WASHINGTON eXPERIMENTAL MATHEMATICS LAB

# Spring 2018

## Logistics:

WXML general meetings will be from 5-6pm on the following three Thursdays in Mary Gates Hall (MGH) room 295:

• First meeting: March 29
• Mid-quarter meeting: April 26
• Final meeting: May 24

The WXML Winter 2018 Open House will be on May 31, 4:30-6pm, in Odegaard 136.

WXML has reserved MGH 295 on Tuesdays and Thursdays from 5-7pm for all of Winter quarter for groups to use as a meeting space. The WXML lab in the basement of the Communications building is also available.

## Projects and Teams:

### Rook Placement Games

Faculty Mentor: Dr. Jonah Ostroff

Graduate Mentors: Connor Ahlbach, Sean Griffin

Team Members: Matt Manner, Zian Chen, Shruti Mokate

Here’s a game: two players take turns placing rooks on a chessboard. You can’t place a rook in the same row or column as a previously placed rook. If it’s your turn and you can’t find a valid move, you lose and your opponent wins.

For rectangular boards this is super boring, and the winner is determined solely by the dimensions of the board (regardless of strategy). Add a few holes to the board and it becomes slightly more interesting, but still pretty easy. Add a *lot* of holes to the board and, well, now I have no idea which player has a winning strategy, or what is. Let’s figure it out.

We’ll start by learning about combinatorial game theory: Nim, impartial games, and the Grundy-Sprague theorem. Then we’ll try to figure out what’s going on in this game.

### Stability Spectrum for PDES

Faculty Mentor: Dr. Bernard Deconinck

Team Members: Kush Gupta, Ryan Bushling

An important property for solutions of differential equations is stability. Stability is important in physical applications because it determines whether or not a solution may be seen in the real world. If a solution is unstable, small perturbations of the system will fore the system away from the solution. If a solution is stable, the system will return to that solution after small perturbations.

To determine whether or not a solution is stable, one linearizes the differential equation and examines the spectrum of the resulting linear operator. The geometry of the spectrum reveals stability properties, but is typically very difficult to find analytically. The spectral problem may be truncated to a matrix eigenvalue problem. Once the matrix is formed, we use computers to find the eigenvalues and plot them to understand the geometry of the spectrum.

In this project, students will be given a partial differential equation (PDE) and class of traveling wave solution to that equation. This class of solutions typically has various parameters that may take a continuum of values in some fixed region of space. From a given solution, students will use the above described method to compute the spectrum using a computer. If there is time, the solution parameter space can be examined to determine regions where the spectrum has qualitatively different shapes. If there is still time, students may look at transition regions by altering their program to allow for dynamic solution parameter changes. In doing this, interesting movies and an applet can be created.

### Algebraic Combinatorics

Faculty Mentors: Dr. Sara Billey, Dr. Philippe Nadeau

Team Members: Thomas Browning, Jesse Rivera

In Algebra, a familiar object is the set of cosets G/H for a given group G and subgroup H. There are also many reasons to study double cosets H\G/K. If G is the symmetric group S_n and H and K are Young subgroups, then the double cosets are closely related to representation theory and also to the Bruhat order. In Combinatorics, we are interested in studying efficient formulas for counting discrete objects. Double cosets of symmetric groups are discrete objects, so they deserved to be counted! But, what is the best possible formula? What are the interesting types of subgroups to consider? This work has connections to prior research by the mentors on tanglegrams, parabolic double cosets, and pattern-avoiding permutations.

### Orbit Structures of Crystal Operators

Faculty Mentor: Dr. Jake Levinson

Team Members: Junchen Pan, Yujin Jeong

We’re going to study a transformation (algorithm) involving a type of diagram called a Young tableau. The transformation is combinatorial, applying a sequence of ‘moves’ to the diagram. The algorithm’s significance comes from geometry, where it describes the result of varying a parameter (i.e. moving a point) along a special kind of curve. After iterating the algorithm enough times, the diagram will be transformed back to its original form, which indicates that the point has traveled around a loop, back to where it started. I’d like to understand the orbit structure of this transformation. That is, given a diagram, how many steps does it take for it to return to its original form? And, given a set of diagrams (= points on the curve), how many distinct orbits do they form (= number of loops formed by the curve)? Can we tell if two diagrams are in the same orbit? Goals of the project. Code up the algorithm and run it, collecting data about the orbit structure of the transformation. Test out some conjectures about identifying and counting orbits. And, if there’s interest, learn about some of the geometry as well — where this combinatorics comes from.

### Mathematics of Gerrymandering

Faculty Mentor: Dr. Christopher Hoffman

Team Members: Leon Segovia, Weifan Jiang, Alexander Robkin, Namyoung Kim

Districting and Gerrymandering have been in the news a lot recently, and the mathematical modeling of how to draw districts is a very hot topic. This project will look at the space of all possible redistrictings of a state to see whether the current plan is an outlier. It will involve probability and computing skills.

### Counting k-tuples in Discrete Sets

Team Members: Kimberly Bautista, Andrew Lim, Maddy Brown

We will count pairs, triples, and k-tuples of integer vectors in the plane, 3-space, and higher dimensions, using both number theory and programming. We’ll also explore other interesting sets arising from dynamical systems and billiards. This project will involve number theory, geometry, and probability, and will be a great introduction to those subjects.

### Uniformity of Solutions of Diophantine Equations

Project Mentor: Dr. Amos Turchet

Team Members: Daria Mićović, Blanca Viña Patiño, Bryan Quah, Rohan Hiatt

Given an equation in two variables with integer coefficients, how many solutions can you find where both values of the variables are rational numbers? This apparently easy question turned out to be very deep and very interesting from the point of view of Number Theory and Algebraic Geometry. In this project we will investigate the behavior of such solutions set for special equations representing so called, higher genus curves. We will gather some statistic on the size of the solutions set trying to address an important open problem on the uniformity of these sizes.

### Number Theory and Noise

Project Mentor: Dr. Matthew Conroy