Summer Research in Mathematics

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.
    Mentor: Jordan Watts. Undergraduates: N. Downey, L. Goad, and M. Mahoney.

  • 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).
    Mentor: Elizabeth Gillaspy. Undergraduates: G. Erdenejargal, O. Orejola. First Year Graduate Students: K. Adamyk, S. Salmon.

  • 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.