A Nonnegative Version of Whitney’s Extension Problem

The HADES seminar on Tuesday, November 17th will be given by Kevin O’Neill via Zoom from 3:40 to 5 pm.

Speaker: Kevin O’Neill (UC Davis)

Abstract: Whitney’s Extension Problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $E\to\mathbb{R}$, how can we tell if there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials $P^x$ at each $x\in E$, there is such an extension $F$ if and only if the $P^x$’s are compatible under Taylor’s theorem. However, this leaves open the question of how to tell solely from $f$. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when $f\geq 0$ and we require $F\geq 0$.

Network Control on Scattering Manifolds

The HADES seminar on Tuesday, October 20th will be given by Ruoyu Wang via Zoom from 3:40 to 5 pm.

Speaker: Ruoyu Wang (Northwestern)

Abstract: Lebeau (’93) suggested that on a compact manifold the damped waves decay logarithmically with merely some smooth damping inside a small open set. This phenomenon exploits the Carleman estimate establishing the exponentially weak observability. The natural generalisation of “small” sets to establish such exponential weak observability on noncompact manifolds is the Network Control Condition, formulated by Burq and Joly (’16), an condition requiring an upper bound of distance from the region of observability to any points on the manifold. We will show that this condition guarantees the exponentially weak observability on cylinders and scattering (asymptotically conic) manifolds, and henceforth derive a logarithmic decay for the damped waves in the high frequency regime, via a n-weight Carleman argument.

Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition

The HADES seminar on Tuesday, October 13th will be given by Dongxiao Yu via Zoom from 3:40 to 5 pm.

Speaker: Dongxiao Yu

Abstract: In this talk, I will discuss the long time dynamics of a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition introduced by Lindblad and Rodnianski, and it admits small data global existence which was proved by Lindblad. I will present a proof of the existence of the modified wave operators for this quasilinear wave equation. This is accomplished in three steps. First, we derive a new reduced asymptotic system by modifying Hörmander’s method. Next, we construct an approximate solution to the quasilinear wave equation by solving the reduced system given some scattering data. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.

Fractal Weyl Laws for Scattering Resonances

The HADES seminar on Tuesday, October 6th will be given by Yonah Borns-Weilvia Zoom from 3:40 to 5 pm.

Speaker: Yonah Borns-Weil

Abstract:It is well-known that on a bounded domain, the number of eigenvalues of the stationary Schrödinger equation in a given interval follow asymptotics known as Weyl laws. In scattering theory however, we work in an unbounded domain, and such operators need no longer have any eigenvalues. Instead, they have complex resonances, which satisfy a general upper bound (but not necessarily a lower bound) due to Sjöstrand that is analogous to the Weyl law. We present this bound, and describe how the proof must change if we instead count the eigenvalues in an h-dependent region. Following this, we present a result due to Sjöstrand and Zworski, which says that the exponent in such a Weyl law can depend on the fractal dimension of a hyperbolic trapped set. At the end, we will discuss what can still be said when the trapped set is not hyperbolic. Along the way, we will attempt to point out many of the standard “tricks” that are commonly used in scattering theory.

Singular solutions to the Einstein equations

The HADES seminar on Tuesday, September 22nd will be given by Jonathan Luk via Zoom from 3:40 to 5 pm.

Speaker: Jonathan Luk, Stanford

Abstract: I will discuss the construction of a class of low-regularity (merely $W^{1,2}$) solutions to the Einstein vacuum equations which have the property that the solutions are foliated by $2$-spheres so that the metric is more regular along the tangential directions of the $2$-spheres. I will first discuss a model semi-linear problem, then introduce the relevant geometric setup and give a sketch of the proof. This type of singular solutions is relevant to the problems of impulsive gravitational waves, high-frequency limits, null dust shells and the formation of trapped surfaces in general relativity (discussed in the Analysis and PDE seminar on 9/21).

On forward-in-time finite time blow-up construction for geometric dispersive equations

The HADES seminar on Tuesday, September 15th will be given by Sung-Jin Oh via Zoom (please contact the organizer at “james_rowan at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Sung-Jin Oh

Abstract: Since the pioneering works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz, Raphaël-Rodnianski and Merle-Raphaël-Rodnianski, the construction of finite-time blow-up solutions to geometric dispersive equations has been a topic of immense interest. In this expository talk, after a brief general review, I’ll try to describe in some detail the modulation-theoretic scheme of Raphaël-Rodnianski for constructing a finite time blow-up solution with smooth initial data. If time permits, I’ll present briefly a work-in-progress on an analogous blow-up construction for the self-dual Chern-Simons-Schrödinger equation (joint with Kihyun Kim and Soonsik Kwon).

Some smooth functions are more equal than others

The HADES seminar on Tuesday, September 29th will be given by Maciej Zworski via Zoomfrom 3:40 to 5 pm.

Speaker: Maciej Zworski

Abstract: As analysts we are used to smooth functions of compact support and after constructing one example of a bump function we are happy to apply it for many purposes. We also know that for any sequence of numbers we can construct a smooth function with that sequence as coefficients of its Taylor series. Can that map from sequences to functions be made linear? The answer is no for all sequences but yes for sequences satisfying certain growth conditions. I will prove the Denjoy–Carleman theorem which shows what growth is needed if you want to keep compact support, describe Carleson’s moment problem and talk about characterization of an important subclass of Gevrey functions. Those functions appear naturally in the theories of diffraction, of Landau diffusion for the Boltzmann equation, and of trace formulas for Anosov flows.

Blowup for supercritical nonlinear Schrodinger equations via concentration of an Euler front

The HADES seminar on Tuesday, September 8will be given by James Rowan via Zoom (please contact the organizer at “james_rowan at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker:  James Rowan

Abstract:  I will present a recent paper by Merle, Raphael, Rodnianski, and Szeftel which constructs a new kind of blowup solution for certain supercritical nonlinear Schrodinger equations.  The mechanism is neither a rapid frequency cascade nor concentration of a [quasi]soliton, but rather a highly-oscillatory front blowup coming from a collection of special solutions to the self-similar spherically symmetric Euler equations.  The construction relies on studying the behavior of a wave equation in the phase and modulus variables and a fixed point argument to control the behavior of unstable modes.  Along the way I hope to showcase some common techniques in the study of nonlinear PDEs.

When does the (Marked) Length Spectrum determine geometry of the billiard table?

The HADES seminar on Tuesday, April 21th will be given by Vadim Kaloshin via Zoom (please contact the organizer at “wangjian at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Vadim Kaloshin, University of Maryland, College Park

Abstract: We study the billiard on the plane ask: does the (Marked) Length Spectrum, i.e., the set of lengths of periodic orbits (together with their labeling), determine the geometry of the billiard table? This question is closely related to the well-known question: “Can you hear the shape of a drum?”

We report two results for planar domains having certain symmetry and analytic boundary. First, we consider billiards obtained by removing from the plane three strictly convex analytic obstacles satisfying the non-eclipse condition and a suitable symmetry. We show that under a non-degeneracy assumption, the Marked Length Spectrum determines the geometry of the billiard table. This is a joint work with J. De Simoi and M. Leguil.

Second, we consider billiards inside of a strictly convex planar domain having certain symmetry. We show that under a non-degeneracy assumptions, the Length Spectrum determines the geometry of the billiard table. This is a joint work with M. Leguil and K. Zhang. These results are analogous to results of Colin de Verdière,  Zelditch and Iantchenko-Sjöstrand-Zworski in terms of the (Marked) Length Spectrum.

Ruelle Zeta Function for Anosov Flows

The HADES seminar on Tuesday, April 14th will be given by Yonah Borns-Weil via Zoom (please contact the organizer at “wangjian at berkeley dot edu” for the Zoom ID) from 3:40 to 5 pm.

Speaker: Yonah Borns-Weil, Berkeley

Abstract: We discuss the meromorphic extension of the Ruelle zeta function for Anosov flows on a compact manifold. This was shown under an orientability condition by Giulietti, Liverani, and Policott in 2012, and then again by Dyatlov and Zworski in 2016 using microlocal analysis. We present the microlocal proof, and give a simple argument to remove the orientability assumption.