Title: The asymptotic behavior of eigenfunctions on symmetric spacesAbstract: Let X be a compact locally symmetric space, and Y a locally symmetric subspace. Let f be an eigenfunction of the invariant differential operators on X with eigenvalue tending to infinity. I will present bounds for the period and Fourier coefficients of f along Y, and […]
Title: A fourth-order Escobar-Yamabe problem on a half-ballAbstract: The celebrated Yamabe problem asks us to make a conformal change on a compact Riemannian manifold such that the scalar curvature becomes constant. The (Type-II) Escobar-Yamabe problem is to make a conformal change on a compact Riemannian manifold with boundary so that the scalar curvature vanishes and […]
Title: Stability of perturbed wave equations on Kerr black hole spacetimesAbstract: I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the geometric wave equation with small linear perturbations on Kerr black hole spacetimes.Our proof adapts the framework introduced by Dafermos, Rodnianski, and Shlapentokh-Rothman for the homogeneous wave […]
Title: Maximal Subellipticity Abstract: The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on […]
Title: A Generalization of the Geroch Conjecture with Arbitrary EndsThe Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the […]
Title: On the existence of isoperimetric sets on nonnegatively curved spaces Abstract: In this presentation, I will deal with the isoperimetric properties of spaces with nonnegative curvature, placing special emphasis on the existence of isoperimetric sets in the non-compact case. Isoperimetric sets of any volume do exist in compact Riemannian manifolds. Conversely, it is possible to construct simple examples […]
Title: Carleson ε^2 conjecture in higher dimensionsAbstract: I will talk about a joint work with Xavier Tolsa and Michele Villa on a higher dimensional analogue of the Carleson ε^2 conjecture. In this work, we characterise tangent points of certain domains in Euclidean space via a novel "spherical" square function which measures whether the common boundary of the […]