About Ben Stucky
I am a 5thyear mathematics PhD candidate at the University of Oklahoma. I am working under the direction of Max Forester. I spend most of my time thinking about topics in the field of Geometric Group Theory. See the section for more info. Through Spring of 2018, I organized the Student Geometry and Topology Seminar with Paul Plummer.
This semester, I am a visiting graduate student at Temple University. I have worked as a graduate teaching assistant at the University of Oklahoma since the fall of 2012. See the section for more info.
I have collected some interesting and useful math resources in the
section.In addition to mathematics, my hobbies include card games (some favorites are Hanabi and poker), skateboarding, “jazz” piano, and old video games (some favorites are Zelda: OOT, Zelda: MM, Earthbound and SSBM).
Contact
 Office at Temple University: Wachman 523
 Email: bwstucky AT ou DOT edu
 PGP key: public.txt
Research
Preprints and manuscripts
[arxiv] [abstract] 
Ben Stucky Cubulating onerelator products with torsion 

[pdf] [abstract] 
Ben Stucky Another visual proof of Nicomachus' theorem 
Summary of interests
My graduate research lies in the field of Geometric Group Theory (GGT). 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.
I am interested specifically in generalizations of onerelator groups (ORGs). ORGs are groups which admit a presentation with one or more generators and a single defining relator. These groups bear some similarities to their prototypes  free groups and surface groups  and one can ask how deep these similarities run. I am interested in conditions which ensure that ORG's have “negative curvature,” meaning that they admit nice actions on spaces which resemble hyperbolic space (as do, e.g., closed surface groups) or infinite trees (as do, e.g., free groups). Curiously, groups which admit negative curvature enjoy many nice properties which make them much easier to study than general groups. See my research statement for more information.
HadwigerNelson Problem
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 method due to Achlioptas for generating 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.
 This graph shows that the chromatic number of the plane is at least 4.
 This tiling shows that the chromatic number of the plane is at most 7.
 An example of a graph I would feed to the program.
 What a graph might look like after being run through the program. The green edges are springs which are close to 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 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.
Teaching
I have more than six years of experience teaching at the college level. The courses that I have taught 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.
Evaluations
Most of my students seem to respond well to my approach in the classroom. Here are some select student evaluations.
 During the summer of 2017 I taught Calculus II to a class of 18 students. View their evaluations here.
 During the spring of 2017 I TA'ed an Honors Calculus II course for Dr. Ameya Pitale, leading a discussion section of 24 students. View their evaluations here.
Links
Resources for learning mathematics, visualization, and homework help
 Math Learning Center
Collection of apps to visualize precalculus topics. I particularly like Geoboard.  WolframAlpha
One of the easiest ways to check answers on your homework.  Desmos online graphing calculator
Great if you don't have access to a TI83.  3Blue1Brown's “Essence of Calculus” series
Excellent series that I always require my calculus students to watch. I learned some new things as well!  3Blue1Brown's “Essence of Linear Algebra” series
Same as above, but for linear algebra.  MathStackExchange
One of the best places to ask for homework help or other questions you have (undergrad through graduate topics). You are required to outline your thought process in the question.  Khan Academy
Free online courses, lessons & practice.  MIT Open Courseware
Free online course materials.
Resources for reference
 $\pi$base
A community database of topology concepts and examples with expressive searches.  indiana.edu/~knotinfo/
Online tabulation of knots up to 12 crossings.  House of Graphs
Searchable database of finite graphs.  Numbers Aplenty
Explore interesting properties of the natural numbers.  Online Encylopedia of Integer Sequences (OEIS)
The world's largest collection of integer sequences.  Cuttheknot
Alexander Bogomolny's collection of math miscellany from a variety of topics, with excellent descriptions.  The Geometry Junkyard
David Eppstein's collection of mostly open problems in geometry.  Cardcolm.org
Colm Mulcahy's collection of mathematical card tricks.  Henry Segerman's website
A collection of fascinating (mostly geometric) miscellany with an emphasis on 3Dprinted examples.  quadibloc
John Savard's collection of math miscellany.  akalin.com/quinticunsolvability
Fred Akalin's demonstration of Arnol'd's topological proof of unsolvability of the quintic.  Tilings & Geometric Ornaments
From website: “The goal of this project is to explore the relationship between computer graphics, geometry, and ornamental design.”  Math riddles subreddit
Fun recreational math problems from various sources.
Resources for research
 The arxiv
STEM eprint collection managed by Cornell University. No subscription necessary!  MathSciNet
The AMS's catalog of research articles. Requires a subscription.  MathOverflow
For researchlevel mathematics questions.
Math blogs
 Low Dimensional Topology Blog
 Here There Be Dragons
 Danny Calegari
 Wilton  Geometric Group Theory
 Chiasme
 Alessandro Sisto
 Baking & Math
 Sketches of Topology
Math vlogs
 3Blue1Brown
 Mathologer
 Numberphile
This one is pretty hitormiss...  Ester Dalvit
Relaxing and informative videos about visualizing knots in both low and high dimensions.