About Ben Stucky
I will soon be a Visiting Assistant Professor of Mathematics at Millikin University, where I will teach both mathematics and computer science courses. See the section for more info.
My research interests lie in the field of geometric group theory, and I am interested in becoming involved with undergraduate research projects combining mathematics and computer science. See the section for more info.
I have collected some interesting and useful math resources in thesection.
In addition to mathematics, my hobbies include card games (some favorites are Hanabi, poker, and mafia), skateboarding, “jazz” piano, and playing and watching speedruns of old video games (some favorites are Zelda: OOT, Zelda: MM, Earthbound and SSBM).
I have more than six years of experience teaching at the college level. This fall, I will be teaching College Algebra, Elementary Probability and Statistics, and Introduction to Computer Science at Millikin University.
The courses that I have taught previously include College Algebra, Precalculus and Trigonometry, Math for Critical Thinking (a basic introduction to statistics), Calculus I, and Calculus II. I have also TA'ed Calculus III and Calculus IV and graded Differential Equations. In the summer of 2016 I taught a class called Paradoxes and Infinities to gifted 7th through 10th graders. This unique course introduced middle and high school students to topics not typically covered until college, including Peano Arithmetic and Cantor's Diagonalization, and it challenged me to implement classroom methodology with which I was not as familiar.
I am passionate about teaching and have reflected considerably about what I believe makes a successful teacher. See my teaching philosophy for more information.
Most of my students seem to respond well to my approach in the classroom. Here are some select student evaluations.
Preprints and manuscripts
Cubulating one-relator products with torsion
Another visual proof of Nicomachus' theorem
Summary of interests
Most of the research that I do is in the field of geometric group theory. This is a relatively new field, the origins of which trace back to Henri Poincaré, Max Dehn, and others. The current viewpoint is motivated by influential ideas of Mikhail Gromov and William Thurston, to name a few. Broadly, geometric group theorists seek to understand groups via their presentations by finding nice spaces which encode their symmetry. One then uses the geometry and topology of those spaces to derive algebraic properties of those groups. My dissertation, Cubulating one-relator products with torsion, used these techniques to verify a type of nonnegative curvature for a specific class of groups which generalize one-relator groups. See my research statement for more information.
Research with undergraduates
I am excited to become involved with undergraduate research projects combining aspects of mathematics and computer science. For one, there are questions in group theory which are quite amenable to computational approaches, and in fact part of what makes the field so interesting to me is that there are certain basic questions which are known to be undecidable, in general.
More accessible to most undergraduates are questions in graph theory, combinatorics, game theory, knot theory, low-dimensional topology, or geometry which are amenable to a computational approach, such as the Hadwiger-Nelson problem, described below. As a graduate student, I studied this question with an undergraduate math major at the University of Oklahoma.
If you are an undergraduate who is interested in getting involved with math/CS research, but not sure where to start, please feel free to reach out to me!
Around 1950, Ed Nelson asked the seemingly innocuous question, “What is the smallest number of colors needed to color the plane so that no two points distance one apart are the same color?” It was established quickly and straightforwardly that the answer, which we call the chromatic number of the plane, lies between 4 and 7, inclusive, but the exact answer has proved difficult to come by. See Wikipedia for a nice synopsis of this problem.
This is one of my favorite open problems, and I have written a program to explore how one might raise the lower bound. My approach is to immerse a graph of chromatic number 5 (meaning that one needs to use 5 colors to paint the vertices in such a way that no two adjacent vertices have the same color) in the plane, and then use a method called stochastic proximity embedding to treat the edges like springs which are length 1 when balanced. One picks a spring at random and moves it towards equilibrium by a small amount. After repeating this process several thousand times, we hope that we have made each spring close to length 1. This would be strong evidence that the chromatic number of the plane is actually greater than or equal to 5.
There are several challenges to overcome in order to get this approach to work. First, computing the chromatic number of a graph is a computationally hard problem, so getting lots of good graphs to start with is no simple task. Here, a result due to Achlioptas and Naor which allows us to generate large graphs having a prescribed chromatic number with high probability is useful. Second, the graphs I have used do not seem to come close to having edges of length 1 after running them through the program, so one needs to find a clever way to “shake loose” or perturb a graph when it gets tangled. Third, simply knowing that all of the springs are close to being balanced does not mean that they are actually balanced, so we must find a way to decide when a graph which looks balanced actually is balanced.
Update: Hobbyist mathematician Aubrey de Grey established in April of 2018 that the chromatic number of the plane is at least 5. Read about his fascinating algorithmic approach to the problem here. Efforts to refine his results have been collected in a PolyMath project. One of the smallest currently known graphs which shows that the chromatic number of the plane is at least 5 has 803 vertices and 4144 edges and was found by Marijn Heule. A visualization of this graph is available here.