Location: 384I.

Time: Monday, 4pm.

#### Monday, September 25, BAMAS at Berkeley, Evans 740:

#### 2:40-3:30pm

### Kiril Datchev (Purdue)

### ``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.
and
#### 4:10-5:00pm

### Charles Hadfield (Berkeley)

### ``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.

#### Monday, October 30:

### Jacob Shapiro (Purdue)

### ``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.

### Tuesday, October 31, 3-4pm, Room 383N (Unusual day/time!):

### Sergiu Klainerman (Princeton)

### ``Stability of Schwarzschild under axially symmetric polarized perturbations’'

#### Monday, November 13:

### Dietrich Häfner (Grenoble)

### ``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.#### Monday, November 27:

### Kirk Lancaster

### ``The behavior of Capillary Surfaces along Edges''

#### Monday, December 4:

### Dejan Gajic (Imperial College) (TBC)

### ``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.

#### 2:30-3:30pm

### Jesse Gell-Redman (U Melbourne)

### `` 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).
and
#### 4:00-5:00pm

### Martin Vogel (Berkeley)

### ``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).

#### TBA

### Xuwen Zhu (Stanford)

### ``TBA''

Abstract:
and
#### TBA

### Long Jin (Purdue)

### ``TBA''

Abstract:

- Schedule for 2016-2017.
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