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

Location: 384I.

Time: Monday, 4pm.

  • Tuesday, January 16, BAMAS at Stanford, Room 383N

  • Monday, February 12:

    Yiran Wang (University of Washington)

    ``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.
  • Monday, March 5:

    Charles Fefferman (Princeton)

    ``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
  • Monday, March 12:

    Boyu Zhang (Harvard)

    ``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.
  • Tuesday, March 13, BAMAS at Berkeley:

  • Monday, March 19:

    Kazuo Akutagawa (Tokyo Institute of Technology)

    "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).
  • Monday, April 16:

    Romain Gicquaud (University of Tours)

    ``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.
  • Monday, April 30:

    Jeff Galkowski (Stanford)

    ``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.
  • Tuesday, May 8, 4pm, Room 383N (Note unusual day and room!):

    Andras Vasy (Stanford)

    ``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.