Catalan Numbers: Applications and a Generalization
Speaker: Stefanie G. Wang, Iowa State University
Date & Time:
12:00 PM - 1:00 PM
Owens Science Center (OWS) 250
How many ways are there to assign parentheses to an expression like 1 − 2 − 3 − 4 − 5? A versatile sequence of numbers, called the Catalan numbers, tells us the answer to questions like this! Catalan numbers have many interpretations in mathematics. This talk will discuss several interpretations, including the number of ways to triangulate a convex (n + 1)-gon, and the number of full rooted binary trees with n leaves. Rooted binary trees represent the number of ways to assign parentheses to an n-argument expression involving a single nonassociative binary operation. For instance, consider integers and subtraction. The expressions 2 − (5 − 7) and (2 − 5) − 7 represented by the trees
The talk will also provide an introduction to a nonassociative structure called a quasigroup, and an extension of Catalan numbers that arise in the study of quasigroups, called peri-Catalan numbers.
Biographical Information: Stefanie Wang is a fifth-year graduate student at Iowa State University completing her Ph.D. in algebra under the direction of Jonathan D.H. Smith. She earned her B.A. at Saint Mary's College of California in the Integral Liberal Arts Program with a minor in mathematics. After that, she studied in the post-baccalaureate program for women in mathematics at Smith College. Stefanie is an avid powerlifter, knitter, and yoga practitioner. Stefanie will join the faculty at Trinity College in Hartford, CT as a postdoctoral fellow this fall. She enjoys spending time with her three gerbils, none of whom study math.