# Summer Research in Mathematics

## Math Research Experience 2021

The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2021. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

## Projects

• Computing invariants for expansive dynamical systems. Informally, a dynamic system is any physical system that evolves with time (e.g., a pendulum, a planet orbiting the sun, the weather, etc). From a more mathematically precise perspective, one can consider a function mapping a space to itself. For example, f(x)=x^2 defined on the set of real numbers. Using this formulation, time is represented by iterating the function. In the example f(x)=x^2, if the initial value is 2, then after one unit of time, the value is f(2)=4, after two units of time, the value is f(f(2))=f(4)=16 and so on.

We will study a class of expansive dynamical systems that are examples of chaos. Roughly speaking chaos is characterized by the property that "the present determines the future, but the approximate present does not approximately determine the future." Our investigation will be example based. Each class of examples will involve ideas from an area of math that goes beyond pure dynamical system theory: subshift of finite type will involve combinatorics and graph theory, solenoids will involve covering spaces and number theory, and Anosov diffeomorphisms will involve linear and abstract algebra. Our main objective will be to study and explicitly compute invariants associated with these dynamical systems. These invariants will allow us to distinguish different dynamical systems. A prototypical example of an invariant is the length of time it takes a planet to orbit the sun, which can be used to distinguish the planets in the solar system.

Prerequisites: At least two of Math 2135, Math 3001, and Math 3140 (in particular, no experience with dynamical systems is required)
Dates: May 31 – July 9
Mentor: Robin Deeley

• Invertibility of Randomly Perturbed Matrices. Many fundamental algorithms in data science and machine learning run quickly in practice, but at the same time are known to have horrible running times in the worst case. One explanation of this phenomenon is that noise tends to transform even the worst-case scenarios into more tractable problems. For algorithms that involve matrices, one of the theoretical obstacles is the singularity of the matrix. In this project, we will investigate how noise affects the singularity of a fixed matrix. The singularity of random matrices is a rich and active area of research that involves techniques from combinatorics, geometry and probability. Depending on interest, we can pursue several directions. One is to investigate the restrictions necessary on the fixed matrix and another is to broaden the class of noise that can be tolerated. Furthermore, in practice there are often structural constraints on the fixed matrix and the noise matrix. For example, in the modeling of physical systems, the matrices are often required to be symmetric and when interactions are local, the matrices can take on a band structure. Each of these structural classes poses new obstacles and presents new opportunities.

Prerequisites: Math 4510, Math 2135, Math 3001
Dates: May 10 – June 18
Mentor: Kyle Luh

• Topological Data Analysis and Applications to the Sciences. Two of the main challenges of modern data analysis are how to separate noise and outliers from significant parts of data samples and the high dimensionality of data. One of the most recent and powerful methods to attack these problems comes from topology. It is the goal of this REU to study methods from topological data analysis (TDA) and apply them to examine data sets coming from the sciences. In particular it is planned to consider energy landscape as they appear in chemistry and see in how far TDA can help to better understand them. During the REU an review of simplicial complexes and their homology will be given, afterwards we will study the main tool of topological data analysis, persistent homology. We will also do some hands-on experimental mathematics using software tools for TDA and computational homology. Part of the REU will also be to learn the software/computational tools we need (from MATLAB, python, topology toolkit etc...). In the end we hopefully can apply all this to some more realistic data scenarios coming from chemistry.

Prerequisites: knowledge of linear algebra, fundamentals of simplicial complexes and willingness to learn some abstract math and do computer assisted computations
Dates: May 10 – June 30
Mentor: Markus Pflaum

• What is an n-dimensional shuffle? A matrix is fundamentally a finite sequence of its entries. However, by organizing the sequence into rows and columns we give it a useful additional structure. Suddenly we can talk about upper-triangularity, whether a matrix is diagonal, what its transpose might be, and a procedure for multiplying matrices in a way that encodes linear function composition. This project explores how these concepts might look like in larger dimensional arrays (known as tensors). For example, a cubic matrix might have rows, columns, and a notion of depths.

A problem of particular interest for us examines the situation where we have a set of matrices and a set of allowable matrix operations (i.e. scaling, adding a row to another, etc.). We say two matrices are equivalent if one can get from one to the other using only allowable operations. Classifying the resulting equivalence classes (for example, giving their number) is a classical problem that is provably impossible in some cases, but in other cases has surprisingly elegant solutions. For example, if we allow both scaling columns and adding columns to the right on strictly upper-triangular matrices, then the equivalence classes are indexed by card shuffles. This result is particularly mysterious, and the goal is to better understand it by replicating it in higher dimensional tensors.

Prerequisites: Math 2135 or equivalent; Math 3140 is recommended, but not required
Dates: May 10 – June 25
Mentor: Nat Thiem

• Stability conditions for genus 2 curves. In this project, we will be interested in the ways Riemann surfaces can degenerate. If one starts with a smooth Riemann surface and deforms it continuously, it eventually breaks into pieces that are not smooth. For a given degeneration, there is not a unique choice of singular limit. We would like to find a way of choosing a singular limit for these degenerations. One way has been known for about 50 years: we can ask for the singularities to be nodes. But we've been discovering other choices lately.

Last year, the summer REU catalogued all ways of choosing singular limits of Riemann surfaces of genus 1 (the genus is the number of holes in the surface). This year, we'll be working on curves of genus 2.

We won't actually be using complex geometry to study this problem, because it can be re-encoded as a combinatorial problem using tropical geometry. We will need to find piecewise linear functions with certain specific properties on metric graphs.

Prerequisites: Not all required, but desirable: some algebra (finitely generated, torsion-free abelian groups), and maybe some category theory; some basic combinatorial notions (i.e., graphs); mathematical maturity (ideally having taken a graduate math class); some programming skills (Python)
Dates: May 10 – June 25
Mentor: Jonathan Wise

• Derived Poisson Structure on sl_2. Poisson algebras appear naturally in many areas of mathematics such as geometry, topology, representation theory and mathematical physics. Classically, the notion of a Poisson structure is defined for commutative algebras. It is therefore, natural to ask how the notion of a Poisson structure may be extended to arbitrary associative algebras. It turns out that the naive extension of the definition of a Poisson structure is too restrictive. As a remedy, in 2005, Crawley-Boevey proposed the notion of an H_0-Poisson structure which could be viewed as the correct definition of a Poisson algebra from the point of view of noncommutative algebraic geometry. In 2012, Y. Berest, X. Chen, F. Eshmatov and A. C. Ramadoss introduced the notion of a derived Poisson structure which can be viewed as a higher homological extension of the H_0-Poisson structure.

It is known that the universal enveloping algebra of a semisimple Lie algebra or an abelian Lie algebra equips with a canonical derived Poisson structure. In this project, we will review the basic knowledge of noncommutative Poisson structure, cyclic/Hochschild homology, etc., and try to compute this derived Poisson structure on U(sl_2) explicitly.

Prerequisites: Solid knowledge of linear algebra; experience with programming would be helpful; preference will be given to students who have taken abstract algebra and have learned basic Lie algebra
Dates: May 10 – June 25
Mentor: Yining Zhang

## Dates

• Starting date

• May

• June

• July

• May 10 (6 weeks)

• Luh

• May 10 (7 weeks)

• Wise