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Abstract

In the summer of 2009, our group developed a computer program that computes Hochschild Homology, a topological invariant. While we must assume that the reader has at least encountered algebraic topology, in this paper we provide the mathematical background and motivation for our algorithm. After presenting a number of definitions, we will explain how the algorithm works. Specifically, we first define the Floer complex of two curves on surface; the resulting homology is invariant under isotopies. Then, we introduce the Fukaya category associated to a sequence of curves. Next, we define the Hochschild complex of the Fukaya category. And finally, we describe an algorithm for computing Hochschild Homology and provide some examples.

Author Bio

Jin Woo Jang is a fourth-year student at Columbia University. He graduated from Incheon Science High School in Korea in 2007. During the summer in 2009, he started working on this project in algebraic topology developing the computer programs for computing Hochschild homology. This project was completed in the summer of 2010 with funding from NSF㤼㸲s RTG grant DMS-0739392, and with guidance from our professors, Robert Lipshitz and Timothy Perutz, and their teaching assistants, Thomas Peters and Jonathan Bloom. After graduating Columbia University in May 2011, he will attend graduate school in mathematics. His research interests are in harmonic analysis, analytic number theory, and algebraic topology.

Xuran Wang graduated from Columbia University with a Bachelor in Mathematics and a Bachelor in Economics in spring 2010 as a Rabi Science Scholar. This research paper is completed in the summer of 2009 as part of the Summer Undergraduate Research Program funded by the mathematics department at Columbia under the guidance of Professor Robert Lipshitz and Professor Tim Perutz.

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