http://hdl.handle.net/1721.1/7680 2013-06-15T18:10:48Z http://hdl.handle.net/1721.1/77807 The generalized Tate construction Stroilova, Olga (Olga Y.) The purpose of this work is give some field notes on exploring the idea that a generalized Tate construction tk reduces chromatic level in stable homotopy theory. The first parts introduce the construction and discuss chromatic reduction. The next section makes a computation and gives the duals of L(n) = L(n)1. The last part looks ahead, mentioning how this computation could be extended to finding the duals of Steinberg summands in corresponding Thom spectra of negative representations, L(n)-q, and presents an equivariant loopspace machine. Finally, observations made are pulled together and brought back to compute the base case of the generalized Tate construction, evaluated on a sphere. Results parallel work of A. Cathcart, B. Guillou and P. May, and N. Stapleton, among others. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.; Cataloged from PDF version of thesis.; Includes bibliographical references (p. 69-70). 2012-01-01T00:00:00Z http://hdl.handle.net/1721.1/77804 Spectral asymptotics for coupled Dirac operators Savale, Nikhil, Jr. (Nikhil A.) In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an n dimensional manifold is of order r n+1/2 followed by a remainder of O(r n/2). We perform computations of spectral flow on the sphere which show that O(r n-1/2) is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show that its odd functional trace exhibits cancellations in its first n+3/2 terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.; Cataloged from PDF version of thesis.; Includes bibliographical references (p. 137-139). 2012-01-01T00:00:00Z http://hdl.handle.net/1721.1/77757 Optimal control of linear hereditary systems with quadratic criterion. Mcalla, Clement Aynsley Waters Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1973.; Vita.; Bibliography: leaves 334-344. 1973-01-01T00:00:00Z http://hdl.handle.net/1721.1/77697 A van Kampen spectral sequence for higher homotopy groups Stover, Christopher Roy Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1988.; Includes bibliographical references. 1988-01-01T00:00:00Z