2021 Project Descriptions

This is the list of projects for 2021.  These should each be looked over before applying.  The application form must be filled out by February 15, 2021.

Spinors and Graph Theory

Spinors are fundamental objects in mathematical physics, and they play a crucial role in quantum mechanics. The space of spinors is formally defined as the fundamental representation of the Clifford algebra. The objective of this project is to study spinors in graph theory, arising from a combinatorial model of the Dirac equation. We will describe explicitly the space of solutions of the Dirac equation on a finite graph, and compare it with known results on the graph version of the Schrodinger equation. We will also explore a gluing formula for the Clifford algebras associated to spinors on graphs.
Mentor: Ivan Contreras (Amherst College)

Cambrian lattice on plabic graphs

The Tamari lattice, introduced by Tamari in the 1960s, is a sublattice and also a lattice quotient of the weak order on the symmetric group. It can be realized as a partial order on triangulations of the polygon with n boundary vertices labeled 1 through n in clockwise order. The Tamari lattice is a special case of c-Cambrian lattices which are sublattices and lattice quotients of the weak order on a finite Coxeter group. A plabic graph is a graph embedded in a disk with colored internal vertices as well as n boundary vertices labeled 1 through n in clockwise order. Plabic graphs were introduced by Postnikov in 2006 to study the totally nonnegative Grassmannian and since then have been connected to various areas of mathematics and physics. There is a natural Tamari lattice structure on a class of plabic graphs. A generalized plabic graph is a plabic graph drawn on a disk with boundary vertices labeled 1 through n but not necessarily in clockwise order. In this REU project, we will study the Cambrian lattice structure on classes of generalized plabic graphs.
Mentor: Emily Gunawan (University of Oklahoma)

Cylinder configurations on flat surfaces

A major theme in low-dimensional geometry and topology is to understand when given combinatorial or topological data can be realized geometrically. For example, any simple (non-self-intersecting) curve on the flat torus R^2/Z^2 can be realized as the core curve of an embedded Euclidean cylinder. For a given flat cone structure on a higher genus surface S (aka the structure of a translation surface) this isn’t always true: there are curves on S that can never be realized as cylinders, even if you allow geometry-preserving deformations of S. Even more strangely, there are examples of configurations of curves that can never be realized by cylinders on any translation surface. This project will explore criteria for when configurations of curves can be realized as cylinders on translation surfaces, focusing both on combinatorial/topological constraints and geometric constructions.
Mentor: Aaron Calderon (Yale)

Hypergraph Containers

We will discuss problems related in some way or another to the beautiful theory of hypergraph containers, a new approach to counting problems in extremal combinatorics which was developed by Balogh-Morris-Samotij and Saxton-Thomasson in the last few years. A collection of containers in a hypergraph H is a collection of subsets of the vertex set such that every independent set in H lies inside a container. Perhaps a bit counterintuitively, it has been discovered that, whenever the edges in H are (sufficiently) evenly distributed, it is always possible to find such a collection for which the containers are not big (being rather close to the independent sets they each capture) and the size of the collection itself is quite small (compared to what one would normally expect). This observation has led to several significant breakthroughs in combinatorics, most notably in the field of extremal graph theory, where the method originates, but also in the adjacent areas of additive combinatorics and discrete geometry, where many problems can also be naturally reformulated in terms of questions about independent sets in certain hypergraphs. The main goal of this project will be to find new applications of these recent ideas (in several different directions).
Mentor: Cosmin Pohoata (Yale) and Dmitriy Zakharov (MIPT)

Diffusion Geometry and Topology 

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 of the data. However topological data analysis methods, which are really a combination of data geometry and computational homology, have been slow in adoption. Here we seek to tackle several of the challenges involved in making these methods effective and efficient for use. The topics are in three main categories. First, there are general aspects of persistent homology that can be vastly improved by investigation into different underlying metrics, filtrations or incorporation of time-varying data. Second, we have found diffusion-based manifold learning very effective in high dimensional data, and we explore its use as a scaffold for topological analysis. Finally, we focus on trajectories or progression created by  several processes such as cellular differentiation or neural network training and how characterizing these trajectories can allow us to categorize them, for example predict when a neural network will generalize well.
Mentors: Smita Krishnaswamy, Jeff Brock, Ian Adelstein, Bastian Reick, Dhananjay Bhaskar (Yale)

Problems in Homogeneous Dynamics

Dynamical systems study the long-term behavior of a transformation acting on some kind of mathematical space. A Lie group is a group that is also a smooth manifold, i.e. a space that looks locally like Euclidean space. The field of homogeneous dynamics studies the actions of subgroups of a Lie group G on the quotient of G by a discrete subgroup. For an example of this kind of system, let v be a nonzero vector in R^2 and consider the left multiplication action of the linear subgroup generated by v on the torus, R^2/Z^2. It turns out that the trajectory of every point on the torus under this flow is closed and homeomorphic to a circle if and only if the entries of v are rationally dependent, and dense otherwise. In this example, we see the interplay between dynamics (the flow), geometry (the torus), and number theory (rationality).
In this project, we will study other problems motivated by number theory and geometry using the tools of homogeneous dynamics.  In particular, we will look at gap distributions for the slopes of saddle connections on translation surfaces, a problem that can be translated into a question about a particular flow on the quotient of SL(2,R) (the space of 2x2 real matrices with determinant one) by a discrete subgroup.  We will develop the necessary background to understand these terms and attack the problem, and the precise direction of research will be guided by the interests of the group.
Mentor: Taylor McAdam (Yale)