**Applications are now open for the fall 2018 projects!** The deadline for full consideration is August 31, 2018. A preliminary list of fall 2018 MCL projects can be found below. Information on how to apply can be found on the **application page**.

# 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: Two-radius Planar Disk Packings

**Project supervisor:** Ali Mohajer

**Description:** The densest packing of disks of the same radius in the plane is the hexagonal lattice. If disks of two different radii are allowed, it is sometimes possible to pack disks more densely than the hexagonal lattice, however this is only possible if the ratio of the radii is not very close to one. Exactly how close is “very close” is an area of active research, and much remains to be discovered about the behavior of two-radius packings. In this research project we will study the densities of randomly generated two-radius disk packings with various ratios of radii, with a goal of obtaining experimental insights into the variation of the optimal density as the ratio of radii changes.

This project will build on the results of the Spring 2018 MCL semester project *Efficiency of Planar Disk Packings*.

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

# Project: Robot Kinematics

**Project supervisor:** Julius Ross

**Description:** Robot kinematics considers the following kinds of problems. Imagine a robot arm made up of a number of straight segments joined together by hinges. One end of the arm is fixed on a table and at the other end is a tip (or finger) of some kind. Suppose the robot wants to move its finger to a give point, then what angles must it choose for each of the joints? Is it the case that the robot can move its finger to any position, or are there some that it cannot reach? What about if it wants to move its finger to a given position and make it point in a given direction? One can ask this question in either two dimensions or three dimensions (or even more!).

This project will consider these questions by understanding the problem as finding solutions to a system of simultaneous polynomial equations, which then becomes a problem in numerical algebraic geometry.

**Prerequisites:** Math 310 or Math 320 (Linear Algebra).