- Lower dimensional flag triangulated manifolds
- Origami on Lattices
- Finding real polynomials
- SET Marble-Run Computer

**Lower dimensional flag triangulated manifolds**

Faculty advisors: Christin Bibby and Hailun Zheng

Grad student mentor: Andy Odesky

Undergraduate researchers: Mengmeng Wang, Shuyang Wang, Ziyi Zhang

[Final poster]Project description: In this project, we will explore some open problems on the face enumeration of lower dimensional flag triangulated manifolds. A triangulation is a simplicial complex which, roughly speaking, describes how to build a space (such as a sphere or a torus) out of triangles. Understanding the combinatorics of triangulations has always been an important research aspect of geometric combinatorics. A fundamental combinatorial invariant of a triangulated manifold is its f-vector, which encodes the number of faces in each dimension. Not surprisingly, the topology of the manifold affects the f-vectors of its triangulations.

The 1-skeleton of a simplicial complex can be viewed as a graph, and a simplicial complex (or triangulation) is said to be flag if the faces correspond to cliques (complete subgraphs) in this graph. Examples of flag complexes include barycentric subdivisions of simplicial complexes and of Coxeter complexes. The additional structure of flag complexes makes their combinatorics quite different from those of general simplicial complexes. For example, an observation by Gromov leads to an elegant combinatorial interpretation of the celebrated Hopf conjecture from differential geometry, known as the Charney-Davis conjecture. This conjecture states that for odd-dimensional flag simplicial spheres, a certain linear combination of the face numbers is nonnegative.

This project will start by exploring (minimal) flag triangulations of surfaces: It is already known that an octahedron gives a flag triangulation of the sphere with the minimum number of vertices, but less is known for other surfaces (such as a torus). Once comfortable with flag surfaces, we will then explore the relationship between the face numbers and topological invariants of flag triangulated 3-manifolds, in the spirit of the Charney-Davis conjecture and related works.

** Pre-requisites ** are Math 217 or equivalent. Experience in Math 490 and/or Math 465, along with basic coding skills (Sage or Python) are strongly suggested.

** Additional expectations**: Students will be expected to keep track of hours worked on timesheets.

An extended abstract is available here

**Origami on Lattices**

Faculty advisors: Zachary Hamaker and Ian Tobasco

Grad student mentor: Harry Richman

Undergraduate researchers: Anamaria Cuza, Yuqing Liu, Osama Saeed

[Final poster]Project description: Origami, or the ancient art of paper folding, is filled with interesting mathematical questions. Suppose, for instance, that you were given a collection of lines across which a sheet of paper could be folded: could you describe the class of shapes that could be made? The inverse problem has many practical implications, since its solution would entail an algorithm for generating a folding pattern to yield a desired shape. The ultimate goal of this project is to classify all origami folding patterns that a given lattice of folds can produce. We will start with simple examples and build from there. A second goal is to write a computer program to help visualize the folding patterns that result. This work will support further research so that we can better understand the shape space of origami — what fold configurations are possible — and how this space depends on a given energy budget. The initial work will use three main methods in accessible ways: geometric, combinatorial and computational. As we progress, student researchers will have the option of specializing in one of these methods.

**Prerequisites**: Students should have had prior experience with multivariable calculus and linear algebra (at the level of 217 or beyond). Experience with combinatorics, differential geometry, and/or partial differential equations is a bonus. If students choose to specialize in computational methods, programming experience would be helpful.

**Additional expectations**: An end of the semester write-up (in article format) documenting the project’s findings for future reference.

**Finding real polynomials**

Faculty advisor: Becca Winarski

Grad student mentors: Alex Kapiamba and Jasmine Powell

Undergraduate researchers: Tredon Austin, Allen Macaspac, Hannah Moon

[Final poster]Project description: Take a quadratic polynomial z^2+c (where c is a complex number). Now compose z^2+c with some continuous function g(z) and call the map f. The map f is probably not a polynomial, but under certain conditions it is a “topological polynomial”: meaning there is some change of (possibly nonlinear!) coordinates under which f is a polynomial z^2+d. There are only finitely many options for d (based on the c we started with) and d depends both on c and g(z). In this project, we’ll write a program to estimate the percentage of the continuous maps (the g(z) we composed by) that lead to d being real. The answer has both topological and dynamical significance.

Early in the semester, we’ll meet regularly to learn the relevant topology, dynamics, and group theory. As the semester progresses, more time will be spent creating the program.

**Additional expectations**: Weekly journals will be expected and will eventually be used to create a writeup at the end of the semester.

**Prerequisites**: Math 217, programming experience – the language is flexible, but Sage (python based) or Mathematica are particularly useful.

Complex dynamics, topology, and algebra/group theory would all be helpful (in that order), but aren’t necessary.

**SET Marble-Run Computer**

Faculty advisor: Martin Strauss

Grad student mentors: Lizbee Collins-Wildman and Salman Siddiqi

Undergraduate researchers: Lingcong Xu, Amanda Yao, Yi Zhou

[Final poster]Project description: In the card game SET, cards have one, two, or three symbols, with a shape, shading, and color chosen from a collection of three values. The goal is to find tricks of three cards (called SETs) in which, for each attribute, the cards all have the same value of the attribute or have all three values.

It turns out that the number of SETs in a collection of cards can be computed fairly simply, suitable for a mechanical computation, e.g., a marble run with parts produced on a 3-d printer or similar. The final computer might take the form of a human turning a dial for each face-up card dealt (thereby moving some marble run infrastructure), then letting loose the marbles. The final count of marbles in a bin reveals the number of SETs among face up cards. The project will consider the structure of finite fields and Fourier transforms. See, for example, this paper, for the Fourier transform in this context.

**Prerequisites**: Students should have experience making parts with a 3-d printer or otherwise–the final construction might involve other designs, e.g., paper folding, or other mechanical or electrical system. Background in algebra is not necessary; we’ll learn as we go, but students should be ready to learn.