Location: 384I.

Time: Monday, 4pm.

#### Wednesday, August 24, 11am, Room 383N (unusual day/time/room):

### Jan Derezinski (Warsaw)

### ``Almost homogeneous Schrödinger operators''

Abstract: First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions. I try to market them under the name "a toy model of renormalization group". Then I will discuss 1-dimensional Schroedinger operators with a 1/x^2 potential with general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.#### Wednesday, August 31, 2pm, Room 383N (unusual day/time/room):

### Frédéric Rochon (UQAM)

### ``QAC Calabi-Yau manifolds''

Abstract: We will explain how to construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). Our strategy consists first in introducing a natural compactification of QAC-spaces by manifolds with fibred corners and to give a definition of QAC-metrics in terms of a natural Lie algebra of vector fields on this compactification. Using this and the Fredholm theory of Degeratu-Mazzeo for elliptic operators associated to QAC-metrics, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. We can then obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation. This is a joint work with Ronan Conlon and Anda Degeratu.#### Monday, October 3:

### Jonathan Luk (Stanford)

### ``Stable shock formation for solutions to the multidimensional compressible Euler equations in the presence of non-zero vorticity''

Abstract: It is well-known since the foundational work of Riemann that plane symmetric solutions to the compressible Euler equations may form shocks in finite time. For a class of simple plane symmetric solutions, we prove that the phenomenon of shock-formation is stable under perturbations of the initial data that break the plane symmetry with potentially non-vanishing vorticity. In particular, this is the first constructive shock-formation result for which the vorticity is allowed to be non-vanishing at the shock. We show in particular that the vorticity remains bounded all the way up to the shock, and that the dynamics are well-described by the irrotational compressible Euler equations. This is a joint work with J. Speck (MIT), which is partly an extension of an earlier joint work with J. Speck (MIT), G. Holzegel (Imperial) and W. Wong (Michigan State).#### Wednesday, October 12, 3:15pm, Room 383N (unusual day/time/room, geometry seminar slot):

### Georgi Raikov (Pontificia Universidad Catolica de Chile)

### `` Resonances and SSF singularities for a magnetic Schroedinger operator''

Abstract: I will consider the 3D Schroedinger operator with constant magnetic field, perturbed by a rapidly decaying electric potential.First, I will discuss the asymptotic behavior of the corresponding Krein spectral shift function (SSF) near the Landau levels which play the role of spectral thresholds. I will show that the SSF has singularities near these thresholds, which admit a fairly explicit description in terms of appropriate Berezin-Toeplitz operators.

Further, I will introduce the resonances of the perturbed Schroedinger operator, and will discuss their asymptotic distribution near the spectral thresholds. I will show that under suitable assumptions on the pertubing potential, there are infinitely many resonances near every fixed Landau level, and the main asymptotic term of the corresponding resonance counting function is again related to the Berezin-Toeplitz operators arising in the description of the SSF singularities.

If time allows, some extensions to Pauli and Dirac operators with non-constant magnetic fields could be briefly outlined.

#### Friday, October 28, BAMAS at Berkeley, Room 736, Evans Hall:

### 2:30-3:30: Peter Hintz (Berkeley)

### ``Non-linear stability of Kerr-de Sitter black holes''

Abstract: In joint work with András Vasy, we prove the stability of the Kerr-de Sitter family of black holes in the context of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will describe the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges. In particular, I will explain our microlocal proof of a key ingredient of this framework, called `constraint damping,' a device first introduced in numerical relativity.-
### 4-5pm: Laurent Michel (Nice/Stanford)

### ``Metastability for semiclassical random walk''

Abstract: We study the return to equilibrium for a semiclassical random walk associated to a multiple well transition density. We describe accurately the small eigenvalues of the associated operator. The proof is based on a supersymmetric approach. As a preliminary we prove a general factorization result on pseudo differential operators. Joint work with J.-F. Bony and F. Hérau

and#### Monday, October 31 (double header), both talks in 384I:

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

### Jussi Behrndt (TU Graz)

### ``Spectral shift functions and Dirichlet-to-Neumann maps''

Abstract: In this talk we discuss a representation formula for the spectral shift function of a pair of self-adjoint operators via an abstract operator valued Titchmarsh-Weyl m-function. The general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. The talk is based on joint work with Fritz Gesztesy and Shu Nakamura.

#### 4-5pm

### András Vasy (Stanford)

### ``The stability of Kerr-de Sitter black holes (Part II: The analysis)''

Abstract: In this lecture, based on joint work with Peter Hintz, I will discuss analytic aspects of our proof of the stability of slowly rotating Kerr-de Sitter black holes, following the October 26 geometry seminar in which geometric aspects were discussed.Kerr-de Sitter black holes are rotating black holes in a universe with a positive cosmological constant, given by an explicit formula, due to Kerr and Carter. They are parameterized by their mass and angular momentum, much like Kerr black holes in a space-time with vanishing cosmological constant.

I will discuss analytic (PDE) aspects of the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted (fixing a gauge), Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data along a Cauchy hypersurface produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. The key aspects are arranging appropriate gauge conditions and the analysis of linear wave equations on asymptotically Kerr-de Sitter backgrounds, including with coefficient possessing limited regularity.

#### Monday, November 14:

### Laurent Michel (Nice/Stanford)

### ``On the small eigenvalues of the Witten Laplacian''

Abstract: The Witten Laplacian was introduced (by E. Witten) in the early 80’s to give an analytic proof of the Morse inequalities. The first optimal description of the small eigenvalues of this operator was obtained in 2004 (Bovier-Gayrard-Klein, Helffer-Klein-Nier) under some generic assumptions on the landscape potential f. In this talk, we present the approach of Helffer-Klein-Nier and show some recent progress to get rid of the generic assumption.#### Monday, December 5 (joint with symplectic seminar, 383N):

### Sergei Kuksin (Paris 7)

### ``Small-amplitude solutions for space-multidimensional hamiltonian PDEs under periodic boundary conditions''

Abstract: I will discuss the problem of studying the long-time behaviour of small solutions for nonlinear Hamiltonian PDEs on T^d, and will explain that in certain sense the behaviour for solutions of space-multidimensional equations (d>1) significantly differs from that for the 1d systems. The talk is based on my recent joint work with H.Eliasson and B.Grebert, to appear in GAFA, arXiv 1604.01657.#### Friday, December 9, BAMAS at Stanford, Room 384H:

#### 1:30-2:30pm

### Semyon Dyatlov (MIT)

### ``Resonances for open quantum maps''

Abstract: Quantum maps are a popular model in physics: symplectic relations on tori are quantized to produce families of N x N matrices and the high energy limit corresponds to the large N limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker's maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems. However in the setting of quantum maps we obtain results which are far beyond what is known in scattering theory.We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as N tends to infinity for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures. This talk is based on joint work with Long Jin.

and
#### 2:45-3:45pm

### Maciej Zworski (Berkeley)

### `` Ruelle zeta function at zero for surfaces of variable curvature''

Abstract: For surfaces of constant negative curvature the Selberg trace formula shows that the order of vanishing of the Ruelle zeta function at 0 is given by the absolute value of the Euler characteristic of the surface. Using simple microlocal arguments we prove that this remains true for any negatively curved sufficiently smooth surface. Joint work with S. Dyatlov.

#### Monday, January 9:

### Dean Baskin (Texas A&M)

### ``TBA''

Abstract: TBA

- Schedule for 2015-2016.
- Schedule for 2014-2015.
- Schedule for Spring 2014.
- Schedule for Winter 2014.
- Schedule for Autumn 2013.
- Schedule for Spring 2013.
- Schedule for Winter 2013.
- Schedule for Autumn 2012.
- Schedule for Winter/Spring 2012.
- Schedule for Autumn/Winter 2010-2011.
- Schedule for Winter-Spring 2010.
- Schedule for Autumn 2009.
- Schedule for Spring 2009.
- Schedule for Winter 2009.
- Schedule for Spring 2008.
- Schedule for Winter 2008.
- Schedule for Autumn 2007.