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 $h \to 0+$.

The APDE seminar on Monday, 1/29, will be given by Joonhyun La (Korea Institute for Advanced Study) 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: 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.

The APDE seminar on Monday, 12/4, will be given by Albert Ai (UW Madison) in-person in Evans 736, 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: 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.

The APDE seminar on Monday, 11/27, will be given by Rita Teixeira da Costa (Princeton) in-person in Evans 736, 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: 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).

The APDE seminar on Monday, 11/13, will be given by Thierry Laurens (UW Madison) in-person in Evans 736, 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: 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.

The APDE seminar on Monday, 11/06, will be given by Mihaela Ifrim (UW Madison) in-person in Evans 736, 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: 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.

The APDE seminar on Monday, 10/23, will be given by Kevin Ren (Princeton) in-person in Evans 736, 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: 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.

The APDE seminar on Monday, 10/30, will be given by Yuzhou (Joey) Zou (Northwestern) in-person in Evans 736, 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: 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.

The APDE seminar on Monday, 9/25, will be given by Benjamin Pineau (UC Berkeley) in-person in Evans 736, 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: Sharp Hadamard well-posedness for the incompressible free boundary Euler equations

Abstract: I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equation on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions.

Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.

The APDE seminar on Monday, 9/11, will be given by Zhongkai Tao (Berkeley) in-person in Evans 736, 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: Solution operators for divergence type equations and relativistic initial data gluing

Abstract: Given two solutions of the Einstein vacuum equation, can you glue them together along a spacelike hypersurface? Since the pioneering work of Corvino and Corvino–Schoen, we know it is possible to glue two initial data on an annulus in the asymptotically flat regime, modulo a 10-parameter obstruction, given by the energy, momentum, center of mass and angular momentum. Recently, Czimek–Rodnianski showed that the 10-parameter obstruction can be removed: instead they only need certain positivity assumptions on the energy-momentum tensor! Their proof of the obstruction-free gluing involves the null-gluing technique developed recently by Aretakis–Czimek–Rodnianski.
We develop a new, simple, spacelike method to obtain the above gluing results, which also optimizes the positivity, regularity and decay assumptions. It is based on solution operators for divergence type equations with nice support properties. I will explain the construction of such solution operators, and how the underlying positivity in the nonlinear part of scalar curvature enters the story. This talk is based on joint work with Yuchen Mao and Sung-Jin Oh.