Summer Research in Mathematics

Math Research Experience 2019

The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2019. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

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.

    Mentor: Agnès Beaudry. Prerequisites. Math 3140, and no fear of linear algebra.

  • 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: Sebastian Casalaina-Martin. 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).

  • 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!

    Mentor: Jeanne Clelland, Prerequisites. Calculus 3, Linear algebra, and some experience with Python. (Minimal experience okay as long as you're willing to learn more!)

  • 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.

    Mentor: Magdalena Czubak. 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.

  • 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.

    Mentor: Peter Mayr and Athena Sparks, Prerequisites. Ideally MATH 3140, but MATH 2135 and one or more MATH 3000+ level courses may suffice.

  • 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.

    Mentor: Markus Pflaum. Prerequisites. Some basic knowledge in Linear Algebra and willingness to learn some abstract math and do computer assisted computations.

  • 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.

    Mentor: Nat Thiem. Prerequisites. Math 3140 and no fear of linear algebra.

  • 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.

    Mentor: Martin Walter. Prerequisites. Math 3140.

Dates

  • Starting date

  • May

  • June

  • July