Department of Mathematics at MIT | Graduate Study : Summer 2003 Thesis Defenses


July 2003

Candidate:	Francois Blanchette	Abstract: We study the interaction between settling particles and a stratified ambient in a variety of contexts. We first describe problems where large scale fluid motions are generated by the localized release of a finite mass of particles in the form of plumes or gravity currents. We then focus on suspensions, where particles are initially present throughout the fluid. The simultaneous presence of particles and of a stratified ambient may lead to behavior analogous to double--diffusive systems, with particles playing the role of a diffusing component. We examine the linear stability of the settling of a particle concentration gradient in a stratified fluid. Numerical simulations allow us to determine the stability of the system for a broad range of particle settling speeds and diffusion coefficients. We report on layering arising from sedimentation in a density stratified ambient beneath an inclined wall before presenting an experimental and theoretical study of the combined influence of hindered settling and settling speed variations due to an ambient stratification. We develop a criterion for the stability of a suspension settling in a stratified ambien t and experimental observations allow us to qualify the main features of this instability.
Title:	Sedimentation in a stratified ambient
Date:	Thursday, July 17, 2003
Time:	2:00 pm
Location:	Room 2-132
Committee:	John Bush, thesis advisor Martin Bazant Anette Hosoi (Mech.Eng., MIT)

Candidate:	Lijing Wang	Abstract: In this thesis, we introduce a notion of asymptotic stability for a holomorphic vector bundle with a global holomorphic section on a projective manifold. We prove that the special metric on the bundle studied by Bradlow is the limit of a sequence of balanced metrics that are induced from the asymptotic stability. Conversely, assuming the convergence of a sequence of balanced metrics, we show that the sequence converge to a special metric in the sense of Bradlow. One feature of our notion of asymptotic stability is that it also depends on a real parameter, as the case for the stability defined by Bradlow. Another feature is that our convergence result is proved under a weak condition. The proof uses the asymptotic expansion of the Bergman kernel for general holomorphic vector bundle and machineries about moment maps involving two group actions developed by Donaldson.
Title:	Bergman Kernel and Stability of Holomorphic Vector Bundles with Sections
Date:	Monday, July 21, 2003
Time:	2:00 pm
Location:	Room 2-131
Committee:	Gang Tian, thesis advisor Richard Melrose Jeffrey Viaclovsky

Candidate:	Peter Clifford	Abstract: Motivated by results and conjectures of Stanley involving minimal border strip tableaux of partitions, we present three results. First we generalize the rank of a partition $[lambda]$ to the rank of a shifted partition `S`( $[lambda]$ ). We show that the number of bars required in a minimal bar tableau of `S`( $[lambda]$ ) is max(`o, e` + ( `l` ( $[lambda]$ ) \ mod \ 2)), where `o` and `e` are the number of odd and even rows of $[lambda]$ . As a consequence we show that the irreducible projective characters of \tilde`{S}_n` vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's `Q` symmetric functions in terms of the power sum symmetric functions. The second result gives a basis for the space spanned by the lowest degree terms in the expansion of the Schur symmetric functions in terms of the power sum symmetric functions. These lowest degree terms studied by Stanley correspond to minimal border strip tableaux of $[lambda]$ . The Hilbert series of these spaces is the generating function giving the number of partitions of `n` into parts differing by at least 2. Applying the Rogers-Ramanujan identity, the generating function also counts the number of partitions of `n` into parts 5`k` + 1 and 5`k` - 1. Finally for each $[lambda]$ we give a relation between the power sum symmetric functions and the monomial symmetric functions; the terms are indexed by the types of minimal border strip tableaux of $[lambda]$ .
Title:	Algebraic and combinatorial properties of minimal border strip tableaux
Date:	Thursday, July 31, 2003
Time:	2:00 pm
Location:	Room 2-131
Committee:	Richard Stanley, thesis advisor Daniel Kleitman Igor Pak

August 2003

Candidate:	Wei Luo	Abstract: In this thesis, I study pseudo-holomorphic curves in symplectisation of the unit cotangent bundle of a Riemann surface of genus greater than 1. The contact form and compatible almost complex structure are both constructed from a metric on the Riemann surface whose curvature is constant -1. I related the pseudo-holomorphic curve equation to harmonic map equations and a Cauchy-Riemann type equation perturbed with quadratic terms for functions on a punctured Riemann sphere. Then I prove a Theorem that gives one to one correspondence between solutions to the perturbed Cauchy-Riemann equation and finite energy pseudo-holomorphic curves.
Title:	On Contact Homology of the Unit Cotangent Bundle of a Riemann Surface with Genus Greater than One
Date:	Friday August 15, 2003
Time:	4 pm
Location:	Room 2-132
Committee:	Shing-Tung Yau (Harvard), thesis advisor I.M. Singer Denis Auroux

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