Research Interests

Broadly, I am interested in Data Science, Graph Theory, and Operations Research. However, my research has always bounced between my love for Computer Science and my love for Mathematics. Recently, I have been fascinated with the advances in Artificial Intelligence through Machine Learning and Deep Learning. My current research is on using a deep learning with natural language processing to convert mathematical papers into the language of Homotopy Type Theory for the purpose of automatic proof checking by Coq.

My Academic Research

Graphs are an amazing way to represent data. This is codified in mathematics. In 1980, Ales Pultr and Věra Trnková proved that every countable concrete category can be represented in the category of graphs. However, Graph Theory has many open problems and little in the way of an overarching theory. My academic research is geared toward the generation of such a theory.

For my Ph.D. dissertation, I studied six categories of graphs, discovered an extension of the Noether Isomorphism Theorems to two of these categories, and provided an elementary axiomatization of four of these categories in the style of F. W. Lawvere's Elementary Theory of the Categories of Sets. With Dr. Tien Chih, the extension of the Noether Isomorphism Theorems was applied to the Graph Reconstruction Conjecture, creating an existence formulation of the conjecture.

Researching the graph ecosystem in the categories of graphs has provided me many insights. So too has studying the anatomy of graphs. In our recent publication, Graphs and Their Inverse Semigroups, I worked with Dr. Tien Chih to show that graphs are defined by their local symmetries. The idea behind this approach is the same idea that rejects that a triangle and a Sierpinski Triangle are the same because they have the same group of symmetries. As seen above, the Sierpinski Triangle has a wealth of local symmetries that define it. Graphs find themselves in the same situation, and the language of inverse semigroups gives us a natural framework to understand graph local symmetries.

I am also a fan of graph applications. One such application is the PageRank algorithm that Google uses for ranking websites. Working with Dr. Tien Chih, we applied a modification of the PageRank algorithm to rank champion boxers.

Publications and Preprints

• D. Plessas, Special Morphisms in the Categories of Graphs, currently submitted for peer review.
• T. Chih and D. Plessas, A Search for Champion Boxers, currently under peer review.
• T. Chih and D. Plessas, The Fundamental Morphism Theorem in the Categories of Graphs with an Application to Graph Reconstruction Conjectures, currently under peer review.
• T. Chih and D. Plessas, Graphs and Their Inverse Semigroups, Discrete Mathematics, Vol. 340, Iss. 10, Oct. 2017.
• D. Plessas, The Categories of Graphs, Ph.D. dissertation, The University of Montana, 2011.
• D. Plessas, Topos-like Properties in Two Categories of Graphs, Master’s Thesis, The University of Montana, 2008.

Presentations

• Catching Quasars with Neural Nets - Machine Learning in Astrophysics, Y0 Physics Seminar at Northeastern State University, February 27th, 2017.
• What Good Is Homework? A Case Study in Data Analysis, Oklahoma Alpha Chapter of Kappa Mu Epsilon, February 20th, 2017.
• Homotopy Type Theory: The Most Important Development in Mathematics You Have Never Heard Of, Oklahoma Alpha Chapter of Kappa Mu Epsilon, September 23rd, 2016. 
• Learning from the New Masters: Using YouTube to Inspire, Community and Collaboration Day at Northeastern State University, August 10th, 2016.
• Graph Theory in Quantum Mechanics, Y0 Physics Seminar at Northeastern State University, March 21st, 2016.
• Graphs Are Uniquely Identified by Their Inverse Semigroup, 2016 AMS-MAA Joint Mathematics Meetings, January 7th, 2016.
• Discovery at a Distance: Technology Solutions for Research Collaboration, Community and Collaboration Day at Northeastern State University, August 12th, 2015.
• Avoiding Gödel’s Incompleteness Theorems, 2015 Annual Meeting of the OK-AR Section of the MAA, April 11th, 2015.
• From Gödel to Turing: What Incompleteness Tells Us About the Limitations of Computation, Y0 Physics Seminar at Northeastern State University, March 23rd, 2015.
• An Elementary Theory of the Categories of Graphs, 2015 AMS-MAA Joint Mathematics Meetings, January 12th, 2015.
• Is NASA Strangling Us? Examples of Bad Science and Bad Statistics, Y0 Physics Seminar at Northeastern State University, November 17th, 2014.
• My Favorite Unsolved Problems, Oklahoma Alpha Chapter of Kappa Mu Epsilon, October 29th, 2014.
• Generating Caves in Video Games, Oklahoma Alpha Chapter of Kappa Mu Epsilon, April 20th, 2014.
• The Mathematics of Procedural Generation in Video Games, 2014 Annual Meeting of the OK-AR Section of the MAA, April 12th, 2014.
• How Recreational Mathematics Becomes Applied Mathematics, Oklahoma Alpha Chapter of Kappa Mu Epsilon, February 26th, 2016.
• Euclid Versus Lobachevsky, What Shape Is Local Space?, Y0 Physics Seminar at Northeastern State University, February 24th, 2014.
• How Recreational Mathematics Becomes Applied Mathematics in Graph Theory, The University of Minnesota Morris, April 22nd, 2013.
• Using the Binomial Distribution to Model Gender Selection in Beehives, Montgomery College, April 17th, 2013.
• What Do Games Tell Us About Graphs?, Special Session on Games, 2013 Annual Meeting of the Pacific Northwest Section of the MAA, April 13th, 2013.
• How Recreational Mathematics Becomes Applied Mathematics in Graph Theory, Carroll College, January 28th, 2013.
• Topos Axioms and the Categories of Graphs, 2013 AMS-MAA Joint Mathematics Meetings, January 10th, 2013.
• Graph Embedding and Hedetniemi's Conjecture, Spokane Regional Mathematics Colloquium at Gonzaga University, November 7th, 2012.
• Graph Embedding and Hedetniemi's Conjecture, Department of Mathematics Colloquium at the University of Idaho, November 1st, 2012.
• The Fundamental Morphism Theorem in the Categories of Graphs, Department of Mathematical Sciences Colloquium at the University of Montana, October 29th, 2012.
• Larson's Independence Decomposition Theorem, Combinatorics and Optimization Seminar at the University of Montana, October 10th, 2012.
• My Favorite NP-Hard and NP-Complete Problems, Computer Science Club at the University of Montana, October 5th, 2012.
• Computers Have Limits: What Is P=NP?, Undergraduate Mathematics Seminar at the University of Montana, October 2nd, 2012.
• A Special Case of Hedetniemi's Conjecture, Montana Academy of Sciences and Montana Sigma Xi Joint Meeting, April 14th, 2012.
• The Preflow Push Algorithm, Combinatorics and Optimization Seminar at the University of Montana, March 14th, 2012.
• The Ford Fulkerson Algorithm, Combinatorics and Optimization Seminar at the University of Montana, March 7th, 2012.
• My Undergraduate in a Mathematics and Computer Science Integrated Department, Computer Science Club at the University of Montana, March 9th, 2012.
• Clever Counting - The Art of Combinatorial Proof, Undergraduate Mathematics Seminar at the University of Montana, March 6th, 2012.
• How Recreational Mathematics Becomes Applied Mathematics in Graph Theory, Undergraduate Mathematics Seminar at the University of Montana, November 1st, 2011.
• Applications of Linear Optimization: Milk Truck Routing in Italy, Combinatorics and Optimization Seminar at the University of Montana, October 12th and 19th, 2011.
• How Recreational Mathematics Becomes Applied Mathematics in Graph Coloring, Graduate Student and Faculty Research Conference at the University of Montana, April 23rd, 2011.
• A Special Case of Hedetniemi’s Conjecture, Department of Mathematical Sciences Colloquium at the University of Montana, April 21st, 2011.
• An Elementary Theory of the Categories of Graphs, Montana Academy of Sciences and Montana Sigma Xi Joint Meeting, April 16th, 2011.
• Graphs and the Fundamental Morphism Theorem, Graduate Student and Faculty Research Conference at the University of Montana, April 24th, 2010.
• Graphs and the Fundamental Morphism Theorem, Montana Academy of Sciences and Montana Sigma Xi Joint Meeting, April 10th, 2010.
• Alfred Tarski and the Completeness of Euclidean Geometry, Montana Academy of Sciences and Montana Sigma Xi Joint Meeting, April 11th, 2009.
• Shortest-Path Algorithms, Combinatorics and Optimization Seminar at the University of Montana, March 4th, 2009.
• Graph-Like Objects in an Abstract Category, Montana Academy of Sciences and Montana Sigma Xi Join Meeting, April 12th, 2008.
• Bipartite Objects in an Abstract Category, Graduate Student and Faculty Research Conference at the University of Montana, April 5th, 2008.
• Simple Loopless Graphs with Strong Morphisms is an Impoverished Category, Montana Academy of Sciences Annual Meeting, April 14th, 2007.