Application Information
APPLICATIONS FOR SUMRY 2026 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 2026: June 1, 2026 to July 24, 2026 at Yale University in New Haven, CT
The due date for applications is February 2, 2026. 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, 2026.
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 2026
The Riordan group and RNA applications
Algebraic combinatorics involves the use of techniques from algebra, topology, and geometry in the solution of combinatorial problems. Because of this interplay with many fields of mathematics, algebraic combinatorics is an area in which a wide variety of ideas and methods come together. Riordan arrays appear in algebraic combinatorics and are useful for proving combinatorial sums and identities. They also appear in various counting problems in enumerative combinatorics. A certain subset of Riordan arrays called proper Riordan arrays, otherwise known as Riordan matrices, form the Riordan group, an infinite noncommutative matrix group, which is the main combinatorial device used in this research. The Riordan group can be characterized as being algebraic and combinatorial. Thus, as a subfield of enumerative and algebraic combinatorics, the Riordan group brings together a wide variety of combinatorial methods and mathematical ideas. This project will investigate open problems related to the Riordan group and its applications to RNA secondary structure sequences. The optimal prediction of RNA secondary structures is an important problem in the area of discrete mathematical biology, subfields of bioinformatics and molecular biology.
Mentors: Asamoah Nkwanta (Morgan State University)
Steinberg module of the braid group
The braid group on n strands, denoted B_n, is the group whose elements are equivalence classes of n-braids and whose group operation is composition of braids. The braid group on n strands can also be viewed as the mapping class group of the closed 2-disk with n marked points, denoted Mod(D_n). Bieri and Eckmann introduced duality groups to describe groups that have a relationship between homology and cohomology groups. Using the arc complex of the disk, Harer proved that for n≤3 the braid group B_n, is a duality group with dualizing module the Steinberg module St(B_n). Furthermore, he established a finite B_n-module resolution of St(B_n), in which using the last two terms of this finite resolution one arrives at a finite free B_n-module presentation of St(B_n), called the Harer presentation. This presentation has 1/n ((2n-2)¦(n-1)) generators and ((2n-2)¦n) relations.
Preliminary work done shows that we can reduce the Harer presentation down to a single generator and ⌈(n+1)/2⌉ relations, with small indices of n=3,4,5 reducing the number of relations down to two. In mathematics, cohomology of mapping class groups is of interest because it can used to define an invariant of surface bundles. In this project we will build upon our preliminary results and use our simplified presentation of St(B_n), to compute the top cohomology of the braid group with any coefficients.
Mentors: Anisah Nu’Man (Spelman College)
The intersection of vertex domination-critical graphs and edge domination-critical graphs
Given a graph G with vertex set V (G), a set D ⊆ 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 γ(G). A vertex v ∈ V (G) is said
to be critical (with respect to domination) if γ(G − v) < γ(G) and G is vertex domination-
critical if every vertex in G is critical. Additionally, G is said to be edge domination-critical
if for every nonadjacent pair u, v ∈ V (G), γ(G + uv) < γ(G). The properties of G being
either vertex domination-critical or edge domination-critical have been studied separately in
the literature. Recently, it was shown that every graph of order at most nine that is both
vertex domination-critical and edge domination-critical is also well-covered, i.e. every maximal
independent set in G is maximum. We will study whether this is in fact true for all graphs in
the hopes that at the very least, our efforts will produce an infinite class of graphs that are
both vertex and edge domination-critical.
Mentors: Kirsti Kuenzel and Ryan Pellico (Trinity College)
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 and Ian Adelstein (Yale)