Identifier	Thesis 1978
Author	Nelson, Julie M.
Title	Researching and understanding the process of proving the Poincaré conjecture
Publisher	Central Connecticut State University
Date of Publication	2008
Resource Type	Master's Thesis
Abstract	What began as a competition to appease a member of royalty's birthday wishes, continued on to not only introduce an entirely new area of mathematical studies but also to leave one particular question to remain without a complete and substantiated answer for a span of centuries. On a much smaller, simpler scale, the consideration of the classification of surfaces can help to model the overall goal within the Poincar Conjecture. By surfaces we mean any two-dimensional figure which can cover a three-dimensional figure. These concepts can be simply manipulated and proven to stem from the combinations of three basic two-dimensional figures. The Poincar Conjecture takes this proposition to a new level and seeks to find a contradiction to a three-dimensional body having the same physical characteristics of what will be considered a 3-sphere while not being able to be molded into the 3-sphere itself. The complexity of the Poincar Conjecture lies in the fact that we are unable to manipulate such three-dimensional bodies because in order to do so we must imagine ourselves in a four-dimensional world. It is this one element which proves to be most challenging to those who have made attempts to answer the unanswerable since without the knowledge of what actions can and cannot be performed in the fourth dimension. This paper is an entirely basic outline of the process which was involved to prove the Poincar Conjecture. It begins with the classification of surfaces in order to have the overall understanding of the basic representations and manipulations that are used throughout the process. This also helps to indicate the structure that many mathematicians sought to follow within their approach due to the fact that it seemed the most logical format. After the classification of surfaces, the paper moves on to consider various topological invariants, as well as the mathematicians whose work supplemented one another in order to arrive at new branches of mathematics. With each new piece of information, more mathematicians were able to find intricate areas of relevance to the problem at hand. Within this section, the Euler characteristic is introduced, along with Thurston, a mathematician who was able to explain how being in the fourth dimension could be considered. He was able to take prior knowledge, along with his own visualizations and formulate some of the most unique and inspiring work on the Poincar Conjecture, along with the area of topology altogether. For the remainder of the paper, the focus is on the actual attempts to prove and disprove the Poincar Conjecture using the knowledge discussed previously in the paper. Finally we are able to revel in the general concept of the proof which was brought together in 2003 by Grigory Perelman, who was able to bring together the work of many mathematicians and fill in the gaps which they had been missing. Every piece of his proof required more proof and he had the reasoning to back it up. To see how everything was able to fit together so beautifully is breath-taking, which is precisely why this paper was able to come together in the end.
Subject	Poincaré conjecture
Department	Department of Mathematical Sciences
Advisor	McGowan, Jeffrey K
Type	Text
Digital Format	application/pdf
Language	eng
OCLC number	713733989
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Researching and understanding the process of proving the Poincaré conjecture

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