# About

I am a 5th-year 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

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 one-relator 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.

### Hadwiger-Nelson 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:** Amateur mathematician and anti-aging expert Aubrey de Grey has
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.

### Visual Proofs

I love geometric proofs. One of my favorite things to do in my Calculus II class is to show my students a visual proof of the "sum of squares" formula \[\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6},\] which comes up when computing Riemann integrals from the definition. The particular proof that I give was shown to me by a professor I knew during my time with Budapest Semesters in Mathematics. I challenged myself to come up with an analagous proof of the "sum of cubes" formula (sometimes called Nicomachus' Theorem), \[\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}.\] The result is the proof described in this write-up from 2015:

# 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 TI-83. - 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. - Cut-the-knot

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 3D-printed examples. - quadibloc

John Savard's collection of math miscellany. - akalin.com/quintic-unsolvability

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 e-print collection managed by Cornell University. No subscription necessary! - MathSciNet

The AMS's catalog of research articles. Requires a subscription. - MathOverflow

For research-level 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 hit-or-miss... - Ester Dalvit

Relaxing and informative videos about visualizing knots in both low and high dimensions.