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#### Abstract

This research project is an investigation into quadratic rational maps, $\vp$, of one complex variable that map the unit disk to itself. Previous research \cite{brittney} shows that for each $\vp$, a corresponding linear fractional map $\zeta$ can be found using the coefficients of $\vp$, and this $\zeta$ can be used to characterize functions in the kernel of the adjoint of the composition operator with symbol $\vp$, defined on a space of analytic functions. In this paper, we show sufficient conditions to ensure that certain cases of $\vp$ map the unit disk to itself and find all the forms of $\zeta$. Finally, for a specific $\vp$, we search for the kernel of its corresponding composition operator by calculating the matrices of $C_\vp$ and of the Toeplitz operator $T_{-z \zeta'(z) / \zeta(z)}$, defined by projecting $\dfrac{-z \zeta^ \prime (z) }{\zeta(z)} f(z)$ onto the space of analytic functions.

#### Author Bio

At the time of the research project, Laura Schlesinger was a student of Coe College. Her hometown is Madison, Wisconsin, and a fun fact about her is that she used to play piano in a competitive marching band. She is a senior, and outside of math, practice yoga, write plays, and bake honey muffins. Upon graduation, she plans to attend graduate school for math education and research education policy.

At the time of the research project, Anna Marek is a senior at Coe College. Her hometown is Anamosa, Iowa. When she’s not enjoying mathematics, she’s usually enjoying computer science, reading a book, or playing with her cat, T. After she graduated, Anna took a software development job and started a Masters program in Human Computer Interaction.

Ella White is a student at Coe College. Her hometown is Minneapolis, Minnesota. Outside of studying mathematics, she enjoys playing tennis, writing papers, reading books, binge watching Netflix, and karaoke with friends. Ella is unsure what awaits her after graduation just yet, but is excited to navigate life after college.

Danqi Yin is a junior at Coe College. She is from Inner Mongolia, China. A fun fact about her is that she has a cute turtle as a pet for 11 years. Outside of math, she enjoys riding horses, reading sci-fi novels, playing Chopin's piano music, and cooking. After she graduates, she plans to take a trip to Hawaii and Japan, and then attend graduate school for mathematical physics.

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