Mathematics - Ph.D. / Sc.D.
http://hdl.handle.net/1721.1/7843
Wed, 29 Mar 2017 07:16:21 GMT2017-03-29T07:16:21ZOn folding and unfolding with linkages and origami
http://hdl.handle.net/1721.1/107547
On folding and unfolding with linkages and origami
Abel, Zachary Ryan
We revisit foundational questions in the kinetic theory of linkages and origami, investigating their folding/unfolding behaviors and the computational complexity thereof. In Chapter 2, we exactly settle the complexity of realizability, rigidity, and global rigidy for graphs and linkages in the plane, even when the graphs are (1) promised to avoid crossings in all configurations, or (2) equilateral and required to be drawn without crossings ("matchstick graphs"): these problems are complete for the class IR defined by the Existential Theory of the Reals, or its complement. To accomplish this, we prove a strong form of Kempe's Universality Theorem for linkages that avoid crossings. Chapter 3 turns to "self-touching" linkage configurations, whose bars are allowed to rest against each other without passing through. We propose an elegant model for representing such configurations using infinitesimal perturbations, working over a field R(e) that includes formal infinitesimals. Using this model and the powerful Tarski-Seidenberg "transfer" principle for real closed fields, we prove a self-touching version of the celebrated Expansive Carpenter's Rule Theorem. We switch to folding polyhedra in Chapter 4: we show a simple technique to continuously flatten the surface of any convex polyhedron without distorting intrinsic surface distances or letting the surface pierce itself. This origami motion is quite general, and applies to convex polytopes of any dimension. To prove that no piercing occurs, we apply the same infinitesimal techniques from Chapter 3 to formulate a new formal model of self-touching origami that is simpler to work with than existing models. Finally, Chapter 5 proves polyhedra are hard to edge unfold: it is NP-hard to decide whether a polyhedron may be cut along edges and unfolded into a non-overlapping net. This edge unfolding problem is not known to be solvable in NP due to precision issues, but we show this is not the only obstacle: it is NP complete for orthogonal polyhedra with integer coordinates (all of whose unfolding also have integer coordinates)
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 121-127).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1075472016-01-01T00:00:00ZThe quantum Johnson homomorphism and symplectomorphism of 3-folds
http://hdl.handle.net/1721.1/107327
The quantum Johnson homomorphism and symplectomorphism of 3-folds
Blaier, Netanel S
introduce a subset K2,A of the symplectic mapping class group, and an invariant ... that associates a characteristic class in Hochschild cohomology to every symplectomorphism ... K2,A. These are analogues to the familiar Johnson kernel X9 and second Johnson homomorphism - 2 from low-dimensional topology. The method is quite general, and unlike many abstract tools, explicitly computable in certain nice cases. As an application, we prove the existence of symplectomorphism ... of infinite order in symplectic mapping class group ... where Y is the blow-up of P3 at a genus 4 curve. The classical connection between such Fano varieties and cubic 3-folds allows us to factor ... as a product of six-dimensional generalized Dehn twists.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 345-354).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1721.1/1073272016-01-01T00:00:00ZHigher limits via subgroup complexes
http://hdl.handle.net/1721.1/106719
Higher limits via subgroup complexes
Grodal, Jesper (Jesper Kragh), 1972-
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000.; Includes bibliographical references (p. 42-45).
Sat, 01 Jan 2000 00:00:00 GMThttp://hdl.handle.net/1721.1/1067192000-01-01T00:00:00ZCompactness results for pseudo-holomorphic curves in symplectic cobordisms
http://hdl.handle.net/1721.1/106044
Compactness results for pseudo-holomorphic curves in symplectic cobordisms
Young, Carmen M
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000.; Includes bibliographical references (p. 69-70).
Sat, 01 Jan 2000 00:00:00 GMThttp://hdl.handle.net/1721.1/1060442000-01-01T00:00:00Z