CombinatorialLaplacians - Combinatorial Laplacians of...

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Combinatorial Laplacians of Simplicial Complexes A Senior Project submitted to The Division of Natural Science and Mathematics of Bard College by Timothy E. Goldberg Annandale-on-Hudson, New York May, 2002
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Abstract In this paper, we study the combinatorial Laplacian operator on the vector space of oriented chains over R of a Fnite simplicial complex. We develop an easy method of computing the matrix of this operator from the adjacencies of simplices in the simplicial complex, and then apply this and results from linear algebra and simpli- cial homology to study properties of the Laplacian operator and its spectrum. We examine and explore connections between the combinatorial structure of simplicial complexes and their Laplacian spectra. SpeciFc examples studied include certain classes of graphs and higher dimensional simplicial complexes, in particular cones of simplicial complexes, especially simplicial cones of dimension 2.
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Dedication To my parents, for supporting me on whatever path I take, and To Paul, Grandma, Judy, and Grandmom, who probably would not have understood a word of this, but who would have loved it anyway.
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Acknowledgments Let me warn you, I am not known for my brevity. This project would not have been possible without my project advisor, Ethan Bloch, who provided incalculable expertise, energy, cheerfulness, and boundless en- thusiasm, even when I started talking about graph theory. His meticulous and un- tiring attention to detail and careful craftsmanship, expressed as ominous and inde- structible spider webs of red ink, was absolutely invaluable and very nearly painless, and was never expressed without the deepest kindness. This project would also not have been possible without my other project advisor, Lauren Rose, who was the Frst mathematician I ever worked with at Bard, and whose passion, experience, knowledge, and skill taught me as much by example as by explicit instruction. Both of them have my eternal and deepest gratitude. I owe a great deal to the entire extended mathematics department family, in- cluding Ethan and Lauren, but also Mark Halsey, Robert McGrail, Ranny Bledsoe, Rebecca Thomas, Sven Anderson, Robert Cutler, and Matthew Deady, who each provided a di±erent viewpoint into this wacky world of mathematics, but who have one and all demonstrated to me an inspiring passion for their subjects. Mark, thank you for showing me that mathematicians are fully capable of being almost preternaturally serene and calm, unbelievably organized, and still supremely enthusiastic and, well, great. Bob, thank you for showing me some of the most fascinating and beautiful mathematics I have ever seen, which I probably would not have seen but for you. Ia l sowan ttothankP ro fe s so rV ic to rReiner from the University of Minnesota and Professor Robin ²orman from Rice University, for being nice enough to meet with us and talk about Laplacians, and for pointing me in the right direction.
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4 Thank you Professors Christopher Lindner and John Ferguson, for luring me into Bard Asylum in the ±rst place.
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CombinatorialLaplacians - Combinatorial Laplacians of...

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