**Gluing tetrahedra**

Faculty advisors: Diana Hubbard and Dohyeong Kim

Grad student mentor: Mark Greenfield

Undergraduate researchers: JunCheng An, Yingsi Jian, Yichen Liu, Yi Zhou

Gluing tetrahedra along their faces, one can build various kinds of spaces which locally look like the ordinary 3-space. These are called 3-manifolds, which we aim to explore through this project.

Given a combinatorial description of a 3-manifold, namely a finite collection of tetrahedra and the pairings between their faces, one can write down a sequence of finite-dimensional vector spaces together with linear transformations between them. The sequence is called the cochain complex of the manifold. Despite its elementary nature, the dimensions of the involved vector spaces are often too large to admit a manual computation.

We plan to employ the computing power of a modern computer to handle these cochain complexes numerically. The goal is to find a basis for harmonic cochains and to compute the discrete Hodge star operator, a linear transformation that maps a harmonic cochain into another.

**Piercing d-intervals, d-convex sets, and other geometrical hypergraphs**

Faculty advisor: Shira Zerbib

Grad student mentor: Yiwang Chen

Undergraduate researchers: Connor Puritz, Bennet Sakelaris, Billy Warner

A classical theorem of Helly asserts that if F is a family of convex sets in R^d in which every d+1 members have a non-empty intersection, then all the members in F intersect. Helly’s theorem initiated the broad area of research in discrete geometry, dealing with questions regarding the number of points needed to “pierce” (or “stub” or “cover”) families of convex sets in R^d satisfying certain “local” intersection properties.

What are those local intersection properties? Given integers p ≥ q > 1, a family F of sets is said to satisfy the (p,q) local intersection property (or just “the (p,q) property”) if among any p elements of F there exist some q elements with a non-empty intersection. We denote by τ(F) the piercing number of F, namely the minimal number of points needed to intersect every set in F. In this language, Helly’s theorem is that if a family F of convex sets in R^d satisfies the (d+1, d+1) property, then τ(F)=1.

In 1992 Alon and Kreitman resolved a long standing conjecture of Hadwiger and Debrunner, proving that for every p ≥ q ≥ d + 1 there exists a constant c = c(d,p,q) such that if a family F of convex sets in R^d satisfies (p,q) property then τ(F) ≤ c. An extensive research has been done since trying to improve the bounds on the piercing numbers given by the Alon-Kleitman proof, as well as generalizing their result to other special geometrical objects.

In this Log(M) project we will add to these research efforts by examining the piercing numbers in families of d-intervals, d-trees and d-convex sets.

**A missing entry in Sullivan’s dictionary?**

Faculty advisor: Dylan Thurston

Grad student mentors: Patrick Haggerty, Didac Martinez-Granado and Maxime Scott

Undergraduate researchers: Colby Kelln, Sean Kelly, Jung Suk Lee

Sullivan’s dictionary highlights some similarities between 1-dimensional complex dynamics and 3-dimensional hyperbolic geometry. Many mathematicians attempted to use this “correspondence” to translate proofs from one field to the other. Some succeeded, some failed, but nonetheless this dictionary has driven enormous expansion in both fields since the 1980’s. More recently, some mathematicians conjectured that there might be some missing entries. One of these possible missing entries is the degeneracy of Julia sets and degeneracy of limit sets of quasi-Fuchsian groups. The goal of this project is to use computer programs to generate images of degenerate Julia sets and limit sets that share similar behavior.

**The chromatic number of flip graphs**

Faculty advisor: Nick Vlamis

Grad student mentor: Francesca Gandini

Undergraduate researchers: John Paul Koenig, Sanjana Kolisetty, Zihui Qi

A flip graph is associated to a finite-sided convex

polygon P living in the plane. Given the polygon P, the vertices of

the flip graph associated to P correspond to a triangulation of P

(i.e. a choice of disjoint diagonals that decompose P into triangles).

Two vertices are connected by an edge if the associated triangulations

differ by a flip (i.e. if one triangulation can be obtained from the

other by changing a single diagonal).

A proper coloring of a graph is a coloring of the vertices such that

any two adjacent vertices have distinct labels. The chromatic number

of a graph is the minimum number of colors needed to give a proper

coloring of the graph.

The question we will try to answer is: What is the chromatic number

associated to a convex polygon? Observe that the structure of the flip

graph only cares about the number of sides of the polygon. So we will

try to understand how the chromatic number changes as the number of

sides increases. Here is an open research question that we will focus

on: As the number of sides goes to infinity does the chromatic number

go to infinity?