# Analysis and PDE Seminar Homepage, Autumn-Winter-Spring 2017/2018

Location: 384I.

Time: Monday, 4pm.

### Semiclassical resolvent estimates away from trapping''

Abstract: Semiclassical resolvent estimates relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. Roughly speaking, more trapping of particles corresponds to a larger resolvent near the trapping. If the trapping is mild, then propagation estimates imply that the larger norm occurs only there. However, in this talk I will show how the effects of heavy trapping can tunnel over long distances, implying that the resolvent can be very large far away as well. This is joint work with Long Jin.
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### Resonances on asymptotically hyperbolic manifolds; the ambient metric approach ''

Abstract: On an asymptotically hyperbolic manifold, the Laplacian has essential spectrum. Since work of Mazzeo and Melrose, this essential spectrum has been studied via the theory of resonances; poles of the meromorphic continuation of the resolvent of the Laplacian (with modified spectral parameter). A recent technique of Vasy provides an alternative construction of this meromorphic continuation which dovetails the ambient metric approach to conformal geometry initiated by Fefferman and Graham. I will discuss the ambient geometry present in this construction, use it to define quantum resonances for the Laplacian acting on natural tensor bundles (forms, symmetric tensors), and mention an application showing a correspondence between Ruelle resonances and quantum resonances on convex cocompact hyperbolic manifolds.

### Resolvent estimates and local energy decay in low regularity via Carleman estimates and distribution of resonances''

Abstract: We study weighted resolvent bounds for semiclassical Schrödinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter. This extends the works of Burq, Cardoso-Vodev, and Datchev. Our main tool is a global Carleman estimate. Applying the resolvent estimates along with the resonance theory for blackbox perturbations, we show logarithmic local energy decay for the wave equation with a wavespeed that is a Lipschitz perturbation of unity. The decay rate is the same as that proved by Burq for a smooth wavespeed outside an obstacle.

We also present what appears to be the first explicit resolvent bound for compactly supported $L^\infty$ potentials, which holds in the semiclassical limit. This is a step toward showing the optimal bound as conjectured by Vodev.

### Some mathematical aspects of the Hawking effect for rotating black holes''

Abstract : The aim of this talk is to give a mathematically rigorous de- scription of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We consider a simplified model of the collapse of the star. We formulate and prove a theorem about the Hawking effect for fermions within this setting.

### Price’s polynomial late-time tails for waves on asymptotically flat spacetimes''

Abstract: The late-time behaviour of solutions on asymptotically flat spacetimes typically does not obey the strong Huygens principle that is valid in the Minkowski spacetime. In Reissner–Nordström black hole exteriors, for example, it is governed by polynomially decaying “tails”, first discovered heuristically by Price. A mathematically rigorous derivation of these polynomial tails is instrumental in understanding the nature of singularities that might be present in black hole interiors. I will discuss a method for proving the existence of polynomial tails and furthermore uncovering the precise leading-order asymptotics in time for the wave equation on Reissner–Nordström spacetimes. The method involves new hierarchies of weighted energy estimates and exploits the existence of conserved quantities along future null infinity. This talk is based on joint work with Yannis Angelopoulos and Stefanos Aretakis.

###  Dirac type operators on pseudomanifolds''

Abstract: We study elliptic differential operators on iterated wedge spaces. These are incomplete Riemannian manifolds on which the metric undergoes iterated conical degeneration; they include cones, cone edges, and, products of cone edges, and they live on pseudomanifolds -- topological spaces characterized by an analogous topological notion of iterated conical degeneration. We determine the structure of the generalized inverses and the heat kernels of such operators using the radial blow up program of Melrose. In particular, we extend the edge calculus of Mazzeo to manifolds with corners with iterated fibrations structures (resolutions of pseudomanifolds), for both pseudodifferential and heat kernel type operators. We go on to prove an index theorem for those Dirac type operators on pseudomanifolds associated to (iterated) wedge metrics. Joint with Pierre Albin (UIUC).
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### Spectrum of random non-selfadjoint operators''

Abstract: The spectrum of non-selfadjoint operators can be highly unstable even under very small perturbations. This phenomenon is referred to as "pseudospectral effect". Traditionally this pseudosepctral effect was considered a drawback since it can be the source of immense numerical errors, as shown for instance in the works of L. N. Trefethen. However, this pseudospectral effect can also be the source of many new insights. A line of works by Hager, Bordeaux-Montrieux, Sjöstrand, Christiansen and Zworski exploits the pseudospectral effect to show that the (discrete) spectrum of a large class of non-selfadjoint pseudo-differential operators subject to a small random perturbation follows a Weyl law with probability close to one. In this talk we will discuss the local statistics of the eigenvalues of such operators (in dimension one). That is the distribution of the eigenvalues on the scale of their average spacing. We will show that the pseudospectral effect leads to a partial form of universality of the local statistics of the eigenvalues. This is joint work with Stéphane Nonnenmacher (Université Paris-Sud).

### Inverse problems for the Einstein-Maxwell equations’'

Abstract: We study the inverse problem of determining space-time structures using Einstein equations in general relativity. This type of problem was introduced by Kurylev, Lassas and Uhlmann in 2013. We consider the Einstein-Maxwell model and show the unique determination of the Vacuum space-time structure in a neighborhood of an observer. The solution of the problem is based on the understanding of the nonlinear interaction of gravitational and electromagnetic waves, especially from the microlocal point of view as the interaction of singularities for nonlinear hyperbolic equations. This is a joint work with M. Lassas and G. Uhlmann.

### The Muskat Problem’'

The Muskat equation describes the evolution of oil and water in sand. The system is linearly stable if the water lies below the oil, and linearly unstable if the oil lies below the water. The nonlinear system can start in the linearly stable regime, but "turn over" in finite time, so that neither fluid lies entirely below the other. After this happens, the solution continues to exist for some additional time even though it is in the unstable regime. Finally, a singularity appears. Joint work with A. Castro, D. Cordoba, F. Gancedo and M. Lopez-Fernandez

### Rectifiability and Minkowski bounds for the zero loci of Z/2 harmonic spinors’'

Abstract: We prove that the zero locus of a Z/2 harmonic spinor on a 4-manifold is 2-rectifiable and has finite Minkowski content. This result improves a regularity result by Taubes in 2014. It gives more precise descriptions to the limit behaviors of non-convergent sequences of solutions to many gauge-theoretic equations, such as the Kapustin-Witten equations, the Vafa-Witten equations, and the Seiberg-Witten equations with multiple spinors.

### Control and stabilization on hyperbolic surfaces''

Abstract: In this talk, we discuss some recent results concerning the control and stabilization on a compact hyperbolic surface. In particular, we show that
• the Laplace eigenfunctions have uniform lower bounds on any nonempty open set;
• the linear Schrödinger equation is exactly controllable by any nonempty open set; and
• the energy of solutions to the linear damped wave equation with regular initial data decay exponentially for any smooth damping function.
The new ingredient is the fractal uncertainty principle for porous sets by Bourgain–Dyatlov. This is partially based on joint work with Semyon Dyatlov.
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### Deformation of constant curvature conical metrics''

Abstract: In this joint work with Rafe Mazzeo, we aim to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of such conical metrics has a nice compactification as the cone points coalesce. This is a key ingredient of understanding the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.

### "A gap theorem for positive Einstein metrics on the four-sphere"

Abstract: On the $4$-sphere $S^4$, the standard round metric $g_S$ is, to date, the only known Einstein metric (up to rescaling and isometry). One of best gap theorems for Einstein metrics on $S^4$ is the following due to M. Gursky: Let $g$ be a positive Einstein metric on $S^4$ whose Yamabe constant $Y(S^4, [g])$ satisfies $$Y(S^4, [g]) \geq \frac{1}{\sqrt{3}} Y(S^4, [g_S]).$$ Then, up to rescaling, $g$ is isometric to $g_S$. In my talk, we will give an extension of the above Gursky's gap. This is a joint work with H. Endo (Tokyo Tech., Japan) and H. Seshadri (Indian Inst. Sci., Bangalore, India).

### Mass-like covariants for asymptotically hyperbolic manifolds’'

Abstract: The mass of an asymptotically hyperbolic manifold is a vector in Minkowski space defined in terms of the geometry at infinity of the manifold. It enjoys covariance properties under the change of coordinate chart at infinity. In this talk we classify covariants satisfying similar properties. This is a joint work with J. Cortier and M. Dahl.

### Concentration of Eigenfunctions: Sup-norms and Averages’'

Abstract: In this talk we relate concentration of Laplace eigenfunctions in position and momentum to sup-norms and submanifold averages. In particular, we present a unified picture for sup-norms and submanifold averages which characterizes the concentration of those eigenfunctions with maximal growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur.

### Fredholm theory and the resolvent of the Laplacian near zero energy on asymptotically conic spaces’'

Abstract: We consider geometric generalizations of Euclidean low energy resolvent estimates, such as estimates for the resolvent of the Euclidean Laplacian plus a decaying potential, in a Fredholm framework. More precisely, the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$ on asymptotically conic spaces $(X,g)$ of dimension at least $3$, and the main result is uniform estimates for $P(\sigma)^{-1}$ as $\sigma\to 0$ on microlocal variable order spaces under an assumption on the nullspace of $P(0)$ on the appropriate function space (which in the Euclidean case translates to $0$ not being an $L^2$-eigenvalue or having a half-bound state). These spaces capture the limiting absorption principle for $\sigma\neq 0$ in a lossless, in terms of decay, manner.