I will discuss solutions of the higher order nonlinear Schrödinger equation (HNLS) on intervals with boundary from the points of regularity and control. We will consider weak solutions defined via special transform methods. At the level of wellposedness, the presence of multiple spatial derivatives raises certain concerns related with complex analyticity and estimates of integrals involving singularities. In addition, the failure of the Banach algebra property in the setting of low regularity solutions forces one to establish rather uncommon boundary type Strichartz estimates. Regarding the control of solutions, we will consider the finite interval framework and review construction of integral type boundary feedbacks that leads to exponential stabilization with prescribed or unprescribed decay rates.

The question of persistence of dynamics of completely integrable systems as one transitions to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this talk, we discuss the proximity between the solutions to (i) the integrable focusing nonlinear Schrödinger (NLS) equation and (ii) a broad class of non-integrable generalized NLS counterparts of that equation, in the framework of the Cauchy (initial value) problem on the real line. We consider two main settings: the traditional scenario of zero boundary conditions at infinity, and the scenario of nonzero boundary conditions at infinity, which is less studied but very relevant in terms of physical applications, as it is directly related to the phenomenon of modulational instability.

This talk introduces some new results on the structure of the singular set of some degenerate and nonlocal elliptic equations defined on Poincaré-Einstein manifolds. We will mainly exhibit a series of sharp dimension bounds and Hausdorff measure estimates for the singular set of the solutions. The setup is based on the quantitative differentiation theory for degenerate elliptic operators, and the proof involves subtle regularity issues on the underlying Poincaré-Einstein space.

Conformally variational Riemannian invariants (CVIs) such as scalar curvature and Q-curvature are homogeneous scalar invariants which appear as the conformal gradient of a Riemannian functional. In this talk, I will talk about constructing a formally self-adjoint conformally covariant multilinear operator associated with a given CVI. This construction recovers the relationship between GJMS operator and higher order Q-curvature $Q_{2k}$. I will also discuss a complete classification of tangential bi-differential operators in terms of ambient Laplacian. This result is a curved analogue of such operators on spheres classified by Ovsienko-Redou and Clerc. In addition, when choosing suitable weights, these operators are formally self-adjoint on conformal manifolds. At the end of the talk, I will present a family of sharp, fully nonlinear Sobolev inequalities involving the Paneitz operator and Ovsienko-Redou operator. This talk is based on joint works with Jeffrey Case and Wei Yuan.

I will discuss the weak* framework for solutions of hyperbolic systems of conservation laws in several space dimensions. The flexibility of these solutions appears to provide a natural way of ruling out several examples of non-unique weak solutions. I will re-express the PDE as an equation in function spaces, describe a Newton iteration for finding solutions, and illustrate with well-known examples.

The Riemann curvature tensor on a Riemannian manifold induces two curvature operators: the first acting on two-forms and the second acting on (traceless) symmetric two-tensors. The curvature operator of the second kind recently attracted a lot of attention due to the resolution of Nishikawa's conjecture by X.Cao-Gursky-Tran and me. In this talk, I will survey some recent works on the curvature operator of the second kind on Riemannian and Kähler manifolds and mention some interesting open problems. The newest result, joint with Harry Fluck at Cornell University, is an investigation of the curvature operator of the second kind in dimension three and its Ricci flow invariance.

In the study of completely integrable nonlinear wave equations interesting physical phenomena are often uncovered by careful analysis of the governing equation in certain singular limits. One such example of this is the notion of ``soliton gases'' in dispersive hydrodynamics. In this talk I will: (i) introduce ``solitons'' as special traveling wave solutions to certain integrable nonlinear evolution equations (pdes); (ii) discuss the spectral theory of soliton gases; (iii) introduce families of elliptic multi-phase solutions for the Korteweg-de Vries (KdV) and focusing nonlinear Schrödinger (NLS) equations, and show how, in certain singular limits, they give rise to soliton gases; and finally, (iii) present numerical and analytical solutions of the corresponding nonlinear dispersion relations which have some interesting physical consequences.

We will see an introcuction to the concept of Critical Threshold Phenomena (CTP) and how it plays a role in the Euler-Poission systems. We will go over some of the existing results in this area and how the techniques have developed over time. In the end, we will see some new results on the multidimensional spherically symmetric Euler-Poisson systems.

In this talk, I will present the Cauchy problem of the Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations: One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend previous work for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this work, a particular contribution is a systematic treatment of a parabolic PDE with only Hölder continuous diffusion coefficient and rough (worse than Hölder) nonhomogeneous terms.

The higher-order nonlinear Schrödinger (HNLS) equation is a more accurate alternative to the standard NLS equation when studying wave pulses in the femtosecond regime. It arises in a variety of applications ranging from optics to water waves to plasmas to Bose-Einstein condensates. In this talk, we consider the initial-boundary value problem for HNLS on a finite interval in the case of a power nonlinearity. We establish the local well-posedness of this problem in the sense of Hadamard (existence and uniqueness of the solution as well as its continuous dependence on the data) for initial data in the Sobolev space $H^s$ on a finite interval and boundary data in suitable Sobolev spaces determined by the regularity of the initial data and the HNLS equation. The proof relies on a combination of estimates for the linear problem and nonlinear estimates, which vary depending on whether $s > 1/2$ or $0 \leq s < 1/2$. The linear estimates are established by using the explicit solution formula obtained via the unified transform method of Fokas.