The APDE seminar on Monday, 2/26, will be given by Ethan Sussman (Stanford) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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.

The APDE seminar on Monday, 2/12, will be given by Steve Shkoller (UC Davis) online via Zoom from 4:10pm to 5:00pm PST (in particular, there will be no in-person talk). To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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.

The APDE seminar on Monday, 2/5, will be given by Haoren Xiong (UCLA) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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 $h \to 0+$.