Brendan Sullivan

Assistant Professor

I studied mathematics and physics in college, and went on to graduate school to study applied math. While there, I found a passion for teaching and learning, and my dissertation instead focused on teaching students to read and write mathematical proofs. I still love mathematics for its usefulness in the sciences, and I do enjoy teaching students about the utility of mathematical thinking in the real world. But I also think of mathematics as a gigantic puzzle to play with, and I enjoy helping students explore how the pieces of that puzzle can fit together. Indeed, some of my research and teaching focuses on practical applications (like the theory of voting) while some of it is more recreational in nature (like pursuit-evasion games and crossword puzzles). Overall, I love thinking about and doing mathematics, and I hope that passion comes across and inspires my students to learn and use mathematics in their own lives.

What I Love about Emmanuel:

The students, faculty, and staff I’ve worked with are so supportive. I appreciate the Emmanuel community’s focus on helping each other to learn and to make positive changes in the world.

D.A., Mathematics, Carnegie Mellon University, 2013.
Dissertation: Textbook and Course Materials for 21-127 Concepts of Mathematics.
Advisors: Dr. Jack Schaeffer, Dr. John Mackey.

M.S., Mathematics, Carnegie Mellon University, 2010.

B.A., Mathematics & Physics, Hamilton College, 2007.

  • MATH1101 College Algebra
  • MATH1103 Precalculus
  • MATH1105 Math of Everyday Life
  • MATH1111 Calculus I
  • MATH1112 Calculus II
  • MATH1121 Applied Math for Management
  • MATH2103 Calculus III
  • MATH2109 Introduction to Proofs
  • MATH3101 Real Analysis
  • MATH3113 Special Topics in Math
    (Intro to Combinatorics & Graph Theory)

I have constructed some mathematics-themed crossword puzzles for the MAA's Mathematics Magazine:

  • "What Do You Study?", Math. Mag. 86 (2013) 370-371 [pdf link]
  • "Types Theory", Math. Mag. 87 (2014) 186-197 [pdf link]
  • "Award Winners", Math. Mag. 87 (2014) 360-361,402 [pdf link]
  • "2D or not 2D", Math. Mag. 88 (2015) 52-54 [pdf link]
  • "Books for a Math Audience", Math. Mag. 88 (2015) 154-155 [pdf link]
  • More can be found on my Google Scholar profile 

Recent publications:

  • Brendan W. Sullivan (2018) Open Problems in the Game of Lazy Cops and Robbers on Graphs, DOI: 10.13140/RG.2.2.15688.42246
  • Brendan W. Sullivan (2017) Lazy Cops & Robbers on Graphs: Extremal Problems, Algorithms, and Capture Times, DOI: 10.13140/RG.2.2.12315.16160
  • Brendan W. Sullivan, Nikolas Townsend & Mikayla L. Werzanski (2017) An Introduction to Lazy Cops and Robbers on Graphs, The College Mathematics Journal, 48:5, 322-333, DOI: 10.4169/college.math.j.48.5.322
  • Brendan W. Sullivan, Nikolas Townsend & Mikayla L. Werzanski (2016) The 3x3 rooks graph is the unique smallest graph with lazy cop number 3, arXiv:1606.08485 [math.CO]
  • "Dominos on a rectangular board", Math Problems 87 (2014) 299-302 [pdf link]

Past publications:

  • R. Bedient, M. Frame, K. Gross, J. Lanski, B. Sullivan, 2011: Higher Block IFS 2: Relations between IFS with different levels of memory. Fractals, 18, 399-408. [pdf link]
  • R. Bedient, M. Frame, K. Gross, J. Lanski, B. Sullivan, 2011: Higher Block IFS 1: Memory reduction and dimension computations. Fractals, 18, 145-155. [pdf link]
  • A.J. Silversmith, N.T.T. Nguyen, B.W. Sullivan, et al., 2008: Rare-earth ion distribution in sol-gel glasses co-doped with Al3+. Journal of Luminescence, 128, 931-933. [pdf link]

For my doctoral thesis, I wrote a textbook for a course that introduces students to mathematical proofs and problem-solving, entitled Everything You Always Wanted to Know about Mathematics (but didn't even know to ask): A Guided Journey into the World of Abstract Mathematics and the Writing of Proofs. Publication details forthcoming. I use this text in our course MATH2109 Introduction to Proofs, and it has been in use at Carnegie Mellon University for their course 21-127 Concepts of Mathematics for several years.

  • Hugh D. Young Graduate Student Teaching Award for 2013, Carnegie Mellon University (Mellon College of Science)

Current research: Pursuit-evasion games on graphs, especially Lazy Cops & Robbers.

  • Graph theory studies network connections. Think of Facebook, with nodes (called "vertices") representing each person and connections (called "edges") amongst the nodes representing who is friends with whom; in this sense, Facebook is a big graph.
  • "Cops & Robbers" is a game played on graphs. A team of Cops place themselves on the vertices of a given graph, and then the Robber places himself on a vertex. The two sides alternate turns: on their turn, the Cops get to move along the edges, then the Robber does the same, and they go back and forth like this. If a Cop lands on the Robber, he is caught and the Cops win. If the Robber is able to evade the Cops indefinitely, then he wins.
  • This game has been studied extensively since its introduction in the 1980s. Mathematicians have made great strides towards understanding what kinds of graphs allow one Cop to win, which graphs require more and how many are required, etc. However, there remain many open questions and unproven conjectures. This is a very active research area!
  • Since this is a game, this research can be purely recreational (as it is for me). But these results also have important applications and implications in computer science and programming. A good example is writing programs to search and organize large datasets efficiently.
  • "Lazy Cops & Robbers" is a variant of the game wherein, on the Cops' turn, only one of them is allowed to move. The main question becomes: Does this rule change the game significantly? For a given graph, do the same number of Cops suffice, or might we need more (and how many)?

I recently conducted a summer research project with two Emmanuel students. We investigated the Lazy Cops game on particular graphs, including graphs based on the movements of Chess pieces. We obtained several new and significant results and are currently working on publications to be submitted to scholarly journals. In addition, the two students (Niko Townsend & Mikayla Werzanski) will presented some of our results at the Young Mathematicians Conference at Ohio State University in Auguat 2015. Here is a link to the abstract for their presentation which summarizes our results.

Past research: In my undergrad and graduate careers, I worked on projects in a variety of math branches:

  • Fractals, specifically Iterated Functions Systems with memory. Suppose we have a set of transformations defined on the unit square. What if we disallow specific sequences of transformations? What sort of images result? Can such situations be reformulated by changing the set of transformations and removing the disallowance of certain sequences? This was investigated in previous publications: see links here and here. (This grew from a senior distinction project during my undergraduate years.)
  • Partial differential equations and the finite element method. I studied these topics in graduate school and am still interested in them.
  • Combinatorics and graph theory, specifically tiling problems and games on graphs (see, especially, "Cops & Robbers" above in Current research.)

Overall, I typically find myself interested in problems where several (perhaps seemingly unrelated) branches of mathematics collide.

I also maintain an active interest in the Basel Problem and the always-growing number of proofs of this centuries-old fact. I am compiling a list of all published proofs of this fact and have plans to write a book about them.