Page 31 - Undergraduate Research Opportunities Program Proceedings 2017-18

School of Science
Department of Mathematics

The title of this UROP project is “Statistical Analysis of Competition Data”. In this UROP project, we collect millions of
posts from Twitter and combine them with the corresponding odds data, to analyze and predict the outcomes of World
Cup football matches. First of all, we wrote programs to crawl data from the bookmaker website and twitter each time
there exists a world cup soccer match. After collecting all the data we need, we clean and integrate large amount of data
and do some simple analysis. Finally, we use statistical software to analyze these data. During this UROP project, I learnt
more about statistical modeling and how to deal with large amounts of data.

Geometric Flows
Supervisor: FONG Tsz Ho / MATH
Student: CHEN Xinyi / SSCI Course: UROP1100, Summer

The paper written b yM.GAGE & R.S. HAMILTON and my report are all playing around with this main equation. We
consider a initially convex curve and shrink it under the heat equation mentioned above. We study the curve when the
evolution is undergoing and prove it would goes to a circular point after a finite time T. Moreover, after scaling the curve,
we can see clearer that it goes to a circle. This report is mainly a summary and supplementary of the original paper,
which completes the proofs that were omitted by Hamiltion.

Supervisor: FONG Tsz Ho / MATH
Student: GUTIERREZ Mauricio Antonio Course: UROP1000, Summer
RODRIGUEZ / SSCI

This report will summarize the paper by Gage and Hamilton while also going more in detail into the explanations. The
numbering and structure will follow that of our main reference. The first section will give the basic context of the
problem, while proving that simple curves remain so during evolution. On the same chapter is a proof of a weaker
isoperimetric inequality without the equality case. The next chapter will restate the problem into a parabolic partial
differential equation and expose bounds on the curvature and its derivatives to prove the long term existence. The last
chapter will prove the ratio of the maximum and minimum curvatures to approach, and finally prove the higher
derivatives uniformly converge to 0 through computations.

Supervisor: FONG Tsz Ho / MATH
Student: HUNG Yui Chi / SSCI Course: UROP1000, Summer

In the paper The Heat Equation Shrinking Convex Plane Curves by M. Gage and R. S. Hamilton, it proves that the heat

equation (note: F is the parametrization, k is the curvature and N is the unit normal vector) will shrink a
convex plane curve into a point with several characteristics. Nevertheless, the authors have omitted several details and
have left them as exercises for the readers. This report mainly highlights the main steps leading to the main theorem
and ”fills” the details that have been omitted by the authors. In this report, section 2 mainly outlines the structure of the
paper, while section 3 constitutes some details of the parts that the authors have omitted.

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