**Title:** Energy decay for the damped wave equation

**Abstract:** The damped wave equation describes the motion of a vibrating system exposed to a damping force. For the standard damped wave equation, exponential energy decay is equivalent to the Geometric Control Condition (GCC). The GCC requires every geodesic to meet the positive set of the damping coefficient in finite time. A natural generalization is to allow the damping coefficient to depend on time, as well as position. I will give an overview of the classical results and discuss how a time dependent generalization of the GCC implies exponential energy decay. I will also mention some results for unbounded damping when the GCC is not satisfied.

**Title:** Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

**Abstract:** We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one(!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

**Title:** Damping for Fractional Wave Equations

**Abstract:** Motivated by highly successful numerical methods for damping the surface water wave equations proposed by Clamond et al. (2005), we study the following leading order linear model for damped gravity water waves

\[ \partial_t^2 U + |D| U + \chi \partial_tU = 0 \]

We show that the energy of the solution has polynomial decay by proving a resolvent estimate. Joint work with Thomas Alazard and Jeremy L. Marzuola.

**Title:** Wave propagation on rotating cosmic string backgrounds

**Abstract: **A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: there exist closed timelike curves near the “string”. Nonetheless, I will describe joint work with Katrina Morgan in which we explore the possibility of obtaining forward solutions to the wave equation, appropriately interpreted.

**Title**: Nonlinear global stability of Kerr black holes with small angular momentum

**Abstract**: I will be talking about recent results which settle the stability conjecture of Kerr black holes in the case of small angular momentum.

**Title:** Singularity formation in the Boussinesq equations

**Abstract:** In this talk, I will first review existing results on singularity formation in incompressible and inviscid fluids. I will then describe a new mechanism for singularity formation in the Boussinesq equations. The initial data we choose is smooth except at one point, where it has Hölder continuous first derivatives. Moreover, the singularity mechanism is connected to the classical Rayleigh–Bénard instability. This is joint work with Tarek Elgindi (Duke University).

**Title:** Irreversible phenomena in 2D perfect fluids

**Abstract:** We discuss a number of results that reveal a form of irreversibility in the two-dimensional Euler equations governing the motion of an incompressible and inviscid (perfect) fluid. Specifically, we establish results on aging, wandering, and filamentation in perfect fluids. The main tool in establishing these results is an all-time stability of twisting result for 2d Hamiltonian flows, which allows us to deduce infinite-time results in settings where only static bounds are available.

**Title:** Sticky Kakeya sets, and the sticky Kakeya conjecture

**Abstract:** A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have dimension n. This conjecture is closely related to several open problems in harmonic analysis, and it sits at the base of a hierarchy of increasingly difficult questions about the behavior of the Fourier transform in Euclidean space. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate self-similarity at many scales, and sets of this type played an important role in Katz, Łaba, and Tao’s groundbreaking 1999 work on the Kakeya problem. In this talk, I will discuss a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have dimension n. I will discuss the proof of this conjecture in dimension 3. This is joint work with Hong Wang.

**Title:** Retiring the third law of black hole thermodynamics

**Abstract:** In this talk I will present a rigorous construction of examples of black hole formation which are exactly isometric to extremal Reissner–Nordström after finite time. In particular, our result can be viewed as a definitive disproof of the “third law of black hole thermodynamics.” This is joint work with Christoph Kehle.

**Title**: Codimension one stability of the catenoid under the hyperbolic vanishing mean curvature flow

**Abstract**: The catenoid is one of the simplest examples of minimal hypersurfaces next to the hyperplane. In this talk, we will view the catenoid as a stationary solution to the hyperbolic vanishing mean curvature flow, which is the hyperbolic analog of the (elliptic) minimal hypersurface equation, and study its nonlinear stability under no symmetry assumptions. The main result, which is a recent joint work with Jonas Luhrmann and Sohrab Shahshahani, is that with respect to a “codimension one” set of initial data perturbations of the n-dimensional catenoid, the corresponding flow asymptotes to an adequate translation and Lorentz boost of the catenoid for n greater than or equal to 5. Note that the codimension condition is necessary and sharp in view of the fact that the catenoid is an index 1 minimal hypersurface.

In a broader context, our result is a part of stability theory for solitary waves (i.e., stationary solutions) in the presence of modulation (i.e., symmetries, such as translation and Lorentz boosts, create nearby stationary solutions), which was mostly studied for semilinear PDEs in the past. Among the key challenges of the present problem compared to the more classical context are: (1) the quasilinearity of the equation, (2) slow (polynomial) decay of the catenoid at infinity, and (3) lack of symmetry assumptions. To address these challenges, we introduce several new ideas, such as a geometric construction of modulated profiles, smoothing of modulation parameters, and a robust framework for proving decay for the radiation part, which are hoped to be useful in the larger context of stability theory for solitary waves for quasilinear wave equations in the presence of modulation.