Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 18 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 145-146). / This thesis studies rational curves in the complex projective plane that are homeomorphic to their normalizations. We derive some combinatorial constraints on such curves from a result of Borodzik-Livingston in Heegaard-Floer homology. Using these constraints and other tools from algebraic geometry, we offer a solution to certain cases of the Coolidge-Nagata problem on the rectifiability of planar rational cuspidal curves, that is, their equivalence to lines under the Cremona group of birational automorphisms of the plane. / by Tiankai Liu. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002. / Includes bibliographical references (p. 35-37). / We show that for a large class of torsionfree classifying spaces, K-theory filtered ring is an invariant of the genus. We apply this result in two ways. First, we use it to show that the powerseries ring on n indeterminates over the integers admits uncountably many mutually non-isomorphic [lambda]-ring structures. Second, we use it to study the genus of infinite quaternionic projective space. In particular, we describe spaces in the genus of infinite quaternionic projective space which occur as targets of essential maps from infinite complex projective space, and we compute explicitly the homotopy classes of maps in these cases. / by Donald Y. Yau. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999. / Includes bibliographical references (p. 69-70). / This thesis presents some practical methods for doing model order reduction for a general type of nonlinear systems. Based on quadratic or even higher degree approximation and tensor reduction with assistance of Arnoldi type projection, we demonstrate a much better accuracy for the reduced nonlinear system to capture the original behavior than the traditional linearization method. / by Yong Chen. / S.M.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 53-55). / by Tibor Beke. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 143-149). / by Yihao Lisa Zhang. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992. / Includes bibliographical references (leaves 64-66). / by Carl C.Z. Dou. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995. / Includes bibliographical references (leaves 38-39). / by Michelle Wilson. / M.S.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 205-211). / We consider the problem of computing the order of an element in a generic group. The two standard algorithms, Pollard's rho method and Shanks' baby-steps giant-steps technique, both use [theta](N^1/2) group operations to compute abs([alpha])=N. A lower bound of [omega](N^1/2) has been conjectured. We disprove this conjecture, presenting a generic algorithm with complexity o(N^1/2). The running time is O((N/loglogN)^1/2) when N is prime, but for nearly half the integers N..., the complexity is O(N^1/3). If only a single success in a random sequence of problems is required, the running time is subexponential. We prove that a generic algorithm can compute [alpha] for all [alpha]... in near linear time plus the cost of single order computation with N=[lambda](S), where [lambda](S)=lcm[alpha] over [alpha]... For abelian groups, a random S...G or constant size suffices to compute [lamda](G), the exponent of the group. Having computed [lambda](G), we show that in most cases the structure of an abelian group G can be determined using an additional O(N^[delta]/4) group operations, given and O(N^[delta]) bound on abs(G)=N. The median complexity is approximately O(N^1/3) for many distributions of finite abelian groups, and o(N^1/2) in all but an extreme set of cases. A lower bound of [omega](N^1/2) had been assumed, based on a similar bound for the discrete logarithm problem. We apply these results to compute the ideal class groups of imaginary quadratic number fields, a standard test case for generic algorithms. the record class group computation by generic algorithm, for discriminant -4(10 +1), involved some 240 million group operations over the course of 15 days on a Sun SparcStation4. We accomplish the same task using 1/1000th the group operations, taking less than 3 seconds on a PC. Comparisons with non-generic algorithms for class group computation are also favorable in many cases. We successfully computed several class groups with discriminants containing more than 100 digits. These are believed to be the largest class groups ever computed / by Andrew V. Sutherland. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008. / Includes bibliographical references (p. 100-104). / In the first part of this work, we study a long standing open problem on the mixing time of Kac's random walk on SO(n, R) by random rotations. We obtain an upper bound mix = O (n2.5 log n) for the weak convergence which is close to the trivial lower bound [Omega] (n2). This improves the upper bound O (n4 log n) by Diaconis and SaloffCoste 1131. The proof is a variation on the coupling technique we develop to bound the mixing time for compact Markov chains, which is of independent interest. In the second part, we consider a generalization of the coupon collector's problem in which coupons are allowed to be collected according to a partial order. Along with the discrete process, we also study the Poisson version which admits a tractable parametrization. This allows us to prove convexity of the expected completion time E T with respect to sample probabilities, which has been an open question for the standard coupon collector's problem. Since the exact computation of E - is formidable, we use convexity to establish the upper and the lower bound (these bounds differ by a log factor). We refine these bounds for special classes of posets. For instance, we show the cut-off phenomenon for shallow posets that are closely connected to the classical Dixie Cup problem. We also prove the linear growth of the expectation for posets whose number of chains grows at most exponentially with respect to the maximal length of a chain. Examples of these posets are d-dimensional grids, for which the Poisson process is usually referred as the last passage percolation problem. In addition, the coupon collector's process on a poset can be used to generate a random linear extension. / (cont.) We show that for forests of rooted directed trees it is possible to assign sample probabilities such that the induced distribution over all linear extensions will be uniform. Finally, we show the connection of the process with some combinatorial properties of posets. / by Sergiy Sidenko. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 217-218). / In this thesis, we analyze a variant of the slice spectral sequence of [HHR (or SSS) called the regular slice spectral sequence (or RSSS). This latter spectral sequence is defined using only the regular slice cells. We show that the regular slice tower of a spectrum is just the suspension of the slice tower of the desuspension of that spectrum. Hence, many results for the RSSS are equivalent to corresponding results for the SSS. However, the RSSS has many multiplicative properties that the SSS lacks. Also, the slice towers that have been computed prior to this thesis happen to coincide with the corresponding regular slice towers. Hence, we find the RSSS to be much better behaved than the SSS. We give a comprehensive study of its basic properties, including multiplicative structure, Toda brackets, interaction with the norm functor of [HHRJ, vanishing lines and preservation of various kinds of extra structure. We identify a large portion of the first page of the spectral sequence algebraically by relating the RSSS to the homotopy orbit and homotopy fixed point spectral sequences, and determine the edge homomorphisms. We also give formulas for the slice towers of various families of spectra, and give several sample computations. The regular slice tower for equivariant complex K-theory is used to prove a special case of the Atiyah-Segal completion theorem. We also prove two conjectures of Hill from [Hill concerning the slice towers of Eilenberg MacLane spectra, as well as spectra that are concentrated over a normal subgroup. / by John Richard Ullman. / Ph.D.