The APDE seminar on Wednesday, 3/13, will be given by Jeremy Marzuola (UNC) in-person in Evans 748, 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). Please note the special time and location of this talk.

Title:  Spectral minimal partitions, nodal deficiency and the Dirichlet-to-Neumann map

Abstract:  The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy function on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.  This is joint work with Greg Berkolaiko, Yaiza Canzani and Graham Cox.

The APDE seminar on Monday, 3/11, will be given by Hezekiah Grayer (Princeton) 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:  On the distribution of heat in fibered magnetic fields

Abstract: We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for “nearly integrable” magnetic fields up to a “critical” size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and related the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field.

This is joint work with Theodore D. Drivas and Dan Ginsberg.

The APDE seminar on Monday, 3/4, will be given by John Anderson (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: Shock formation for the Einstein–Euler system

Abstract: In this talk, I hope to describe elements of proving a certain stable singularity formation result for the Einstein-Euler system, which is the topic of work in progress with Jonathan Luk. I’ll first describe some motivating phenomena. Then, I will describe some of the main mathematical difficulties which present themselves when studying multidimensional shocks and why it is appropriate to call this a shock formation result. Finally, I will try to describe some of the main ideas that go into mathematically understanding shock formation, and the main difficulty in the case of Einstein–Euler.

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 $h\to 0+$.

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.