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February 2019

Jacob Shapiro, Australian National University – Analysis & PDE Seminar

February 6, 2019 @ 4:00 pm - 5:00 pm

Title: Standing wave solution for the nonlinear Schrödinger equation on an asymptotically Euclidean manifold. Abstract: We construct a Poisson operator for the non-linear Helmholtz equation with prescribed incoming boundary data at infinity. We do this by estimating the resolvent between suitable weighted anisotropic Sobolev spaces. This talk is based on joint work in progress with Jesse Gell-Redman, Andrew Hassell, and Junyong Zhang.

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Graham Cox – Analysis & PDE Seminar

February 20, 2019 @ 4:00 pm - 5:00 pm

Title: Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map Abstract: Courant's nodal domain theorem provides a natural generalization of Sturm–Liouville theory to higher dimensions; however, the result is in general not sharp. It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set.…

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Christopher Sogge – Analysis & PDE Seminar

February 27, 2019 @ 4:00 pm - 5:00 pm

Title: "Sharp local smoothing estimates for Fourier integral operators" Abstract: We present joint work with D. Beltran and J. Hickman on sharp local smoothing estimates for general Fourier integral operators.  Local smoothing bounds imply major estimates in harmonic analysis, including Bochner-Riesz estimates, oscillatory integral estimates and bounds for the size of Besicovitch sets and related problems involving Kakeya maximal functions.  Our work gives a sharp resolution to the most general form of the local smoothing problem formulated by the speaker in the early 1990s.  We rely on decoupling estimates of Bourgain and…

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March 2019

Brett Kotschwar, Arizona State University – Analysis & PDE Seminar

March 6, 2019 @ 4:00 pm - 5:00 pm

Title: The determination of a shrinking Ricci soliton from its geometry at infinity. Abstract: Shrinking solitons are self-similar solutions to the Ricci flow and models for the geometry of a solution near a developing singularity. The geometric behavior of a complete noncompact shrinking soliton near infinity is highly constrained; in dimension four, it has been conjectured that every such soliton is either smoothly asymptotic to a cone or to a (quotient of) a generalized cylinder.  I will describe some uniqueness…

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Mihai Tohaneanu – Analysis & PDE Seminar

March 20, 2019 @ 4:00 pm - 5:00 pm

Title: Local energy estimates on black hole backgrounds Abstract: Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is…

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April 2019

Yaiza Canzani, UNC-Chapel Hill – Analysis & PDE Seminar

April 3, 2019 @ 4:00 pm - 5:00 pm

Title: Understanding the growth of Laplace eigenfunctions. Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we…

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Yaiza Canzani, UNC-Chapel Hill – Analysis & PDE Seminar

April 10, 2019 @ 4:00 pm - 5:00 pm

Title: Understanding the growth of Laplace eigenfunctions. Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we…

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Suresh Eswarathasan, Cardiff – Analysis & PDE Seminar

April 17, 2019 @ 4:00 pm - 5:00 pm

Title: $L^p$ restriction of eigenfunctions to random Cantor-type sets Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions of this type arise in physics as modes of periodic vibration of drums and membranes. They also represent stationary states of a free quantum particle on a Riemannian manifold. In the first part of the lecture, I will give a survey of results which demonstrate how the geometry of $M$ affects the behaviour of…

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August 2019

Matt Blair, University of New Mexico – Analysis & PDE Seminar

August 28, 2019 @ 4:00 pm - 5:00 pm

Title: L^p bounds for eigenfunctions at the critical exponent Abstract: We consider upper bounds on the growth of L^p norms of eigenfunctions of the Laplacian on a compact Riemannian manifold in the high frequency limit. In particular, we seek to identify geometric or dynamical conditions on the manifold which yield improvements on the universal L^p bounds of C. Sogge.  The emphasis will be on bounds at the "critical exponent", where a spectrum of scenarios for phase space concentration must be…

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October 2019

Dean Baskin, Texas A&M University – Analysis & PDE Seminar

October 2, 2019 @ 4:00 pm - 5:00 pm

Title: Diffraction for the Dirac equation by Coulomb-like potentials Abstract: The Dirac equation describes the relativistic evolution of electrons and positrons. We consider the (time-dependent!) Dirac equation in three dimensions coupled to a potential with Coulomb-type singularities. We prove a propagation of singularities result for this equation and show that singularities are typically diffracted by the singularities of the potential. We finally compute the symbol of the diffracted wave and show it is typically non-zero. This talk is based on…

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