**Title:** Full asymptotics for Schrodinger wavepackets

**Abstract:** Since the work of Jensen–Kato, the theory of the Schrodinger–Helmholtz equation at low energy has been used to study wave propagation in various settings, both relativistic and nonrelativistic (i.e. the Schrodinger equation). Recently, Hintz has used these methods to study wave propagation on black hole spacetimes. Part of Hintz’s result is the production of asymptotics in *all* possible asymptotic regimes, including all joint large-time, large-radii regimes. We carry out the analogue of this analysis for the Schrodinger equation. Based on joint work with Shi-Zhuo Looi.

**Title:** The geometry of maximal development and shock formation for the Euler equations

**Abstract:** We establish the maximal hyperbolic development of Cauchy data for the multi-dimensional compressible Euler equations throughout the shock formation process. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region such that at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-$2$ surface of “first singularities” called the pre-shock set; second, a downstream hypersurface emanating from the pre-shock set, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock set, which the Euler solution cannot reach. This talk is based on joint work with Vlad Vicol at NYU.

**Title:** Toeplitz operators, semiclassical asymptotics for Bergman projections

**Abstract:** In the first part of the talk, we discuss boundedness conditions of Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with H. Xu), we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We shall review a direct approach to the construction of asymptotic Bergman projections, developed by A. Deleporte – M. Hitrik – J. Sj\”ostrand in the case of real analytic weights, and M. Hitrik – M. Stone in the case of smooth weights. We shall explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit $h \to 0+$.

**Title**: Null shell solutions – stability and instability

**Abstract**: In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.

**Title:** Low Regularity Solutions for the Surface Quasi-Geostrophic Front Equation

**Abstract:** In this talk, we consider the well-posedness of the surface quasi-geostrophic (SQG) front equation in low regularity Sobolev spaces. By observing a null structure, we obtain access to a normal form transformation for the equation. Applying this normal form in the context of a paradifferential analysis with modified energies, we are able to prove balanced cubic energy estimates and thus local well-posedness at just half a derivative above the scaling-critical regularity threshold. This is joint work with Ovidiu-Neculai Avadanei.

**Title**: The Teukolsky equation on Kerr in the full subextremal range |a|<M

**Abstract**: The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. We show that solutions arising from suitably regular initial data remain uniformly bounded in the energy space without derivative loss, and satisfy a suitable “integrated local energy decay” statement. A corollary of our work is that such solutions in fact decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters, |a|<M. This is joint work with Yakov Shlapentokh-Rothman (Toronto).

**Title:** Sharp well-posedness for the Benjamin–Ono equation

**Abstract:** We will discuss a sharp well-posedness result for the Benjamin–Ono equation in the class of H^s spaces, on both the line and the circle. This result was previously unknown on the line, while on the circle it was obtained recently by Gérard, Kappeler, and Topalov. Our proof features a number of developments in the integrable structure of this system, which also yield many important dividends beyond well-posedness. This is based on joint work with Rowan Killip and Monica Visan.

**Title:** The small data global well-posedness conjecture for 1D defocusing dispersive flows

**Abstract:** I will present a very recent conjecture which broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. This conjecture was recently proved in several settings in joint work with Daniel Tataru.

**Title:** Pinned Distances in R^d

**Abstract:** Given a set E in R^d with Hausdorff dimension > d/2, Falconer conjectured that the set of distances between any two points in E has positive Lebesgue measure. This conjecture remains open in all dimensions, despite significant progress in the last 30 years. Building upon this progress, we show that if d >= 3 and dim_H (E) > d/2 + 1/4 – 1/(8d+4), then the distance set of E has positive Lebesgue measure. The proof uses a new radial projection theorem in R^d applied to a variant of a decoupling framework of Guth-Iosevich-Ou-Wang. Joint work with Xiumin Du, Yumeng Ou, and Ruixiang Zhang.

**Title:** Weighted X-ray mapping properties on the Euclidean and Hyperbolic Disks

**Abstract:** We discuss recent works studying the sharp mapping properties of weighted X-ray transforms and weighted normal operators. These include a C^\infty isomorphism result for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We also discuss further work studying additional self-adjoint realizations of this operator, using the machinery of boundary triplets. In addition, we discuss ongoing work which applies these results to the X-ray transform on the hyperbolic disk by using a projective equivalence between the Euclidean and hyperbolic disks. Joint works with N. Eptaminitakis, R. K. Mishra, and F. Monard.