Brendan W. Sullivan: Publications

Journal Papers

• 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.
  
  A continuation of the paper listed immediately below. This paper discusses the relationships between systems with different memory levels, and considers their fractal dimensions.


• R. Bedient, M. Frame, K. Gross, J. Lanski, B. Sullivan, 2011: Higher Block IFS 1: Memory reduction and dimension computations. Fractals, 18, 145-155.
  
  The first in a series of papers on the topic of my undergraduate math research project with my advisor, R. Bedient. We studied a particular Iterated Function System framework with "memory" conditions that restrict the order in which transformations may be applied. The goal was to classify when such a system can be represented by a similar one with "shallower" memory levels. This paper describes the conditions that determine when this can be done and how to do it.


• 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.
  
  Work by my undergraduate physics advisor, A.J. Silversmith, and her collaborators, on work related to my thesis project. I studied the mathematical model used to predict dispersion of aluminum and terbium ions as dopants in rare-earth injected sol-gel glasses, and experimentally studied the resulting effect on fluorescence under ultraviolet light.

Presentations

• Textbook and course materials for 21-127 Concepts of Mathematics: Thesis defense for a D.A. in Mathematical Sciences
  
  This is the presentation I gave to (successfully! yay!) defend my doctoral dissertation work. As you can read in the slides, I wrote a textbook for the course 21-127 Concepts of Mathematics. It is a flagship course of the CMU math undergrad education experience, meant to guide students in the transition from rote, computation-based coursework to higher math, abstract thinking, and writing proofs.


• "The Convergence of a Random Walk on Slides to a Presentation" / slides here
  
  This talk was the culmination of a few weeks of effort based on a crazy idea that some fellow grad students had. One week, I was running late to give my presentation, but my friends happened to know I had already posted the slides on my website to share with others. They joked that someone should just get up and give the talk, not knowing really where the slides would be going. I showed up a few minutes late to "save the day" and gave the actual talk. But the joking idea planted a seed ...
  Afterwards, everyone was talking and it was decided that the following was not only a good and humorous idea, but that we would enact it:
  Someone would oversee the creation of a series of slides for a presentation via "the telephone game" (I have since learned that "exquisite corpse" is a better descriptor, but the idea is similar.) She created the title slide, then passed that to the next person. That person created the first slide, having only seen that title. Then, the organizer forwarded only that slide to the next person, who created the second slide. Then, the third person made their slide having seen only the second slide, and so on.
  Eventually, they had created a presentation with 13 slides, each person having only seen their own and the one before it (except for the organizer, who saw them all). This is where I entered the picture. I volunteered to give the actual presentation, having never seen any of the slides before!
  Knowing that the entire humor of the talk would come from the actual talk itself (particularly the transitions between slides, I was presuming), and not the visual record of the slides, I thought it would be a good idea to record the talk. My friend sat in the front row and used my Flip cam to record the event. And here it is!


• The Basel Problem: Numerous Proofs
  
  Beamer slides for a seminar talk given on Thursday, April 11, 2013 at the CMU Math Grad Student Seminar series. I presented 5 hand-picked solutions of the Basel Problem, which show that the sum of the reciprocals of the squares is ¹⁄₆ π². For each proof, I provide a historical context, a sketch, the proof details, and then a summary.


• How Many Ways Can We Tile a Rectangular Chessboard With Dominos?: Counting Tilings With Permanents and Determinants
  
  Beamer slides for a seminar talk given on Wednesday, February 20, 2013 at the CMU Undergraduate Math Club. It presents an algorithmic solution for computing the number of distinct domino tilings of an m x n rectangular chessboard by reformulating the problem in terms of perfect matchings in graphs and computing the determinant of an appropriate matrix.
  (Note: Here is a series of exercises, with solutions, meant to guide you into discovering the closed-form solution for the problem that I mention at the end of the talk. I did not create these files.)


• I Have Counting Problems!: Three equivalent formulations of a simple counting problem that relate combinatorics, generating functions, and differential equations
  
  LaTeXed .pdf forthcoming! This is a seminar talk given in October 2012 at the CMU Math Grad Student Seminar series. I discussed a combinatorics puzzle I had worked on recently that found me having to solve a variety of equivalent formulations, none of which I was able to tackle. I presented the problem and my approaches and showed how they were all related.


• Memory Reduction In Iterated Function Systems: Closing off (kind of) an avenue of measuring fractal complexity.
  
  Beamer slides for a seminar talk given on Wednesday, February 1, 2012 at the CMU Math Grad Student Seminar series. I discussed the above paper (Higher Block IFS 1) on fractals generated by Iterated Function Systems with memory, and when and how memory reduction can be achieved in those systems.


• Mathematical Diversity and Elegance in Proofs: Who will be the next Renaissance man?
  
  LaTeXed .pdf of a seminar talk given on April 18, 2012 at the CMU Math Grad Student Seminar series. I discussed diversity and elegance in proofs, exhibiting several examples: proofs of the infinitude of primes via topology, Turán's Theorem via probability, and Cayley's Formula via linear algebra.


• Numerous Proofs of ζ(2) = ¹⁄₆ π²
  
  LaTeXed .pdf of a seminar talk given on Thursday, November 19, 2009 at the CMU Math Grad Student Seminar series. I discussed about a half dozen different solutions of the Basel Problem.


• IFS With Memory
  
  Powerpoint slides for a presentation at the Hudson River Undergraduate Mathematics Conference at Sienna College (Loudonville, NY) in April 2007. I discussed IFS with memory (Note: The above file is a .pdf and ruins the "transition effects" of the presentation; email me if you want the .ppt file.)

Books

• R(3,3) = 6
  
  An excerpt from my forthcoming textbook project. This is from Chapter 1, wherein I introduce the idea of abstract mathematics and problem-solving and attempt to motivate the reader into developing cogent arguments for their intuitions. The ending section is a series of "puzzles" for the reader that are posed and then answered, following thorough explanations of my own thought processes as I would go about solving them. This particular puzzle amounts to proving a fact about Ramsey Numbers, namely R(3,3)=6, although this terminology isn't used, of course :-)