We found the image of a subgroup of the mapping class group under the symplectic representation. In particular, this subgroup is the normalizer of a finite-order element of the mapping class group.
Grad student mentor: Sarah Davis
Undergrad researchers: Laura Stordy, Ziyi Zhang
We studied a variation of the "twisted rabbit problem": a polynomial can be viewed as a branched cover of the complex plane. Start with a polynomial. Then compose it with a Dehn twist (in the complex plane with marked points). The new map f will be equivalent to a polynomial p, meaning there is a change of coordinates of the complex plane that takes f to p.
Grad student mentor: Sarah Davis
Undergrad researchers: Janet Huffman, Ruotong Zhai
We studied the Magnus representation of the mapping class group, particularly by identifying a class of elements that are not in the kernel of the Magnus representation.
Undergrad researcher: Mira Wattal
We studied pseudo-Anosov mapping classes of nonorientable surfaces with punctures.
Grad student mentor: Sayantan Khan
Undergrad researcher: Caleb Partin
(Co-mentoring with Jun Li)
Project description: The main goal of the project is to understand the construction of mapping classes of surfaces coming from curves, provide visualization, and produce new constructions.
We will begin with reviewing the geometry and topology of surfaces, and learn some basic properties (include the important Nielson-Thurston classification and pseudo-Anosov maps) of mapping class group, which describes the symmetry of the surfaces. We will also study how to construct the elements of the latter from the former. Questions we are going to explore are:
1. What kinds of important structures (we’ll focus on lamination and train tracks) about curves are used in the geometry/topology of surface, and why are they interesting? We’ll learn some of Thurston’s seminal work.
2. What’s interesting and significant about pseudo-Anosov maps?
3. What are the existing constructions using structures in Question 1 to construct pseudo-Anosov maps in Question 2, and how to visualize them? In particular, we’ll focus on Penner’s construction.
4. It was recently shown that Penner’s construction does not cover all pseudo-Anosov maps. We will learn about what’s known about pseudo-Anosov maps that are not coming from Penner’s construction. An interesting open problem is to find new constructions for these maps using laminations and train tracks.
Grad student mentors: Bradley Zykoski
Undergraduate researchers: Anthony Morales, Zijian Rong, Wendy Wang
We wrote a program that approximates the proportion of "airplane polynomials" that result when we construct polynomials by composing the "rabbit polynomial" with a mapping class.
For mapping classes of word length 10 or less (over 100,000 mapping classes), about 57% of polynomials are airplanes.
The grad mentors for the project are Alex Kapiamba and Jasmine Powell.
The undergraduate team members are Trey Austin, Allen Macaspac, and Hannah Moon.