Jarred J. Wieser

“Math is the science of patterns.”

[Keith Devlin, Mathematics: The Science of Patterns (3)]

I can’t pinpoint when my fascination with patterns started, but I do know that the learning I enjoy the most (and when my mind functions the best) involves recognizing patterns: I excelled in foreign language classes when I noticed the patterns in verb conjugations. I recognized patterns in and through every competitive sport in which I took part (football, basketball, and track). Yet the area in which the patterns spoke to me most intensely was music. The patterns in the pieces I performed on the trombone and upright bass stimulated both my ears and my intellect. I became proficient in jazz, where chord patterns proliferate . . . as a matter of fact, I kept my mother awake for hours at night because I could not stop jamming through jazzy chord progressions. That teenage engagement with patterns and rhythms resurfaced in yet another form in college, namely mathematics: Sequences and Series became my new jam. Finally I found myself in Discrete Math, taught by Dr. Manda Riehl; little time passed before I started work on what led to my current research collaboration with her, a study of permutation pattern avoidance in posets.

That was the path: this is the research. We have researched an extension of pattern avoidance to a new structure: multiple task-precedence posets with three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered several interesting bijections between: diamonds avoiding 132 and certain generalized Dyck paths, diamonds avoiding 132 and 213, and diamonds avoiding 231 and 312. We have also found the generating function for descents in these permutations for the majority of collections of patterns of length three. In the near future, we aim to find closed formulas for avoiding 231 (we currently have a recursive formula) and avoiding 321 and their descent generating functions. Furthermore, an interesting application of this work can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures that two heavier packages are never stacked on top of a lighter package.

Patterns--whether in music, sports or mathematics--also led me to enjoy competition and collaboration. Friendly team competition in Calculus II and III met a team environment in Abstract Algebra and Real Analysis. Both settings relied on participants’ articulate communication, tactics, and cleverness to produce high team scores or generate solutions: perfect training for venues beyond the classroom, namely, the Wisconsin Math Modeling Conference (WMMC) and the Midwest Undergraduate Data Analytics Competition (MUDAC). We took no prize home, but we won knowledge about available software, comparative models, techniques to present one’s model clearly, and self-confidence. Smarter and seasoned, the UWEC team(s) returns to MUDAC in 2016!

Initially a mathematics education for secondary school major, I have learned much math pedagogy, and while I redirected my major at the student teaching phase to mathematics liberal arts, I appreciate the insights into mathematical operations that all of the instructional experiences have provided. I saw, for example, that my approach and effective teaching built successful relationships with students and my cooperating teacher, Mrs. Janelle Yeakey (Chippewa Falls High School) in Algebra II; there I taught lessons, facilitated daily reviews, or went over exams with students who wanted to make test corrections. At the University of Wisconsin-Eau Claire I have tutored students in physical science, general physics, and elementary statistics. My understanding of Calculus II and Abstract Algebra increased tremendously as I assisted Drs. Simei Tong and aBa Mbirika [sic.]. In addition, during Spring 2015, I taught the final review for Calculus II with upperclassmen Sooki Liu. While working in the University Math Lab during the summer of 2015, I have also assisted students in Intermediate Algebra, Calculus I, and Calculus II.

In graduate school I want to pursue a doctorate in either applied or pure mathematics, or Math/STEM Education. Within applied math I have interests in Computational Biology and Fluid Dynamics. In pure Math I am very interested in Enumerative Combinatorics. Within Mathematics Education I see an imminent need for secondary curricula to include Discrete Math and Combinatorics--subjects that would prepare students for computer science and other STEM fields. (I myself would have enjoyed math in high school had I operated more with patterns than with computation.)

I have a knack for tinkering with web programming and other computer-oriented tasks. This could make the work of faculty in my graduate department much easier and simpler. Going forward into graduate school and pursuing a PhD is a high goal and a dream of mine. At this time as I contemplate a dissertation project I could see myself creating a mathematical model of malaria-carrying mosquitos in Africa, or a computational model of some aspect of cancer cells, or integrating discrete mathematics into the K-12 math curriculum.

Pattern recognition, collaboration, and competition drew me in to study mathematics as an undergraduate. Competitive sports and music caused me to love pattern recognition, collaboration, and competition: yes, I see a pattern developing here in my own life. As we increase our familiarity with the patterns that proliferate in our world, I want to be on board studying what these patterns will reveal.

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