# Project: Subrings of ℤ^{n}

**Project supervisor:** Ramin Takloo-Bighash

**Description:** The set of all n-dimensional integer vectors, denoted ℤ^{n}, under componentwise addition is a commutative group. It is not very hard to determine all subgroups of ℤ^{n}, and in fact to count the number of subgroups which have index equal to a fixed integer *k*.

One can also think of ℤ^{n} as a ring equipped with componentwise addition and multiplication. The question of understanding subrings of ℤ^{n} has baffled mathematicians for a few decades. More generally, one can consider those rings *R* which are isomorphic to ℤ^{n} as additive groups, but have other multiplicative structures, and ask to understand their subrings. The only cases of these problems that are reasonably well understood are for n ≤ 5 (by works due to Manjul Bhargava and his collaborators, as well as Kaplan, Marcinek, and Takloo-Bighash, using completely different methods).

These problems have produced a large number of projects that are accessible via computer-based investigation. Here are two sample projects:

- There is reason to believe that the degenerate ring ℤ
^{n}/(x^{n}) has the highest number of subrings of a given index among all rings that are additively isomorphic to ℤ^{n}. Provide numerical evidence for this expectation. - Given a prime number p, let 𝔽
_{p}be the field with p elements. Investigate the number of subrings of 𝔽_{p}[x]/(x^{n}). How does it compare to the number of subrings of 𝔽_{p}^{n}as n and p vary?

**Prerequisites:** Abstract algebra I (Math 330) and Foundations of Number Theory (Math 435) or Codes and Cryptography (MCS 425).

# Project: 3D Printing Surfaces

**Project supervisor:** Daniel Groves

**Description:** This project will explore surfaces in three-dimensional space by 3D printing models of them. The goal is to start by choosing appropriate equations and end with physical models which would be appropriate for use in multivariable calculus classes. This will involve learning about function graphing software, 3D modeling software and 3D printing. Time and student interest permitting, we can investigate such things as curvature, ruled surfaces and other things which touch upon ideas from topology, geometry and algebraic geometry.

**Prerequisites:** It is possible to succeed in this project whilst enrolled in Math 210 (Calculus III).

# Project: Efficiency of Planar Disk Packings

**Project Supervisor:** Ali Mohajer

**Description:** A popular chemistry experiment shows that a mixture of certain pure liquids can be denser than each of the constituents. Mixing a liter of methanol with a liter of ethanol gives a solution with volume measurably less than 2 liters! A mathematical analog of this experiment is the fact that a packing of unequal disks in the plane can be denser than a packing of equal disks, as long as the radii of the disks aren’t very close. Exactly how close is “very close” is an area of active research. In fact, much remains to be discovered about the behavior of two-species packings, except at a handful of very special ratios of radii where everything fits together very nicely. In this research project we will study randomly generated two-species packings in order to gain insight into the shape of the density bounding function.

**Prerequisites:** Math 210 (Calculus III) and some computer programming experience (equivalent to MCS 260).