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Title: "Computational Geometry" Abstract: We know that the Greeks knew a lot about geometry more than 100 generations ago, measured from teacher to student. It began as the study of the space we live in and had numerous motivations such as building and navigation. Whilst still incomplete, the geometry we know today was created by thousands of human minds over centuries. Most classes these days still focus on the processes of correct computation. Many students find that vector calculus is the first course where the ambition to be able to calculate any calculation that comes to mind starts to gain momentum. We learn to calculate by following the pros, and despite the inevitable rise of educational videos on the subject, the functional units of education in modern geometry are the lecture and the textbook, and these essentials are deconstructed and reconstructed by every mathematician who joins in the process of developing the subject. Mathematics has been a very open and available arena supporting sciences and engineering across the country and around the world, because that is how mathematics gets preserved and not lost. When I took up an associate professor position at UCSD in 1997, I was fortunate that Peter Teichner had just been hired in topology. Both of us wanted to learn more geometry, me from the analysis perspective and him from the topological. We soon started running a little graduate seminar together styled on Mike Freedman's famous graduate topology seminar at UCSD, and the graduate analysis seminar at UCLA. I thought about some small points of difficulty that Peter and I spent time trying to explain in our seminar, and in retrospect we could have organized it differently to better clarify those points instead of trying to follow standard text books. Therefore, instead of letting the work go to waste, I thought I would give you the first lecture of the improved version.