### Program Description

Hydrodynamics is a large area of Mathematics that has many interdisciplinary connections and comprises many different sub-fields. It revolves around classical sets of problems in Analysis and Applied Mathematics, and it has been an area of major recent advances. The problems principally involve the study of nonlinear partial differential equations (PDEs) that describe fluid motion in various physical settings. The most prominent examples are the Euler and Navier-Stokes equations, which were already introduced in the 18th century. They have attracted the attention of countless researchers and still remain at the center of some of the most important open scientific questions, such as the \theory of turbulence" and the Clay Millenium problem on the regularity of the Navier-Stokes equations.

The semester will revolve around four main themes:

1. Euler and Navier-Stokes equations: Regular and singular solutions

The Euler and Navier-Stokes equations are among the most important and well-studied PDEs in the sciences. It is an understatement to say that these equations have attracted the attention of many of the best mathematical analysts of the last few generations. Nevertheless, fundamental questions such as the regularity/singularity of solutions, stability/instability of certain flows, and the theory of hydrodynamic turbulence still remain wide open.

The last few years have seen the appearance of incredible works shedding light on old and difficult problems. Among these we mention the series of works by De Lellis-Szekelyhidi who have built surprising non-unique solutions to the Euler equations. Their ideas, inspired by apparently unrelated concepts of convex integration in classical Riemannian geometry, have also been extended by Isett to obtain a complete resolution of Onsager's conjecture in the theory of hydrodynamic turbulence. Related techniques have been employed by Buckmaster-Vicol to construct non-unique solutions to the Navier-Stokes equations. Recent major advances also include the proof of inviscid damping for the Couette flow by Bedrossian-Masmoudi, and the construction by Kiselev-Sverak of double-exponentially growing solutions in 2 dimensions. Some ideas for this latter work were in turn inspired by numerical simulations of Hou-Luo, who proposed a concrete scenario with numerical evidence for the singularity formation in the 3d Euler equations.

The results above are just a few examples of the depth and breadth of the mathematical work that is currently going on in this area, and indicate great potential for more signicant discoveries in the near future.

2. Free surface hydrodynamics

The equations of free surface hydrodynamics describe the motion of waves, such as those on the surface of the ocean. This is a classical problem that has attracted the interest of mathematicians and engineers over the last two centuries, with early studies going back at least to Cauchy, Poisson and Lagrange. The understanding of the motion of waves in the ocean and the atmosphere is clearly a fundamental endeavor for mathematicians, engineers as well as numerical analysts and computational scientists.

Because of the complex nature of the nonlinear problems involved in this area, theoretical advances on some of the basic mathematical questions (e.g., the local-in-time well-posedness of the water waves equations) have only been made relatively recently, starting in the `70s and progressing with major breakthroughs in the late `90s and early `00s.

A spur of activity in the last two decades has brought major advances, such as, just to name a few, the discovery of certain classes of singular solutions, the constructions of global-in-time dispersive solutions by Fourier analytical methods, and proofs of long-time stability for spatially periodic solutions via dynamical systems and KAM methods.

Despite all these recent advances, several fundamental questions still remain poorly understood, including the interplay of surface waves with non-trivial geometries, the role of vorticity, the existence of global multi-dimensional periodic motions, and the dynamic formation of singularities.

3. Vortex filaments

Vortex filaments are a prominent feature of both classical and quantum fluids. Numerous basic phenomena exhibited by concentrated vortex filaments in ideal incompressible fluids (as described by the incompressible Euler equations) have identied and studied from a physical or formal point of view, starting with a groundbreaking 1858 paper of Helmholz. Since about the late 1960s, physical arguments have suggested that essentially the same phenomena should also occur in quantum fluids as described by the Gross-Pitaevskii equations. Some of these phenomena, such as reduced laws of motion for rectilinear vortices, have been proved to occur only within the past 20 - 25 years. Others, such as "leapfrogging vortex rings", have been veried only very recently for superfluids and remain open for classical fluids. Still others, most notably a conjectural asymptotic law of motion, the binormal curvature flow, for general slender vortex laments, have not been rigorously derived in any generality in any setting, and are the subject of prominent open problems.

Several recent results have shed a startling new light on these classical questions. These include powerful new approaches to the construction of 2d Euler flows with concentrated vorticity, affording more hope than earlier work for extension to 3 dimensions; well-posedness results for the initial value problem for the Navier-Stokes equation, with vorticity initially concentrated on a ring; recent progress on the Gross-Pitaevskii equations, via new techniques that might be relevant to the Euler equations. At the same time, the binormal curvature flow has developed as a topic of great interest in its own right. For example, work of de la Hoz and Vega has presented a great deal of evidence, based on a combination of rigorous analysis, numerical results, and formal considerations, for extremely rich behaviour of certain rough solutions of the binormal curvature flow, including a sort of fractal/polygonal dichotomy.

These results provide a new impetus to the study of these classical questions and suggest that this area is ripe for new developments.

4. Complex interactions and Hydrodynamics of active matter

Mathematical fluid dynamics provides a useful framework for the modeling and analysis of very large systems of interacting particles in the continuum limit. This point of view has been fruitful for understanding active matter, whose behavior emerges from the electrical, chemical, or mechanical interaction between between many individual agents and their surroundings. At macroscopic scales, active matter can resemble classical fluids, or develop large-scale patterns. Frequently, patterns arise from a competition between short- and long-range forces, and can change dramatically when external parameters such as the particle density or the availability of nutrients are varied. Important examples are chemotaxis and social interactions in bacterial colonies as described by the Keller-Segel model and its generalizations. Other applications include flocking and swarming, the formation of tissues, and tumor growth.

The program will in particular seek to bring together leading researchers who are developing the notion of hydrodynamics of active matter as a unifying theme that sheds light on numerous distinct fluid dynamical problems of extreme current interest.