# MCM 2009: Contest Question B

# Senior Thesis

### Abstract:

The abelian sandpile model, or chip-firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of selforganized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sandpile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a $2n\times 2n$ grid.

https://www.math.hmc.edu/seniorthesis/archives/2009/ndurgin/

# PCMI: Summer 2008

The Park City Mathematics Institute topic for the summer of 2008 is Algebraic Geometry. More coming Soon!

Seahorse: $$\small{(x^2-y^3)^2=(x+y^2)z^3}$$

### Courses Taken

- Introduction to Algebraic Geometry: David Perkinson

Our solution to challenge problem #1(*.nb) - Algorithmic Fewnomial Thoery: Maurice Rojas
- Blowing Up Surfaces and Ideals: Herb Clemens

Resolving the PCMI T-Shirt (Seahorse) - Toric Varieties: Jessica Sidman

### Lectures Attended

- Resolution of Singularities: Mircea Mustata
- Introduction to Multiplier Ideals: Robert Lazarsfeld
- An Introduction to Minimal Models and Flips : Janos Kollar

# Summer 2008

This research in geometric combinatorics was conducted under the guidence of Professor Francis Edward Su and sponsored by the Baker Foundation. We are hoping to submit our paper (in progress) to a discrete mathematics journal.

### Abstract

A simplex of the 4-dimensional cube is the convex hull of any 5 distinct vertices of the cube. We find there are exactly 27 isomorphism classes of these simplices under symmetries of the cube, 10 of which are degenerate. Our methodology is based on geometric considerations that produce insights beyond computational enumeration. In particular, we provide a complete description of these simplex facets and how they can fit together in any triangulation. Using this, we are able to provide a graph representation of several triangulations of the 4-cube, leading to a new understanding of Mara's minimal triangulation.

# Independent Study, Spring 2008

This spring I enrolled in a reading course with Professor Su. The topic was Geometric Combinatorics. Some results are documented below.

# MSRI: Summer 2007

This summer I participated in the new MSRI undergraduate program. The six-week REU topic was computational mathematics. In particular, our project was from experimental mathematics, the practice of using the computer to form conjectures or gain insight into a problem. Our advisor was Dr. David H. Bailey.

# MCM 2007: Contest Question A

# Summer 2006

After participating in Harvey Mudd Summer Math, I had the privilege to join this project with the Engineering Department. It was a fluid mechanics experiment seeking to verify a conjectured transition point from bubbly to slug flow in zero gravity. The apparatus was flown in the NASA vomit comet (or Weightless Wonder as their PR department insists on calling it.)