Determining stationary compact configurations of vorticity described by the 2D Euler equations is a classic problem dating back to the late 19th century. The aim is to find equilibrium distributions of vorticity, in the form  of point vortices, vortex sheets, vortex patches, and hollow vortices. This endeavour has driven the development of mathematical and numerical techniques such as Hamiltonian vortex dynamics and contour dynamics.

In the case of vortex sheets, methods and results are presented for finding rotating equilibria, some in the presence of point vortices. To begin, a numerical approach based on that recently developed by Trefethen, Costa, Baddoo, and others for solving Laplace's equation in the complex plane by series and rational approximation is described. The method successfully reproduces the exact vortex sheet solutions found by O'Neil (2018) and Protas & Sakajo (2020). Some new solutions are found.

The numerical approach suggests an analytical method based on conformal mapping for finding exact closed-form vortex sheet equilibria. Examples are presented.

Finally, new numerical solutions are computed for steady, doubly-connected vortex layers of uniform vorticity surrounding a solid object such that the fluid velocity vanishes on the outer free boundary. While dynamically unrelated, these solutions have mathematical analogy and application to the industrial free boundary problem arising in the dip-coating of objects by a viscous fluid.

I will discuss the recent progress in the numerical bootstrap of the 3d Ising CFT using the correlation functions of stress-energy tensor and the relevant scalars. This numerical bootstrap setup gives excellent results which are two orders of magnitude more accurate than the previous world's best. However, it also presents many significant technical challenges. Therefore, in addition to describing in detail the numerical results of this work, I will also explain the state-of-the art numerical bootstrap methods that made this study possible. Based on arXiv:2411.15300 and work in progress.

The beautiful displays exhibited by fish schools and bird flocks have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of debate in the scientific literature. I will present a model for flapping wings that interact hydrodynamically in an inviscid fluid, wherein each wing is represented as a plate that executes a prescribed time-periodic kinematics. The model generalizes and extends thin-airfoil theory by assuming that the flapping amplitude is small, and permits consideration of multiple wings through the use of conformal maps and multiply-connected function theory. We find that the model predictions agree well with experimental data on freely-translating, flapping wings in a water tank. The results are then used to motivate a reduced-order model for the temporally nonlocal interactions between schooling wings, which consists of a system of nonlinear delay-differential equations. We obtain a PDE as the mean-field limit of these equations, which we find supports traveling wave solutions. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.

I will discuss two classes of effective, macroscopic models in elasticity: (i) 1D models applicable to thin structures, and (ii) homogenized 2D or 3D continua applicable to materials with a periodic microstructure. In both systems, the separation of scales calls for the definition of macroscopic models that slave fine-scale fluctuations to an effective, macroscopic deformation field. I will show how such models can be established in a systematic and rigorous way based on a two-scale expansion that accounts for nonlinear and higher-order (i.e. deformation gradient) effects. I will further demonstrate that the resulting models accurately predict nonlinear effects, finite size effects and localization for a set of examples. Finally, I will discuss two challenges that arise when solving these effective models: (1) missed boundary layer effects and (2) negative stiffness associated with higher-order terms.