# Summer Research in Mathematics

## 2021 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).

Mentor: Robin Deeley. Undergraduates: A. Fahrner, J. Giornazi. First Year Graduate Students: M. Reardon, M. Welsh.

• 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.

Mentor: Kyle Luh. Undergraduates: R. Huq, A. Yu, R. Vogel First Year Graduate Students: S. Shortt.

• 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.

Mentor: Markus Pflaum. Undergraduates: M. Walker, I. Jorquera, X. Liu, S. Basrur. First Year Graduate Students: A. Doumont, J. Macula, E. Montelius.

• 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.

Mentor: Nat Thiem. Undergraduates: E. Young, C. Schwarzman, C. Weisburg, F. Collins. First Year Graduate Students: J. Gensler, N. Jamesson.

• 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).

Mentor: Jonathan Wise. First Year Graduate Students: E. Orvis, R. Storms, M. Watson.

• 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.

Mentor: Yining Zhang. Undergraduates: E. Senkoff, D. Mann, E. Hodges.

## 2020 Projects

• Equivariant Cohomology of Topological Spaces. In this project, we will first learn about G-spaces (for a finite group G) and the associated equivariant cohomology groups (key words are G-CW Complexes, Bredon cohomology and Mackey functors). This is a relatively young field and there are a lot of G-spaces whose equivariant cohomology has yet to be computed. In this project, we will explore new computations. It would also be interesting for students to explore ways to illustrates their computations using computer visualization tools, but this part of the project will be completely up to the participants. Prerequisite: MATH 6220 Topology 2.

Mentors: Agnès Beaudry and Cherry Ng. First Year Graduate Students: R. Chaiser, J. Moreno.

• Rewiring matrices using combinatorial tools. Linear algebra teaches us that every invertible matrix can be built up from the identity matrix using row and column operations. An advantage to this approach —which we will use extensively— is that we can turn the process into a mathematical game by associating each matrix to a network, where row and column operations correspond to particular rewiring choices.

If we restrict the rewiring choices that we are allowed to use, then arbitrary invertible matrices no longer come from the identity, and the set of all invertible matrices split into subsets which share increasingly many characteristics (as the number of restrictions grow). A fundamental problem is to find a representative of each subset which is emblematic of these shared characteristics, as the identity matrix is for the set of all invertible matrices. Unfortunately, classifying these representatives can be difficult, and some important cases of this problem are even "wild," a mathematical synonym of "impossible." In this project, we work with similar, but possible classification problems which come out of representation theory.

We will use tools from linear algebra and discrete math. As we make progress, we'll also discuss the implications of our work in group theory and combinatorics, but the only prerequisite in these subjects is a willingness to learn them. Prerequisites. Math 2135 with preference to students who have taken Math 3140 or Math 3170.

Mentors: Lucas Gagnon and Nat Thiem. Undergraduates: J. Gaiter, J. Hawkes, T. Kauffman, A. Nelson First Year Graduate Students: R. Deland, J. Willson, B. Wilson.

• Sums of random polynomials. A random polynomial is a polynomial whose coefficients are random variables. One goal of studying random polynomials is to understand the "typical" (or "average") behavior of a polynomial. For example, the roots of a random polynomial describe the typical behavior of roots of all polynomials.

In this project, we will study the roots of sums of random polynomials. We will consider questions such as: How many real roots does a random polynomial have? What is the behavior of a typical root for the sum of two polynomials? We will use tools from analysis and probability theory. Part of the project will involve numerical simulations using software such as Mathematica, but no prior programming knowledge is required. Prerequisites. Some experience with probability and/or analysis.

Mentor: Sean O'Rourke. Undergraduates: F. McWilliams, K. Schneider First Year Graduate Students: G. Lopez, L. Lorenzo.

• Topology, Data Analysis and Chemistry. 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, topoology 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.

Mentor: Markus Pflaum. Undergraduates: E. El Sai, P. Gara, S. Serra First Year Graduate Students: H. Jordan.

• Regression models using pathological data. Given a set of data, the canonical regression model is obtained using ordinary least squares. But how do we proceed in the analysis if anomalies such as colinearlity between variables, badly conditioned or singular coefficient matrices, or outliers exist? In this project, we will use split-apply-combine strategy for data analysis (often used in machine learning), and implement various well-known models such as ridge regression, lasso, elastic net, and importantly support vector regression. Our approach will involve writing or using existing built-in solvers to investigate various regression models. Additional topics such as classification, clustering, and density estimation may come into play. Prerequisites. Linear algebra; R / Python programming. Preference will be given to students who have a knowledge of optimization (e.g., taken MATH 4120).

Mentor: Divya Vernerey. Undergraduates: M. Howard, W. Qu, A. Sanchez.

• Classifying tropical surfaces. Tropical geometry refers to a not-yet-precisely-understood analogue of algebraic geometry in which rational functions are replaced by piecewise linear functions. Tropical curves are now understood very well, so the next frontier is tropical surfaces.

There are two classes of tropical surfaces that we will investigate: abelian surfaces and K3 surfaces. In algebraic geometry, these are the 2-dimensional analogues of elliptic curves. Unlike general tropical surfaces, abelian and K3 surfaces have underlying 2-dimensional (real) manifolds (a torus and a sphere, respectively), which makes their geometry particularly elegant. In addition to the manifold structure, they have a flat integral structure (the flat structure of the K3 surface will have 24 singular points because of the curvature of the sphere). The goal of this project will be to classify these flat integral structures (as well as some related data) as well as possible. Prerequisites. The only truly necessary prerequisite is some comfort thinking about lattices in real vector spaces and some basic topology. However, it will be helpful to be able to rely on tools from abstract algebra and algebraic topology (monoids, exact sequences, cohomology). We can learn what we need from these subjects as we go, but it might be difficult to learn all of them without having had a course. Background in algebraic geometry is not necessary.

Mentor: Jonathan Wise. Undergraduates: T. Aldape, M. Foucher, S. Scheeres First Year Graduate Students: B. Kuo, C. Meredith, A. Neff.

• Fully commutative Kazhdan–Lusztig cells in Coxeter groups. Coxeter groups form a large family of groups with natural connections to algebra, combinatorics and geometry. In particular, the partitioning of a Coxeter group into certain subsets called "Kazhdan–Lusztig cells" is very interesting to representation theory, a branch of mathematics which studies symmetry in linear spaces. A Coxeter group is always generated by a small set, so we can view the generating set as an alphabet and represent the group elements by words on the alphabet. In this project we will study a type of elements called "fully commutative" or "FC" elements. These elements possess especially nice combinatorial properties: the set of words representing a fixed FC element will always satisfy a certain nice condition, and to each FC element we may often associate two combinatorial gadgets—a partially ordered set called a "heap", and a diagram called a "Temperley–Lieb diagram".

The goal of the project is to use heaps and Temperley–Lieb diagrams to describe "FC cells", the Kazhdan–Lusztig cells consisting of FC elements. For example, we will seek to generalize the following fact to more Coxter groups: the familiar symmetric groups are a type of Coxeter groups, and two FC elements in a symmetric group are in the same "left Kazhdan–Lusztig cell" if and only if their associated Temperley–Lieb diagrams share the same south side (each diagram consists of a north and south side); consequently, we can use the possible sides of diagrams to parameterize and count left FC cells. Prerequisites. Having taken Math 3140 or seen symmetric groups would be ideal, but having taken Linear Algebra may also suffice. Experience with Python or a similar programing language would be beneficial.

This project will likely build upon a similar Math Lab project on FC elements of Coxeter groups, and we will spend a significant amount of time writing code to study heaps and Temperley–Lieb diagrams, but experience with Coxeter groups is not necessary.

Mentor: Tianyuan Xu. Undergraduates: R. Castro (highschool), J. Courtney, T. Magnuson, N. Schoenhals First Year Graduate Students: C. Meadors.

## 2019 Projects

• Extensions of Modules over the Steenrod Algebra. The Steenrod algebra is a large non-commutative ring which arises in the study of invariants of topological spaces. It gives rise to powerful tools for studying spaces, which have important applications in topology, differential geometry, and even mathematical physics.
In this project, we will learn about the Steenrod algebra, learn some of the fundamental concepts in a field called homological algebra, and then study questions which arise in doing homological algebra over the Steenrod algebra. There will be the possibility of exploring software used to do computations in this area.
If you like algebra, are not afraid of computations, and find the webpage http://www.math.wayne.edu/~rrb/art/ intriguing, then this project is for you.Prerequisites. Math 3140, and no fear of linear algebra.
Mentor: Agnès Beaudry. Undergraduates: N. Downey, H. Lembeck, Q. Sabathier. First Year Graduate Students: E. Knutsen.

• Graphs, quadratic forms, and period maps. Period maps provide a method of parameterizing certain algebraic and geometric data arising from the Hodge theory of complex projective manifolds. In extending period maps to the boundary of the domains of definition, one is led to study cones of quadratic forms. Specifically, one wants to know whether a given monodromy cone is contained in a "standard" cone of quadratic forms. In certain cases, these monodromy cones can be described from the combinatorics of graphs. In this project, we will investigate this type of situation for the Torelli map and for the Prym map, where the monodromy cones arise from graphs, and where there are several open questions. Our approach will primarily be computational, working out examples, and writing some computer code to investigate. The point is that from previous work there is an expectation as to what graphs will lead to monodromy cones that do not lie in a "standard" cone, and we would like to confirm this computationally in some further examples. Prerequisites. Abstract Algebra 1 MATH 3140 (although Linear Algebra MATH 2135 may be sufficient), and experience coding in Python (although a little coding experience in another language, such as C, or C++, may also be sufficient).
Mentor: Sebastian Casalaina-Martin. Undergraduates: H. Fontana.

• Discrete geometry and applications to redistricting. Gerrymandering refers to the practice of drawing legislative districts so that one political party wins a disproportionate number of seats relative to their share of the electorate. But how can we tell whether districts have been drawn fairly? This is a legal question and, increasingly, a mathematical one, but the mathematical tools used to measure gerrymandering are relatively new and still evolving rapidly.
One common legal requirement is that districts should be "relatively compact," but the word "compact" has no standard, agreed-upon definition. Many different metrics for compactness have been proposed and used in legal contexts; most of them are based on geographic information about districts, such as area, perimeter, etc. These metrics have varying strengths and weaknesses - and importantly, they often disagree about which shapes are more compact than others.
Recently, Duchin and Tenner have proposed a discrete approach to quantifying compactness; their preprint may be found on the arXiv here: https://arxiv.org/pdf/1808.05860.pdf. This project will build on the ideas in this paper, with the goal of better understanding various discrete metrics for compactness, and how they compare to each other as well as to existing metrics. We will start by generating data: We will compute a variety of compactness metrics for a wide variey of district shapes and look for patterns. Depending on what we find - and on the interests of the group - we'll decide together where to go from there! Prerequisites. Calculus 3, Linear algebra, and some experience with Python. (Minimal experience okay as long as you're willing to learn more!).
Mentor: Jeanne Clelland. Undergraduates: N. Bossenbroek, T. Heckmaster, A. Nelson, J. VanAusdall. First Year Graduate Students: P. Rock.

• Pressure in the Navier-Stokes equations. The Navier-Stokes equations are one of the fundamental equations of fluid mechanics. The equations are a system for the vector field, the velocity of the fluid, and for the scalar, which is the pressure. In this project, we will review the known mathematical properties of the pressure and investigate if they are preserved when the fluid flows on curved domains.

All students interested in Analysis and PDE are encouraged to apply. Prerequisites. Preference will be given to students with MATH 4001 or equivalent, and ideally with Math 4230, and 4470. Familiarity with manifolds is not required.
Mentor: Magdalena Czubak. Undergraduates: A. Koek, R. Mike. First Year Graduate Students: I. Miller.

• Promises and Constraints. Constraint Satisfaction Problems (CSP) are everywhere in computer science and the real world: given certain constraints, is there a solution that satisfies them? Particular examples are satisfiability of Boolean formulas, graph colorability, solvability of linear equations, Sudoku, etc. It has recently been shown that any such CSP can be solved efficiently (in polynomial time) if and only if it has some non-trivial underlying symmetry; else it is hard (NP-complete). The critical observation here is that symmetry operations give rise to algebraic structures that can be used to compute solutions; without that one is basically left with trial and error to find solutions.
In this project we consider the even more general Promise Constraint Satisfaction Problems (PCSP): There the question is about how to approximate solutions for CSP, more specifically, to distinguish between whether a given instance of a problem has a solution or even an easier version with relaxed constraints cannot be satisfied. For example, given a finite graph, decide whether you can color its vertices with 3 colors such that adjacent vertices have different colors or not even 10 colors are enough to do that. The `promise' guarantees that each input satisfies exactly one of the two alternatives to be considered. We want to investigate symmetries of specific instances of PCSP and use them for computing solutions. This combines algebra and a some computational complexity. Prerequisites. Ideally MATH 3140, but MATH 2135 and one or more MATH 3000+ level courses may suffice.
Mentor: Peter Mayr and Athena Sparks. Undergraduates: E. El Sai, G. Deng, P. Nakkirt. First Year Graduate Students: T. Manders.

• Topology and Data Analysis. 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 get into topological data analysis (TDA), study its foundations and then apply it to actual situations in mathematics, science or technology where data sets appear which are susceptible to TDA. During the REU an introduction to 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, etc...). If there is interest, we could also write our own code for computational homology/TDA in python or haskell. In the end we hopefully can apply all this to some more realistic data scenario from science or technology. Prerequisites. Some basic knowledge in Linear Algebra and willingness to learn some abstract math and do computer assisted computations.
Mentor: Markus Pflaum. Undergraduates: S. Berman, Y. Cheng, J. Cherry, T. Fara, P. Gara, S. Zhang. First Year Graduate Students: C. McCranie.

• On a tightrope between a rock and a hard place: Schur—Weyl duality on unipotent groups. Linear algebra teaches us that every invertible matrix can be constructed using a combination of four elementary operations: add a column to another column, add a row to another row, scale a row and exchange two rows. The group generated by all these operations is known as the general linear group and its conjugacy classes (or similarity classes) are well understood. However, the group generated by only row and column additions is a much more mysterious group and has conjugacy classes that are famously wild (technobabble for provably impossible to understand).
A fundamental result in representation theory of finite groups is that the size of the group can be written as a sum of squares with one square for each conjugacy class in the group. For our wild group, this means the order is a sum of squares, but no-one knows how many squares we use for this sum. The goal of this project is to combinatorially construct another algebraic matrix ring whose dimension is a sum of squares with one square for each conjugacy class. This ring also lives on the wild side, but perceives the difficulties from an orthogonal point of view. By understanding the structure of this ring we hope to gain valuable insights over the structure of our original group.
We will develop the necessary representation theory during the program, and the (so far unknown) combinatorics will be dictated by the solution. However, familiarity with groups and comfort with linear algebra will be critical. Prerequisites. Math 3140 and no fear of linear algebra.
Mentor: Nat Thiem. Undergraduates: J. Kishinevsky, N. Miesch, C. Walia. First Year Graduate Students: H. Davenport, C. Eblen, M. Muro.

• Positivity and Finite Groups. Given a complex-valued function on a group, there are two notions of positivity: point-wise positive (a "local" property) and positive definite (a "global" property). We will assume here that our groups are finite, although most of what we will say applies in great generality. If a group is abelian, these notions of positivity of a function are Fourier transforms of each other. Also on any group there are two notions of "product" of functions: pointwise product (a local notion) and convolution (a global notion). It is true that the pointwise product of positive definite functions is again positive definite, making the collection of positive definite functions on a group a convex, commutative, semigroup, P(G), even if the underlying group is not abelian. It is a theorem that P(G) is a complete invariant of G. It turns out that the algebraic properties of G are reflected in "geometric" properties of P(G), in the sense that P(G) is a partially ordered cone. This structure is connected to many topics which are open for exploration. For example, the structure theory of finite groups and the representation theory of finite groups are two immense ar- eas where known facts and unsolved problems can be viewed from a new perspective. There are notions from both algebra and analysis that make sense in this context and suggest areas of study. Even the famous unsolved problem: "the Riemann Hypothesis," has an equivalent formulation in the context of the finite symmetric groups. The details of the projects to be pursued will depend on the detailed interests of the students and the depth of their preparation, in particular, knowledge of linear algebra and at least the definition and basic properties of finite groups are essential. Prerequisites. Math 3140.
Mentor: Martin Walter. Undergraduates: D. Salas. First Year Graduate Students: R. Dyer.

## 2018 Projects

• Simplicial Complexes and Connectivity. An n-simplex is an n-dimensional triangle. For example, a 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is a filled in triangle and a 3-simplex is a solid tetrahedron, etc. A simplicial complex is a space built in a very rigid way from n-simplices. We will explore the differences between various simplicial complex arising in the study of an algebraic phenomenon called homological stability. Will ask questions about their connectivity. This will involve learning some concepts from category theory and algebraic topology.
Mentor: Agnes Beaudry. Prerequisite. MATH 3140 or MATH 2135. Undergraduates: McCranie, Meszar, Willson, First year graduate students: Davis.

• Graphs, quadratic forms, and period maps. Period maps provide a method of parameterizing certain algebraic and geometric data arising from the Hodge theory of complex projective manifolds. In extending period maps to the boundary of the domains of definition, one is led to study cones of quadratic forms. Specifically, one wants to know whether a given monodromy cone is contained in a "standard" cone of quadratic forms. In certain cases, these monodromy cones can be described from the combinatorics of graphs. In this project, we will investigate this type of situation for the Torelli map and for the Prym map, where the monodromy cones arise from graphs, and where there are several open questions. Our approach will primarily be computational, working out examples, and writing some computer code to investigate. The point is that from previous work there is an expectation as to what graphs will lead to monodromy cones that do not lie in a "standard" cone, and we would like to confirm this computationally in some further examples.
Mentor: Sebatian Casalaina-Martin. Prerequisite. Ideally MATH 3140, but MATH 2135 may be sufficient. Some experience writing computer code would be beneficial. Undergraduates: Alizadeh, Cates, El Sai, First year graduate students: Wynne.

• Invertible random matrices. A random matrix is a matrix whose entries are random variables. One goal of Random Matrix Theory is to study the "typical" (or "average") behavior of matrices. For example, the eigenvalues of a random matrix describe the typical eigenvalue behavior of all matrices. This project aims to study how often random matrices are invertible and to understand properties of the inverse. We will consider questions such as: What is the probability that a random matrix is invertible? When it is invertible, how large (or small) can the entries of the inverse be? Part of the project will involve numerical simulations using software such as MATLAB, but no prior programming knowledge is required.
Mentor: Sean O'Rourke. Prerequisite. Probability theory and linear algebra. Undergraduates: Atkinson, Collins, Dudley, Pan, First year graduate students: Campbell.

• Liber Mathematicae and Mathematics presentation on the web. Properly presenting Mathematics on the web is still a challenging task. The proposed REU addresses this problem by extending the Liber Mathematicae project and writing appropriate code in ruby on rails. Mathematically it is planned to work on the visualization of singularities within Liber Mathematicae.
Mentor: Markus Pflaum. Prerequisite. Both acquaintance with fundamental algebra and programming. Undergraduates: Zhang, First year graduate students: Loucks.

• Power bases for rings of integers. The rings of integers of algebraic number fields are one of the fundamental objects of study in algebraic number theory. Questions about these rings encompass many of the problems that motivate current research.
For this project, we are interested in when the integral basis of this ring admits a very simple description. Specifically, our question is: When does the ring of integers admit a power basis, a basis consisting of 1 and powers of some other element? Number fields whose ring of integers has this property are called monogenic. The problem of classifying such number fields was posed in the 1960's by Helmut Hasse and is often called Hasse's Problem.
Hasse's problem is far from solved. Most attacks on Hasse's problem have involved classifying monogenic or non-monogenic families of a given degree. Our project will follow in this tradition. Using a tool called the Montes algorithm; we hope to classify infinite families of irreducible monic integer polynomials of degree 5 and greater that generate monogenic fields. Not only will this work help solve Hasse's problem, but the rings of integers we classify will supply other mathematicians with examples that are efficient for computational purposes.
This project may continue during the school year, depending on the interest of the participants.
Mentor: Kate Stange and Hanson Smith. Prerequisite. A strong grade in MATH 3140 required. MATH 4140 even better. Undergraduates: Ibarra, Lembeck, Ozaslan.

• Tinkering with canonical forms. Equivalence relations on sets can give a mathematical way to encode when we believe elements behave the same. Similarity of matrices is a fundamental such relation that preserves eigenvalues, order, invertibility, and many other important properties. If we restrict our attention to invertible n by n matrices with eigenvalue 1, then similarity classes of matrices are in fact indexed by integer partitions of n. This project seeks to better understand the relationship between similarity and another equivalence relation that has classes indexed by set partitions of {1,2,..., n}. At this point we know that the two relations are completely incompatible, but we'd like a more nuanced description. That is, we'd like to convert the basic linear algebraic problem into a nice combinatorial description with broader representation theoretic implications.
Mentor: Nat Thiem. Prerequisite. Ideally MATH 3140, but linear algebra plus one or more MATH 3000+ level courses is probably sufficient. Undergraduates: Bosnich, Bossenbroek, Ekstrom, Martinez, First year graduate students: Gagnon.

• Tropical geometry and lines on cubic surfaces.Tropical geometry is a crude approximation to algebraic geometry that is very effective for making calculations. The idea is to look at a geometry problem from very far away, where the geometry starts to resemble a stick figure. The rules of conventional geometry translate into some unusual combinatorial conditions, different subsets of which are called tropical geometry. By way of tropical geometry, one can sometimes turn difficult geometric problems into tractable combinatorial ones.
(The stick figures of tropical geometry closely resemble Feynman diagrams. This is --- I am told --- not an accident, and that tropical geometry is algebraic geometry seen from far away in the same way Feynman diagrams are string theory seen from far away.)
With hindsight, the advent of tropical geometry can be dated to the 1960s and 70s, and the subject took off more than a decade ago, but its foundations are still very unstable. One of the goals of this project is to get a sense of where these foundations might be shaky and need attention.
Tropical geometry gives insight into algebraic geometry, but it isn't a substitute. Sometimes the solutions to tropical problems come out differently from their algebraic analogues. One such problem is counting the lines on a cubic surface. Cayley proved in 1849 that a surface defined by a cubic polynomial will always contain exactly 27 straight lines, provided it has no singularities. On the other hand, singular cubic surfaces can have infinitely many lines.
In tropical geometry, cubic surfaces often do have 27 tropical lines, but Vigeland constructed examples of cubic surfaces with infinitely many lines. Are Vigeland's examples singular? According to one definition of singularity they are not, but the definitions are still in flux, and we will test for singularity using a more refined definition.
This project will involve a lot of algebra with polynomials, and some convex geometry. We will probably need to write computer programs to facilitate some of our calculations. Mentor:Jonathan Wise.First year graduate students: Kraus, Lee, Quartin.

## 2017 Projects

• An action of the binary tetrahedral group and an application to algebraic topology. A certain group, called the Binary tetrahedral group, is of particular interest to some algebraic topologists. It appears as the group of automorphisms of an elliptic curve defined over the "field with four elements." There is a way to extract an action of this group on a certain power series ring. This action is part of the action of a much larger group, and is one of the key players in beautiful story called chromatic homotopy theory These come into play when computing certain topological invariants. We will study these group actions and use them to make some algebraic computations that have topological meaning. In the process, we will be learning about the p-adic integers, algebra in power series rings, elliptic curves and formal group laws.
Mentor: Agnes Beaudry. Prerequisite. Math 3140. Undergraduates: Downey, McCranie, Meszar, Rock. First Year Graduate Students: Riddle.

• Local and global wellposedness of wave equations. Partial Differential Equations are used to model the world around us. In this project we investigate mathematical properties of wellposedness of the solutions related to systems of equations appearing in the electromagnetism and/or the Standard Model of particle physics. Wellposedness questions ask for example if solutions exist, do they exist globally in time or if they end with a blow-up in some specific mathematical sense.
Mentor: Magda Czubak. Prerequisite. Math 3001. Undergraduates: Ekstrom, Gossett, Richman, Tauber. First Year Graduate Students: Stocker.

• Study of invariants of singularities. The Milnor number is an invariant of a function germ and provides information of the singularity type the germ represents. In this proposed REU project certain singular spaces will be studied, their Milnor numbers computed and it will be attempted to construct further invariants of singularities. Depending on the students interests and knowledge, the project can go more into a computational or more into a theoretical direction.
Mentor: Markus Pflaum. Prerequisite. Math 3001, Math 3450. Undergraduates: Macmaster.

• The Geometry of Number Fields. Motivated by cryptographic applications, we investigate some statistics of number fields. In particular, we consider the ring of integers of a number field embedded into R^n using the Minkowski embedding. We will ask some questions about the geometry of the lattice and the relationship to its ideal sublattices. The project will have a strong experimental/computational component. We will use Sage mathematics software to generate data and conjectures. No pre-requisites besides the algebra pillar sequence; expect to learn some algebraic number theory, some lattice geometry, and how to use Sage and program in Python.
Mentor: Kate Stange. Prerequisite. graduate student only. First Year Graduate Students: Arpin, Ornstein, Wheeler.

• Unipotent Rook Polytopes. Unipotent polytopes are a family of geometric objects (such as the hypercube) that arise naturally out of algebra and combinatorics. Being at the intersection of so many areas gives many possible tools to study the structure of these polytopes. Past REU projects have explored various families of unipotent polytopes from numerous points of view: random walks, face lattice, representation theory, etc. One of the more tractable families seems to be the family of rook polytopes, whose vertices are given by placements of non-attacking rooks on an mxn chessboard. This summer we will use all the hard work of past summers to finally understand the algebraic and combinatorial foundations of rook polytopes.
Mentor: Nat Thiem. Prerequisite. Math 3140. Undergraduates: Evans, Fontana, Sain, Willson. First Year Graduate Students: Dubeau.

• Numerical Solutions of Partial Differential Equations. This project aims to understand the physics behind elliptic, parabolic or hyperbolic PDEs and then to numerically solve them. Convergence and stability properties of algorithms will also be studied. Applications to PDEs are in steady state flow of inviscid fluids, heat conduction, and waves on a drumhead.
Mentor: Divya Vernerey. Prerequisite. Math 3430, and knowledge of programming in MATLAB, Mathematica, Maple or the like. Undergraduates: Liddle, Pei, Wettstein.

• Visualizing the arithmetic of the rational numbers. In this project, we will develop tools for visualizing Conway's topograph as a visual representation of the rational numbers (actually the rational projective line). We will use these tools to illustrate the Euclidean algorithm (for calculating the greatest common divisor of two integers), continued fractions, the arithmetic of polynomial functions, the structure of irrational numbers (especially quadratic irrationals), and some exceptional group isomorphisms. This will involve learning some hyperbolic geometry and will require comfort with linear algebra.
Mentor: Jonathan Wise. Prerequisite. Math 3130, Math 2001 + one higher level, and some programming. Undergraduates: Allen, Cao, Davis, DiPaola. First Year Graduate Students: Wang.

## 2016 Projects

• Eigenvalues of random matrices. A random matrix is a matrix whose entries are random variables. A fundamental problem in random matrix theory is to describe the behavior of the eigenvalues of a given random matrix. The goal of this project is to study the eigenvalues for a class of random matrices whose entries are dependent random variables. Part of the project will involve numerical simulations using software such as MATLAB. Prerequisites: Math 3130, Math 4510 (or equivalent).
Mentor: Sean O'Rourke. Undergraduates: I. Gossett, I. Vance, and K. Zagnoli. First Year Graduate Students: S. Lu and K. Dearborn.

• Applications of Topology in Chemistry and Physics. Topological methods have turned out to play a more and more important role in the study of chemical molecules and modern quantum field theory. The intention of this REU program is to provide an introduction to this theme, mostly to knot and graph theory and its applications, and then consider particular problems, where the toplogical study will help to better describe molecules or symmetries in quantum field theory. Particular themes the students can work on are:
. mathematical descriptions and detection of chiral molecules with the help of knot theory and topology
. numeration problems in chemistry, application of Polya's enumeration theorem
. generalized symmetries in low dimensional QFT
Prerequisite: Math 3130 or 3001. Helpful, but not required: Math 4200.
Mentor: Markus Pflaum. Undergraduates: B. DeMoss, L. Simon, and Y. Tianyi.

• The geometry of complex continued fractions. Every rational number has a continued fraction expansion, which looks something like this:
α = a1 + 1/(a2 + 1/(... + 1/am))
Real numbers also have continued fraction expansions, only we now allow them to be infinite and they converge to the number in question in an appropriate sense:
α = a1 + 1/(a2 + 1/(a3+... ))
The theory of continued fractions has a long and famous history. We can think of the continued fraction expansion of a number as its `address' in the rationals. Determining the continued fraction expansion of a number is an iterative process closely related to the Euclidean algorithm. However, this process can be described in terms of some beautiful geometry, in terms of dynamics, Möbius transformations, fractal structures such as Ford Circles, etc. We will explore some open questions concerning extensions of continued fractions to complex numbers, where the `addresses' are given in terms of certain imaginary quadratic fields. The project will involve working heavily with the geometry of Möbius transformations, hyperbolic geometry, and fractal circle and sphere packings, etc. Prerequisite: Some class requiring rigorous proof (beyond Math 2001).
Mentor: Kate Stange. Undergraduates: C. Gebhart and P. Rock. First Year Graduate Students: R. Li and D. Martin.

• The geometry of set partition statistics. A fundamental problem in combinatorial topology is counting the number of integer lattice contained in an integer polytope. For example, the n-dimensional hypercube has 2n integer lattice points (in this case, it is the same as the number of vertices). In general, this turns out to be a very hard problem without additional tools. This project seeks to study a family of polytopes whose lattice points have an algebraic interpretation in terms of finite unipotent groups. In particular, the goal is to develop concrete relationships via the underlying combinatorics between the algebraic structure of the unipotent groups and the geometry of the lattice points. Prerequisite: Math 3140.
Mentor: Nat Thiem. Undergraduates: A. Allen and K. Murphy.

• Invariants of Quotients by Circle Actions. In many physical applications, one often deals with "nice" spaces, such as Euclidean space, circles and spheres, tori, etc. What all of these spaces have in common is that they are "smooth": they have no corners or cusps. This makes performing basic calculus on them a simple extension of calculus from a standard undergraduate analysis course. However, sometimes "singular" spaces arise: spaces with corners and/or cusps, among other strange artifacts. Think of a cone in 3-space with its pointy apex. A large class of examples of singular spaces consists of what are known as "orbifolds". Locally, these spaces look like some Euclidean space modulo an equivalence relation induced by a finite group action. Understanding how calculus works on these spaces is an old problem, but it is still possible to define "smooth" or "infinitely-differentiable" functions on them. What is interesting is that the various corners, boundaries, and cusps in these spaces can be completely detected by these rings of functions; so much so that these functions can tell the difference between different orbifolds. The purpose of this project will be to consider spaces that (locally) look like some Euclidean space modulo an equivalence relation induced by a circle action. What can the ring of smooth functions tell us in this case? Will the resulting singular spaces have their singularities completely detected by the smooth functions, or are there pieces of information missing? Prerequisites: MATH 2400, MATH 3001, MATH 3130, MATH 3140; Not necessary but helpful: MATH 4001 and MATH 4140.

• The game of SET and its geometric generalizations (or: spaces with many points and few lines). The game of SET is a card game in which you try to identify SETs, which are triples of cards with shared or distinct characteristics. It turns out that finding sets is the same as finding lines in a 4 dimensional vector space over the field with 3 elements. If you play much SET, you will run into a configuration of 12 cards without any SETs---that is, a configuration of 12 points in F_3 without any lines. It is natural to ask how large a collection of points you can find before it must contain a line. The answer turns out to be 20 for the usual came of SET. In "2-dimensional SET", the answer is 4, but it comes with a good reason: the 4 points in question lie on a circle, so they can't contain a line! This leads us to a number of open-ended questions: Can we explain configurations of points that do not contain lines by finding geometric figures that don't contain lines "for a good reason"? What about geometries over other finite fields? How about configurations that contain lines but not planes? We will look into many of these, using computers to gather data and look for patterns. We may also play a lot of SET. Prerequisite: Math 3140.
Mentor: Jonathan Wise. Undergraduates: A. Alnasser and P. Tankslvala. First Year Graduate Students: A. Lotfi and A. Thompson.

## 2015 Projects

• The Schmidt arrangement of the Eisenstein integers. The first picture here shows the Schmidt arrangement of the Eisenstein integers. It's an intricate arrangement of circles that reflects the structure of the Eisenstein integers, a subring of the complex numbers that has properties similar to the integers but a much more geometric flavour. This project will combine geometry with number theory. The required background is Math 3140. You will learn about Mobius transformations and number theory in quadratic fields.
Mentor: Kate Stange. Undergraduates: A. Jensen, E. Oliver. First Year Graduate Students: C. Ng, T. Schrock.

• Random walks on unipotent groups. This project seeks to use algebraic techniques to help understand random walks arising from groups. In particular, we will study a family of unipotent groups called Heisenberg groups and the combinatorial walks they generate. Last summer studied a complementary family and made significant progress; we hope to build on this success. Math 3140 is a required background, but all other mathematics can be learned in the program.
Mentor: Nat Thiem. Undergraduates: D. Anthony, D. Kickbush. First Year Graduate Students: J. Hong, A. Sparks.

• Geometry of composition laws. In 1801 Gauss showed that equivalence classes of 2x2 integer matrices under row and column operations can be given a group structure. Today we can give an elegant geometric description of Gauss's composition law arising from the geometry of the projective line. Recently Bhargava gave a similar composition law on equivalence classes 2x2x2 cubes of integers (this time, equivalence is from row, column, and plane operations). We will study one approach to understanding this composition law geometrically. Comfort with commutative algebras and their modules will be necessary in this project. Math 4140 is a prerequisite.
Mentor: Jonathan Wise. Undergraduates: L. Simon. First Year Graduate Students: S. Bozlee, L. Herr, H. Smith.

• 2-cocycles and higher-rank graphs. Higher-rank graphs are a generalization of directed graphs that can be used to construct examples of C*-algebras (the main focus of my research). 2-cocycles are a particular kind of function on a higher-rank graph; different 2-cocycles give you, in principle, different C*-algebras. The goal of htis project is to improve our understanding of the phrase "in principle" in the previous sentence. That is, we will investigate several different equivalence relations on the set of all 2-cocycles, and try to understand the relationships between them, as well as what they tell us at the level of the C*-algebras. MATH 2001, plus at least one additional proof-based math class, is a prerequisite for this project, but all other necessary background will be covered during the project (including what a higher-rank graph is, and a C*-algebra if you're curious).

• Liber Mathematicae and Mathematics presentation on the web. Properly presenting Mathematics on the web is still a challenging task. The proposed REU addresses this problem by extending the Liber Mathematicae project and writing appropriate code in ruby on rails.
Mentor: Markus Pflaum. Undergraduates: D. Ingoglia.

• Understanding the Recent Proof of the Poincare Conjecture in Simple Examples. The recent proof of Perelman of the long-standing Poincare' Conjecture amazed the mathematical world; indeed (after a lot of deep mathematics) what 'looks like a 3-sphere' was finally actually proven to be a 3-sphere!! We will be looking at several summaries of this astonishing proof, especially in the case of two-dimensional manifolds, to get familiar with the main ideas involved in this astonishing proof. Some knowledge of topology, ODE, and differential geometry is required.
Mentor: Carla Farsi. Undergraduates: N. Downey. First Year Graduate Students: M. Pierson, C. Pinilla.

• Numerical Solutions of Reaction-Diffusion Partial Differential Equations. This project aims to understand the physics behind reaction-diffusion equations and then to numerically solve them. Convergence and stability properties of algorithms will also be studied. Math 4470 is a prerequisite, and some knowledge of programming in MATLAB, Mathematica, Maple or the like is also required.
Mentor: Divya Vernerey. Undergraduates: T. Bisbee, E. Kersgaard.

## 2014 Projects

• Fractals, solenoids, and fractal coefficients. Compact fractal sets, for example the Cantor set and the Sierpinski gasket, have self-similarity properties that give rise to filter functions on the circle and on the two-torus, respectively. The fractal sets can be inflated by dilation and translation to form inflated fractal sets, and both the fractal sets and their inflated versions have measures attached that were first constructed by Dorin Dutkay and Palle Jorgensen. In 2006, Dutkay and Jorgensen used this measure and the associated dilation and translation operators to construct ``wavelets" in the Hilbert spaces associated to these inflated fractal sets. There are filter functions coming from the wavelets associated to the fractals, and these filter functions also give rise to probability measures on compact abelian groups called solenoids. Solenoids are inverse limits of ordinary tori. These latter measures have Fourier coefficients (defined on countable abelian groups) that can be described via two-scale equations. It is of interest whether there are patterns within these non-zero Fourier coefficients in the cases of the two aforementioned fractals as well for measures coming from filters associated to more general fractal sets. This summer project will aim to ultimately analyze and generate the non-zero Fourier coefficients, and attempt to find patterns in them. This research will follow up on previous research of Dutkay, Jorgensen, and L. Baggett, K. Merrill, A. Ramsay, and myself.
Mentor: Judith Packer. Students: E. Tucker.

• Chemical topology. Mentor: Markus Pflaum. Students: J. Duplantis.

• Whitney--de Rham cohomology on singular spaces. Mentor: Markus Pflaum. Students: D. Jones.

• Random walks on abelian unipotent groups. The project considers the superclass variant of a random walk on a particular unipotent group. In particular, we want to use representation theoretic methods to better understand a life and death walk on the integers {0,1,...,m}.
Mentor: Nat Thiem. Students: V. Theplertboon.

• 2-cocycles and higher-rank graphs. The subject of the project is an investigation of the relationship between the topological fundamental group and a better-behaved abstract fundamental group constructed via infinite Galois theory, as defined by Bhatt and Scholze. The two fundamental groups coincide for most topological spaces that arise in practice (namely, those that possess universal covers) but appear to diverge in interesting examples. The object of the project will be to study the relationship between the two definitions in the divergent examples, especially the two known as the Hawaiian earring and the Hawaiian archipelago. We will use these example to develop, support, and hopefully also prove a conjecture about the relationship between the two fundamental groups in general.
Mentor: Jonathan Wise. Students: C. Klevdal.