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\title{Statement of Purpose}

\author{ Shuyun (Conan) Wu }
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In summer 2007, after high-school graduation, I went to the Canada/USA Mathcamp - the place where I first learned about topology, abstract algebra, real analysis and much more. At the time, this was a strange new universe for me. Since then I knew mathematics is the career I will pursue. Later that summer, I was invited to Russia for the Tournament of Towns Summer Conference. While there, I did a project on the carpenter's rule problem - a discrete geometry problem that states the following:
\begin{problem}
Can every carpenter's rule (a finite number of segments with fixed lengths joined by vertices) in $\mathbb{R}^2$ be
straightened? (be deformed to a straight line segment by an isotopy that avoids self-intersection and preserves edge
lengths)
\end{problem}
After being open for thirty years, a positive answer was given by Robert Connelly, Erik Demaine and G\"{u}nter Rote in 2000. Under the guidance of Professor Panina, I worked through a series of problems which leads to the proof. In the process I came up with elegant counterexamples to some of the intermediate problems which were not known to the professor. We expanded on the topic of pointed graphs in the plane and sphere (one of the tools used in the proof of the carpenter's rule problem) and discussed some of her research topics afterwards.

At the end of that summer, I started my undergraduate study in mathematics at University of Toronto. As an undergrad I focused on getting a general idea in all main branches of mathematics as well as working on some more difficult problems that interest me. During my first year, this ``problem part'' was mostly devoted to the three-star problems in Professor Charles Pugh's real analysis textbook -- those problems which he ``did not know the answer to''. I gained much courage and satisfaction by solving six of them.

At the beginning of summer 2008, I got the opportunity to work on a research project under the University of Toronto Excellence Award - a grant offered each year to around forty undergraduate students in all science and engineering for supporting their proposed research projects. This enabled me to work with professor Pugh throughout the summer months on continuum theory and the elusive fixed point conjecture:
\begin{problem}
Does every continuous map from a non-separating plane continuum into itself have a fixed point?
\end{problem}
I explored special cases of the problem while learning about the progress made on the problem to date. Along the way I have been making attempts in constructing counterexamples, although none worked yet. Through doing this I now fully understand many interesting continua such as the pseudo-arc, the bucket-handle, lake of Wada and numerous original constructions I came up with. It was an extremely pleasant experience working with Professor Pugh; he is always inspiring and full of interesting ideas. Under his influence, I found myself being most interested in real analysis and topology among other kinds of mathematics.

During my second and final year, through an introductory graduate course, I became fascinated by the subject of dynamical systems, especially the parts related to topology. In particular, I found Stephen Smale's work on Axiom A dynamics being absolutely beautiful. Perhaps dynamics is the field I will choose to work in.

Aside from classes, I have been participating in the Fields Institute seminar on the Kakeya conjecture since the beginning of this semester. This is a well-known outstanding problem, and yet the statement is extremely simple:
\begin{problem}
Any Kakeya set in $\mathbb{R}^n$ must have Hausdorff dimension $n$.
\end{problem}
I have mainly been attempting to find counterexamples and am currently trying to apply set-theoretical methods to get  consistency results for the existence of such examples. In the process of doing so, I got a more through understanding of the Hausdorff dimension and other related concepts in fractal geometry. Through many discussions with Professor Guth and various other experts in the Kakeya conjecture, I have learned a lot both about Kakeya and analysis in general.

Throughout my undergraduate study, I have always been doing set theory as a side topic. Although I am not planning to become a set-theorist, I believe in the strong connection between set theory and analysis, especially when it comes to forcing and consistency results. At least, having background in set-theory adds an interesting viewpoint when attempting to solve problems.

On the other hand, I believe teaching is a part of mathematician's job. Hence I started to work as a teaching assistant in my second year, delivering two tutorials sessions per week to a freshmen calculus class. I have been trying to find the most efficient way to explain concepts while keep the audiences interested. Due to the success of my tutorials, I am recently given the opportunity to TA the honors third-year real analysis course in the coming spring term. This is going to be a new challenge since the job is usually taken by an upper-year graduate student.

For many reasons, I have always been longing to pursue graduate studies at UC Berkeley. Despite the fact that it's one of the most distinguished departments in the world, I am deeply attracted by its unique culture in mathematics and the diversity of its faculty's research fields. I have a clear vision of myself fitting in Berkeley's broad and active department and getting further in mathematics.

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