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)
The celebrated Robinson-Schensted-Knuth (RSK) insertion algorithm is a bijection between the symmetric group and pairs (P,Q) of standard Young tableaux (arrays of positive integers with increasing rows and columns) of the same shape. The tableaux P and Q are called the insertion tableau and the recording tableau of the corresponding permutation.
A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. At each state, we have a collection of boxes with each integer from 1 to n assigned to a unique box. A box-ball system will reach a steady state after a finite number of steps. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition of the box-ball system. The minimum number of time steps needed to go from a permutation w to a steady state in a box-ball system is called the steady-state time of w.
The main goal of this REU project is to use the pair (P,Q) of a permutation w to study the box-ball system containing w. For example, we can attempt to prove the conjecture that the steady-state time of a permutation is determined by its recording tableau Q. In addition, it is known that the insertion tableau P of a permutation is equal to its soliton decomposition SD if and only if P and SD have the same shape. We can count the number of such permutations in the symmetric group Sn. Such permutations seem to satisfy certain pattern avoidance, so we can try to classify these permutations using pattern avoidance.
Potentially helpful concepts that may be part of the background reading are the weak partial order on the symmetric group, Knuth equivalence, skew tableaux, permutation statistics (for example, descent set and Greene’s theorem), pattern avoidance in permutations, and generating functions.
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)
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 Rieck, Dhananjay Bhaskar (Yale)
Dynamics, Geometry, and Number Theory (formerly 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)