Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. / Includes bibliographical references (p. 83-85). / Consider the unnormalized Ricci flow ...Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times ... then the solution can be extended beyond T. In the thesis we prove that if the Ricci curvature is uniformly bounded under the flow for all times ... then the curvature tensor has to be uniformly bounded as well. In particular, this means that if the Ricci tensor stays uniformly bounded up to a finite time T, a Ricci flow can not develop a singularity at T. We will give two different proofs of that result. One of them relies on Hamilton's estimates on distance changes along the flow and the other one relies on the identities for reduced distances and the monotonicity formula for reduced volumes that has been introduced and proved by Perelman in [29]. Consider the Ricci flow ... on a closed, n-dimensional manifold M. Assume that a solution of the flow exists for all times ... and that the curvatures and the diameters are uniformly bounded along the flow. We will prove that for every sequence ... there exists a subsequence such that g(ti + t) converges to a metric h(t) and h(t) is a Ricci soliton. We will also prove that if one of the limit solitons is integrable, then a soliton that we get in the limit is unique up to diffeomorphisms and the convergence toward it is exponential. / (cont.) We will also prove that in an arbitrary dimension, for a given Kähler-Ricci flow with uniformly bounded Ricci curvatures, for every sequence of times ti converging to infinity, there exists a subsequence such that ... and the convergence is smooth outside a singular set (which is a set of codimension at least 4). Moreover, g(t) is a solution of the flow off the singular set. In the case of a complex dimension 2, for any sequence of times converging to infinity we can find a subsequence of times such that we have a convergence toward a Kähler-Ricci soliton, away from finitely many isolated singularities. / by NataÅ¡a Å eÅ¡um. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / 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. 81-82). / In the framework of fibred cusp operators on a manifold X associated to a boundary fibration ... , the homotopy groups of the space ... of invertible smoothing perturbations of the identity are computed in terms of the K-theory of T*Y . It is shown that there is a periodicity, namely the odd and the even homotopy groups are isomorphic among themselves. To obtain this result, one of the important steps is the description of the index of a Fredholm smoothing perturbation of the identity in terms of an associated K-class in K ... / by Frédéric Rochon. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 133-137). / In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on R3. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of supercritical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition and improved averaging effects for the free evolution of the randomized initial data. In the second part of this thesis, we consider the periodic defocusing cubic nonlinear Klein- Gordon equation in three dimensions in the symplectic phase space H 1/2 (T 3) x H -1/2 (T 3). This space is at the critical regularity for this equation, and in this setting there is no global well-posedness nor any uniform control on the local time of existence for arbitrary initial data. We prove several non-squeezing results: a local in time result and a conditional result which states that uniform bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space implies global-in-time non-squeezing. As a consequence of the conditional result, we conclude nonsqueezing for certain subsets of the phase space and, in particular, we obtain deterministic small data non-squeezing for long times. To prove non-squeezing, we employ a combination of probabilistic and deterministic techniques. Analogously to the work of Burq and Tzvetkov, we first define a set of full measure with respect to a suitable randomization of the initial data on which the flow of this equation is globally defined. The proofs then rely on several approximation results for the flow, one which uses probabilistic estimates for the nonlinear component of the flow map and deterministic stability theory, and another which uses multilinear estimates in adapted function spaces built on UP and VP spaces. We prove non-squeezing using a combination of these approximation results, Gromov's finite dimensional non-squeezing theorem and the infinite dimensional symplectic capacity defined by Kuksin. / by Dana Sydney Mendelson. / Ph. D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2000. / Includes bibliographical references (p. 42-45). / by Jesper Grodal. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 59-60). / In this thesis, we describe the asymptotic behavior of complete Ricci-flat Kihler metrics on open manifolds that can be compactified by adding a smooth, ample divisor. This result provides an answer to a question addressed to by Tian and Yau in [TY1], therefore refining the main result in that paper. / by Bianca Santoro. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D. / Bibliography: leaves 104-106. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1986. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaf 47. / by Edoh Yosef Amiran. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 65). / The criterion of h-positivity corresponds to the criterion that a polynomial representation of the general linear group of V is a sum of tensor products of symmetric powers of V. Expanding the iterated exponential function as a power series yields coefficients whose positivity implies the h-positivity of the characteristic of the symmetric group character whose value on the permutation w is the number of labeled forests with c(w) vertices, where c(w) is the number of cycles of w. Another example of an h-positive symmetric function is the characteristic of the top homology of the even-ranked subposet of the partition lattice. In this case, the positive coefficients of the characteristic refine the tangent number E₂n₋₁ into sums of powers of two. / by Benjamin S. Joseph. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / 56 / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 101-105). / Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. We obtain a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter. We then obtain a fast (nearly linear in the dimension of the matrix) algorithm for the application of the absorbing boundary condition using partitioned low rank matrices. The result, modulo a precomputation, is a fast and memory-efficient compression scheme of an absorbing boundary condition for the Helmholtz equation. / by Rosalie Bélanger-Rioux. / Ph. D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 207-208). / This thesis gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1) we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(St) equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(St) and a localization of the category Op/t,v (the parabolic category 0 for gl(V) associated with the pair (V, 1)). In particular, we obtain an exact contravariant functor SWt from the category Repab(St) (the abelian envelope of the category Rep(St)) to a certain quotient of the category Op/t v. This quotient, denoted by 0 p/t v, is obtained by taking the full subcategory of Op/t v consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the contravariant functor SWt makes Op/t v a Serre quotient of the category Repab(St)OP, and the kernel of SWt can be explicitly described. In the second part of this thesis, we consider the case when V = C[infinity] . We define the appropriate version of the parabolic category 0 and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Op/t1CN with N tending to infinity. The Schur-Weyl functors SWt,CN then give an anti-equivalence between the category Op[infinity]/t C[infinity]and the category Repab(Se). This duality provides an unexpected tensor structure on the category Op[infinity]/t C[infinity]. / by Inna Entova Aizenbud. / Ph. D.