### Research Projects

#### Introduction

Lie algebras and Lie groups are important structures in mathematics whose representation theory has led to many fascination discoveries related to number theory and combinatorics. In particular, many integer partition identities can be interpreted in a representation-theoretic setting, and many new partition identities have been discovered through the study of Lie algebra representations. In this project, we aim to extend work on “principle subspaces” of certain representations of infinite-dimensional Lie algebras. In particular, the principle subspaces in this project can be viewed as quotients of polynomial rings, and in the simplest examples given a Lie-algebraic interpretations to the Rogers-Ramanujan identities.

This project focuses on two topics, briefly outlined below:

##### Lie Algebras

A Lie Algebra is a vector space endowed with a non-associative binary operation, called the “Lie bracket”, which is bilinear, alternating, and satisfies the Jacobi identity. An easy example of such a structure is R^3 equipped with the cross-product as the Lie bracket. Another easy example is the vector space of n x n matrices with Lie bracket given by [A,B] = AB-BA, known as the commutator of A and B. Lie algebras were originally discovered and studied alongside Lie groups, and they have many deep connections to various areas of mathematics and physics.

In many ways, the study of Lie algebras is analogous to the study of groups. For example:

- Group actions play a prominent role in group theory and in describing groups, and when these groups act nicely on a vector space V, we call V a “representation” of the group G. In essence, G is identified with a group of invertible linear transformations on V. Similarly, Lie algebras act on vector spaces V, and we call these vector spaces “representation” of the Lie algebra V. That is, we can represent the Lie algebra as endomorphisms of V.
- Simple groups are those groups which have no non-trivial normal subgroups. In a similar vein, a simple Lie algebra is on which has no non-trivial ideals.

##### Integer Partitions

Given a natural number n, a “partition” of n is a way of writing n as a sum of smaller positive integers where order doesn’t matter. For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1. We say that 4 has 5 partitions. It is an interesting question to study partition identities where the “parts” (the positive integers making up the partitions) satisfy certain conditions. For example, in a famous result, Euler shows that the number of partitions of n where all the parts are odd is equal to the number of partitions of n where the parts are distinct. Other famous partition identities include the Rogers-Ramanujan identities, the Gollnitz-Gordon identities, and many, many more.

#### The Project

This project will primarily focus on the infinite-dimensional affinization of the Lie algebra sl(2), where sl(2) is the Lie algebra of 2 x 2 matrices with trace 0, whose Lie bracket is the commutator given above. In particular, we will study certain important representations of this affine Lie algebra, along with important subspaces of these representations called principle subspaces. Principle subspaces of representations of affine Lie algebras have received considerable study in the past 20 years, and their connections to partitions and other q-series identities have been widely explored. Of particular interest is the fact that these infinite-dimensional principle subspaces can be decomposed naturally into a direct sum of finite-dimensional subspaces whose dimensions are related to various partition and q-series identities arising in number theory.

In this project, we aim to extend certain results on principle subspaces found in the work of Capparelli, Calinescu, Lepowsky, and Milas. Namely, these authors showed that, by constructing certain exact sequences among principle subspaces, the Rogers-Ramanujan and Rogers-Selberg recursions can be given an interpretation in terms of Lie algebra representations. In this project, our aim is to extend and generalize their exact sequences among these principle subspaces, and to explore their connections to other works on q-series related to the Rogers-Ramanujan identities. A particularly nice feature of our setting is that everything about these principle subspaces can be rewritten in the language of polynomial rings and their quotients. In this project, we will work with this interpretation to extend these results.

#### Background

Students are expected to have at least one course on Abstract Algebra and one course on Linear Algebra. This project will draw heavily from certain aspects of Linear Algebra and will make heavy use of polynomial rings and their quotients.

#### How to Apply

#### Introduction

In the field of number theory, the p-adic valuation ν_p(x) is a useful tool in studying the divisibility of an non-zero integer x by powers of a given prime p. The p-adic valuation of x is defined to be the largest positive integer l such that p^l divides x. An interesting question is to analyze a sequence of valuations {ν_p(x_n) : n ∈ N}. Valuations have been studied for the Stirling numbers, the Fibonacci numbers, the alternating sign matrices, and polynomials. [1, 2, 3, 4]

[1] Medina et al., *Periodicity in the p-adic valuation of a polynomial*, J. Number Theory, 2017, 41—54.

[2] A. Berribetzia, L. Medina, A. Moll, V. Moll, L. Noble, *The p-adic valuation of Stirling numbers*, J. Algebra Number Theory Acad. 1, 2010, 1—30.

[3] E. Beyerstedt, V. Moll, X. Sun, *The p-adic valuation of ASM numbers*, J. Integer Seq. 14, 2011, 11.8.7

[4] A. Byrnes, J. Fink, G. Lavigne, I. Nogues, S. Rajasekaran, A. Yuan, L. Almoodovar, X. Guan, A. Kesarwani, L. Medina, E. Rowland, and V. Moll, *A closed-form solution might be given by a tree. Valuations of quadratic polynomials*, 2015.

#### The Project

In this project we will focus on 2-adic and 3-adic valuations of polynomials f(n) with integer coefficients. The sequence {ν_p(f(n)) : n ∈ N} could be periodic or unbounded. For example, for p=2 and f(n)=n^2+3 the sequence of valuations would be alternating zeros and twos, while {ν_p(n^2+7) : n ∈ N} is an unbounded sequence whose terms form a set N\{1, 2}. The 2-adic valuations of {n^2+7} could be represented in an infinite p-adic tree (Fig. 1). The nodes split based on the parity of n. To learn more about the construction of the trees, see [4].

The specific direction of our research will depend on the background and general interest of the students in the group. However, the outline is that we will begin by reviewing the known results for quadratic polynomials, such as f(n)=n^2+a for example. Then the goal is to consider special cases of quadratic polynomials, and determine whether their sequences of valuations are periodic or unbounded. For periodic sequences, it might be interesting to study the length of a period as well as to explore a converse question: Suppose the tree terminates after seven levels, what can be said about the polynomial f(n)?

We will examine the sequences using the trees, an algebraic approach, and numerically. For the computational component, a knowledge of Sage or Mathematica would be helpful. We will also work with the power series representation of p-adic numbers.

#### Background

The students, who wish to apply, should have completed at least an introduction to proofs course (see for example Discrete Math, Math 236W, on Ursinus’s course website) and Calculus II. A course in Number Theory or Abstract Algebra would be helpful. But do not be discouraged if you haven’t had these courses yet: a lot of the background needed for the project could be picked up before or at the beginning of the program. Enthusiasm is a must.

#### How to Apply

#### Introduction

Ordinary webcam videos capture a surprising amount of information, even if it is visually imperceptible. Some recent work [1, 2] showed that incredibly simple techniques based on the Fourier transform can be used to visualize of repetitive hidden motions in these videos, such as color changes and arterial contraction/expansion during bloodflow, breathing in neo-natal units, and mechanical vibrations (Click here and here to see examples from this work).

Even more recently, RGBD (color + depth) 3D cameras are becoming ubiquitous in phones and other devices (e.g. the PrimeSense Kinect and the Tango Tablet). The project mentor (Chris Tralie) has undertaken initial experiments which have revealed that the same subtle changes are present geometrically in this 3D streaming data, as shown below

### 3D Heartbeat Experiment

Below is a video in which the project mentor is sitting still not breathing. So little is happening that it almost appears to be a still image:

However, there is more happening than meets the eye. Below is the same video after the phase-based video magnification techniques of [2] have been applied. The project mentor’s heartbeat is clearly visible as motion around the clavicle and neck.

Below is a 3D version of the same video resulting from the depth stream of the Intel RealSense sensor. As in the ordinary video, very little motion is present, other than some noise of the sensor:

However, after applying some basic periodic amplification to the xyz positions of each point, it is possible to see the entire neck pulsating with the heartbeat

### Reptile Breathing Experiment

Below is an ordinary video of a bearded dragon sitting very still. Again, it almost appears as a still frame

The same can be said of the corresponding 3D depth video

However, when amplifying the depth video, it is possible to see the lizard’s belly expanding and contracting during breathing.

It should be noted, however, that the naive technique applied above also amplifies noise, and it requires the objects to be very still. These are two of the issues that we will tackle in this REU.

[1] Wu, H. Y., Rubinstein, M., Shih, E., Guttag, J., Durand, F., & Freeman, W. (2012). Eulerian video magnification for revealing subtle changes in the world.

[2] Wadhwa, N., Rubinstein, M., Durand, F., & Freeman, W. T. (2013). Phase-based video motion processing. ACM Transactions on Graphics (TOG), 32(4), 80.

#### The Project

The goal of this REU project will be to develop a functioning prototype of a system which takes as input 3D data in the form of streaming depth images, and which outputs a video of a 3D model with hidden periodic motions accentuated geometrically. Applications of such a technology could include, for instance, health monitoring (e.g. monitoring bloodflow in contracting/expanding arteries) and integrity checks of mechanical systems (e.g. hidden oscillations that could grow into unstable chaotic motion).

The bulk of the work will be programming in the Python language, but students will also learn myriad math concepts which enable this technology. Students will start by learning the Discrete Fourier Transform and multiresolution image pyramids to replicate the video-based results in [1]. Students will then apply these techniques to depth videos to replicate the project mentor’s examples on videos of mostly still objects, using naive amplification of raw 3D position. They will build on the project mentor’s techniques by exploring other geometric tools for amplification, including curvature amplification and multiresolution smoothing to avoid amplifying noise. Finally, students will work to extend the tool to factor out global nonrigid motion [3, 4], so that it is possible for subjects of the 3D videos to move around during a scan.

[3] Newcombe, R. A., Fox, D., & Seitz, S. M. (2015). Dynamicfusion: Reconstruction and tracking of non-rigid scenes in real-time. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 343-352).

[4] Slavcheva, M., Baust, M., & Ilic, S. (2018). SobolevFusion: 3D reconstruction of scenes undergoing free non-rigid motion. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 2646-2655).

#### Background

Though it is expected that students will have taken 1-2 computer science courses at the college level before embarking on this REU project, no prior knowledge of Python will be assumed, and students will learn the basics of numerical analysis in numpy “on the job.”

#### How to Apply

*This REU program is supported by the National Science Foundation (NSF), grant number 1851948.*