Schedule for: 21w5222 - Analysis on Singular Spaces (Online)

In this talk I will discuss and compare two approaches via Fredholm theory to resolvent estimates for the Laplacian of asymptotically conic spaces (such as appropriate metric perturbations of Euclidean space), including in the zero spectral parameter limit.

A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, we developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. In this talk, I will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, Lp norms, and Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function on all product manifolds.

Liouville conformal field theory is a conformal field theory quantizing the uniformization of Riemann surfaces. In joint work with Kupiainen, Rhodes, Vargas, we show that Segal axioms are satisfied for Liouville Conformal Field theory on Riemann surfaces, i.e. that the correlation/partition functions can be expressed by cutting the surfaces into surfaces with boundary. This is reminiscent to topological quantum field theory approaches where one associates Hilbert spaces H to boundaries and trace class operators on H to manifolds with boundary, with the property that operators compose when we glue two manifold along one common boundary. Using our previous work on the conformal bootstrap for the 4-point function on the sphere, this allows to express the partition and correlation functions as explicit functions on the moduli space of Riemann surface with marked points in terms of the conformal blocks associated to the Virasoro algebra and the structure constant (called DOZZ). The proof is a combination of probability methods, scattering theory and the representation theory of Virasoro algebra.

Z_2 harmonic spinors arise as limiting objects in gauge theory, and are solutions of an overdetermined boundary problem. I will describe some ongoing work (with Haydys and Takahashi) concerning the index of the associated deformation operator when the branching set is a network of curves in a 3-manifold.

In this talk I will report on continuing work about the spectral geometry of asymptotically complex hyperbolic manifolds. This class of non-compact spaces contain as examples certain quotients of complex hyperbolic space, as well as pseudoconvex domains in Stein manifolds. My focus will be on the resolvent and wave kernel, and how the behavior of closed geodesics in the interior can influence these spectral invariants. Our study of these operators will include discussion of different techniques in microlocal analysis, including radial estimates, complex absorption, and a Fourier Integral Operator calculus modeled on the Theta-calculus of pseudodifferential operators introduced by Epstein-Mendoza-Melrose.

Numerical studies of random plane waves, functions $$u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}$$ where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. First we consider the behaviour of a random variable given by $F(x,\xi)=||u||_{L^{2}(\gamma_{(x,\xi)})}$ where $\gamma_{(x,\xi)}$ is a unit ray from the point $x$ in direction $\xi$. We will see that this random variable is uniformly equidistributed. That is, the probability that for any $(x,\xi)$, $F(x,\xi)$ differs from its equidistributed value is small (in fact exponentially small). This result rules out a strong scarring of random waves. However, when we look at the full phase space picture and study a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ we do see a failure to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

Quasi-fibered boundary (QFB) metrics form a natural class of complete metrics generalizing the quasi-asymptotically locally Euclidean (QALE) metrics of Joyce. After recalling what those metrics are, I will explain how to construct a suitable pseudodifferential calculus containing good parametrices for operators like the Hodge-deRham operator of a QFB metric, allowing us to show that they are Fredholm when acting on suitable Sobolev spaces and yielding results about the decay of L2 harmonic forms. This in turn can be used to study the reduced L2 cohomology of some QFB metrics. This is a joint work with Chris Kottke.

Given a closed and oriented manifold $M$ and Riemannian tensors $g_0 \leq g_j$ on $M$ that satisfy $vol(M, g_j)\to vol(M,g_0)$ and $diam(M,g_j)\leq D$ we will see that $(M,g_j)$ converges to $(M,g_0)$ in the volume preserving intrinsic flat sense. We note that under these conditions we do not necessarily obtain smooth, $C^0$ or even Gromov-Hausdorff convergence. Nonetheless, this result can be applied to show stability of a class of tori. That is, any sequence of tori in this class with almost nonnegative scalar curvature converge to a flat torus. We will also see that an analogous convergence result to the stated above but for manifolds with boundary can be applied to show stability of the positive mass theorem for a particular class of manifolds. [Based on joint works with Allen, Allen-Sormani, Cabrera Pacheco - Ketterer, and Huang - Lee]

The analysis of scattering by thin barriers arises in the study of physical problems involving the confinement of individual electrons by small numbers of atoms. Motivated by work of Galkowski in higher dimensions, we consider a simplified model of such a barrier in the form of a delta function potential on the half-line. Our main results compute quantum decay rates (imaginary parts of resonances) for particles confined by such a potential. In the semiclassical limit, the energy dependence of the decay rates is logarithmic when the barrier is weaker and polynomial when the barrier is stronger. For our computation, we derive a formula for resonances in terms of the Lambert W function and apply a series expansion. This project is joint work with Nkhalo Malawo.

We will discuss a family of four-dimensional non-compact hyperK\"ahler metrics called Tian--Yau metrics, modelled by the Calabi ansatz with inhomogeneous collapsing near infinity. Such metrics were used recently as the scaling bubble limits for codimension-3 collapsing of K3 surfaces, where the study of its Laplacian played a central role. In this talk I will talk about the Fredholm mapping property and L^2 cohomology of such metrics. This is ongoing work joint with Rafe Mazzeo.

The so-called spaces with the Riemannian curvature-dimension condition (RCD spaces for short) are metric measure spaces which are non-necessarily smooth but admit a notion of "Ricci curvature bounded below and dimension bounded above". These arise naturally as Gromov-Hausdorff limits of Riemannian manifolds with these conditions and, in contrast to manifolds, RCD spaces may have topological or metric singularities. Nevertheless, several properties and results from Riemannian geometry can be extended to this non-smooth setting. In this talk I will present recent work, joint with Guido de Philippis, in which we show that the gradients of harmonic functions vanish at the singular points of the space. I will mention two consequences of this result on smooth manifolds: it implies that there does not exist an a priori estimate on the modulus of continuity of the gradient of harmonic functions depending only on lower bounds of the sectional curvature and that there is no a priori Calderón-Zygmund inequality for the Laplacian with bounds that depend only on lower bounds of the sectional curvature.

For a compact negatively curved Riemannian manifold $(\Sigma,g)$, the Ruelle zeta function $\zeta_{\mathrm R}(\lambda)$ of its geodesic flow is defined for $\Re\lambda\gg 1$ as a convergent product over the periods $T_{\gamma}$ of primitive closed geodesics $$\zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}})$$ and extends meromorphically to the entire complex plane. If $\Sigma$ is hyperbolic (i.e. has sectional curvature $-1$), then the order of vanishing $m_{\mathrm R}(0)$ of $\zeta_{\mathrm R}$ at $\lambda=0$ can be expressed in terms of the Betti numbers $b_j(\Sigma)$. In particular, Fried proved in 1986 that when $\Sigma$ is a hyperbolic 3-manifold, $$m_{\mathrm R}(0)=4-2b_1(\Sigma).$$ I will present a recent result joint with Mihajlo Ceki\'c, Benjamin K\"uster, and Gabriel Paternain: when $\dim\Sigma=3$ and $g$ is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely $$m_{\mathrm R}(0)=4-b_1(\Sigma).$$ This is in contrast with dimension~2 where $m_{\mathrm R}(0)=b_1(\Sigma)-2$ for all negatively curved metrics. The proof uses the microlocal approach of expressing $m_{\mathrm R}(0)$ as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott--Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations.