimum cardinality of an edge dominating set and the maximum cardinality of a matching are important graph invariants with applications to network flow, chemical properties of molecules, and scheduling, to name a few. In 2010 Frendrup, Hartnell, and Vestergaard first initiated the study of well-edge-dominated graphs which have the property that all minimal edge dominating sets have the same cardinality. Furthermore, they studied equimatchable graphs which have the property that all maximal matchings have the same cardinality. Since a maximal matching is always a minimal edge dominating set, the set of all well-edge-dominated graphs is contained within the set of all equimatchable graphs. In 2022, it was shown that there are only three well-edge-dominated graphs of girth at least 4 that are not bipartite. In this project, we begin by studying well-edge-dominated graphs that contain exactly one 3-cycle in hopes of completing the classification of all well-edge-dominated graphs.
Mentor: Kirsti Kuenzel and Ryan Pellico (Trinity College)
Finite quotients of groups in low-dimensional geometry
One way to understand an infinite group, such as the fundamental group of a manifold, is by considering how the group acts on finite sets (if it does at all). These actions on finite sets give us finite quotients of the group. Our project will be an open-ended exploration of finite quotients of groups that arise in low-dimensional geometry and topology. In particular, we will be interested in fundamental groups of 3-manifolds, which determine not only the topology, but also the geometry of the 3-manifold. We will begin by thinking about finitely presented groups and groups associated to topological spaces, before thinking about finite quotients of these groups and how to extract meaningful information from these finite quotients.
Mentors: Andrew Yarmola, Franco Vargas Pallete, Tamunonye Cheetham-West (Yale)
Geometric 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 an embedding technique for learning the data manifold called an autoencoder, and will explore how geometry can inform and improve these embeddings. Applications to computational neuroscience and neural representations will be explored.
Mentor: Smita Krishnaswamy, Ian Adelstein, Dhananjay Bhaskar (Yale), Kaly Zhang (MILA) and Tim Rudner (Columbia)
Application Information
UPDATE: We are now finished recruiting for this summer. Thanks to all who applied and best of luck with your mathematics!
Thanks to support from the NSF the SUMRY program is now available both to currently enrolled Yale undergraduates AND non-Yale undergraduate students.
SUMRY 2024: June 3, 2024 to July 31, 2024 at Yale University in New Haven, CT
APPLICATIONS HAVE CLOSED. The due date for applications is February 2, 2024. 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, 2024.
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.