In this talk I will describe the progress that has been made so far concerning the existene of global strong solutions to the L^{2}-critical defocusing semilinear Schroedinger equation. A long standing conjecture in the area is the existence of a unique global strong L^{2} solution to the equation that in addition scatters to a free solution as time goes to infinity. I will demonstrate the proofs of partial results towards an attempt for a final resolution of this conjecture. I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces. Many authors have contributed to the theory of this equation. I will convey my personal involvement to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavolovic, G. Staffilani, J. Colliander and M. Grillakis.

The modern theory of Dynamical Systems is in major part an offspring of celestial mechanics. Poincare proved non-integrability of the three body problem when he discovered the homoclinic picture. Alexeev explained the existence of the oscillatory motions (a planet approaches infinity but always returns to a bounded domain) in Sitnikov model (one of the restricted versions of the three body problem) using methods of hyperbolic dynamics.
We show that the structures related to the most recent results in the smooth dynamical systems (area preserving Henon family and homoclinic bifurcations, persistent tangencies, splitting of separatrices) also appear in the three body problem. In particular, we prove that in many cases the set of oscillatory motions has a full Hausdorff dimension.

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold
of definite or vanishing first Chern class has been the subject of intense
study over the last few decades, following Yau's solution to Calabi's
conjecture. The K\"ahler-Ricci flow is the most canonical way to construct
K\"ahler-Einstein metrics. We define and prove the existence of a family
of new canonical metrics on projective manifolds with semi-ample canonical
bundle, where the first Chern class is semi-definite. Such a generalized
K\"ahler-Einstein metric can be constructed as the singular collapsing
limit by the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension
one. Some recent results of K\"ahler-Einstein metrics on K\"ahler
manifolds of positive first Chern class will also be discussed.

I will discuss a few recent studies on how organisms propel themselves through water, focusing on the appendages that allow them to do so efficiently. I will begin with fish fins, which have evolved over millions of years in a convergent fashion, leading to a highly-intricate fin-ray structure that is found in half of all fish species. This fin ray structure gives the fin flexibility plus one degree of freedom for shape control. I will present a linear elasticity model of the fin ray, based on experiments performed in the Lauder Lab in Harvard's Biology department.
In conjunction with this work, I will present numerical simulations of a fully-coupled fin-fluid model, based on a new method for computing the dynamics of a flexible bodies and vortex sheets in 2D flows. The simulations are applied to the most common mode of fish swimming, based on tail fin oscillations. In the passive case, an optimal flexibility for thrust is identified, and we consider also the optimal distribution of flexibility, with reference to recent measurements of tapering of insect wings and fish fins. We also briefly present work on fundamental
instabilities of a flexible body aligned with a flow (the "flapping flag" problem).
I will then discuss work on the role of bumps on the leading edge of humpback whale flippers, in collaboration with Ernst van Nierop and Michael Brenner at Harvard. Bumps have been shown in wind tunnels to increase the angle of attack at which flippers lose lift dramatically, or "stall." This stall-delay is thought to enable greater agility. In this study we propose an aerodynamic mechanism which explains why the lift curve flattens out as the amplitude of the bumps is increased, leading to potentially desirable control properties.
Finally, I will briefly describe results on a recent problem in self-assembly: the formation of 3D structures from flat elastic sheets with embedded magnets. The ultimate utility of this method for assembly depends on whether it leads to incorrect, metastable structures. We examine how the number of metastable states depends on the sheet shape and thickness. Using simulations and the theory of dislocations in elastic media we identify out-of-plane buckling as the key event leading to metastability. The number of metastable states increases rapidly with increasing variability in the boundary curvature and decreasing sheet thickness.

The phase-field method, also known as the diffuse interface method, has gained popularity in simulating interfacial flows. The interface between two immiscible fluids is treated as a diffuse layer governed by a phase-field variable that obeys the Cahn-Hilliard equation. In my talk, I will first describe two recent contributions to this class of methods: a generalization of the theoretical framework to account for complex rheology of non-Newtonian fluids, and a finite-element implementation with adaptive meshing for highly accurate interfacial resolution. These have allowed us to simulate interfacial dynamics of complex fluids with a refined understanding of the microscopic physics. Then I will discuss numerical simulations of the unique partial coalescence process. This refers to the phenomenon that a drop falling onto a fluid-fluid interface does not merge with it completely but leaves a smaller daughter drop behind, thus forming a cascade of partial coalescence. With a wide range of length scales, this phenomenon highlights the advantages and limitations of the numerical method. The numerical results show qualitative and sometime quantitative agreement with experiments. Furthermore, we develop general criteria for the occurrence of partial coalescence, which are very difficult to explore experimentally.