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.

We discuss new work in a very classical field: the study of branched covers of the Riemann sphere. We first recall the classical picture as developed by Hurwitz, including the relationship between branched covers and group-theoretic monodromy data, and the Hurwitz spaces which parametrize branched covers. We then give two new results: a connectedness result, joint with Fu Liu, for certain Hurwitz spaces in the classical setting, and a result which can be viewed as an analogue of the Riemann existence theorem for certain tamely branched covers of the projective line over fields of positive characteristic.

Sulcal fundi are 3D curves that lie in the depths of the cerebral cortex and are often used as landmarks for downstream computations in brain image processing. In this talk, a sequence of geometric algorithms is introduced to automatically extract the sulcal fundi from magnetic resonance images (MRI) and represent sulcal fundi as smooth polylines lying on the cortical surface. The automatic sulcal extraction can improve the quality and reproducibility of the process as well as yielding considerable time savings. This makes the large number of high-resolution MRI datasets available for analysis.

I will discuss local Holder and W^{1,p} estimates for solutions of some fully nonlinear equations in conformal geometry, and analyze the behavior of singular solutions in punctured balls. I will then show how these estimates are used in the solution of the \sigma_k-Yamabe problem for k > n/2