Date	Speaker	Title	Abstract
01/26	Niels Martin Møller (Princeton U)	The gluing perspective on the non-compactness of moduli spaces of minimal surfaces.	It is expected (Hoffman-Meeks + Ros) that M(g,r), the space of complete embedded minimal surfaces with genus g and r ends, is empty when g + 2 < r and for r ≥ 4 is non-compact whenever non-empty. Candidates for exhibiting such explicit non-compactness come from considering Kapouleas' gluing of coaxial catenoids, but now allowing the intersection angles to degenerate to zero. The simplest (i.e. most symmetric) such case, which viewed at the right scale is really a doubling of the flat plane, has ends that are unfortunately not embedded. That this must be so follows from a uniqueness theorem of Ros, but I will also explain how this effect is observed directly in the gluing procedure, as one can accurately track the influence that the shearing introduced in the gluing process has on the mean curvature equation, and see how the asymptotics will change once perturbing back to minimality. This work is joint with Stephen Kleene.
02/02	Jiuyi Zhu (JHU)	Quantitative uniqueness, doubling lemma and nodal sets.	Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. We also investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions. Using the Carleman estimates, we obtain the doubling estimates and optimal vanishing order of Steklov eigenfunctions, which is the eigenfunctions of the Dirichlet-to-Neumann map. Joint with Xing Wang, a lower bound of nodal sets of Steklov eigenfunctions is derived.
02/09	Timothy Candy (JHU)	Null Frames and the Cubic Dirac equation.	We give an overview of recent work on the problem of small data global well-posedness for the cubic Dirac equation in 2+1 dimensions. The main obstruction is a lack of available Strichartz estimates in low dimensions. To get around this difficulty, there are two key ideas. The first is the observation, originally due to Tataru, is that it is possible to construct null frames in which key endpoint Strichartz estimates can be recovered. The second is the fact that there is a subtle cancellation (or null structure) in the cubic nonlinearity that removes the dangerous parallel interactions. This is joint work with Nikolaos Bournaveas.
02/16	TBA	TBA.	TBA
02/23	Yingying Zhang (JHU)	Complex deformation of Kahler-Einstein manifolds.	Based on the Kodaira-Spencer’s theory, I will discuss the complex deformation of Kahler-Einstein manifolds. As applications, I will show the geometric quantization of the Weil-Petersson metric on the moduli space and also discuss an obstruction of deforming the Kahler-Einstein metric in the Fano case.
03/02	Michał Wrochna (Grenoble 1)	Cauchy data for the Feynman problem on globally hyperbolic spacetimes.	A classical result of Duistermaat and Hörmander provides four parametrices for the wave equation, distinguished by their wave front set. In applications in Quantum Field Theory one is interested in constructing the corresponding exact inverses, satisfying in addition a positivity condition. I will present a method (derived in a joint work with C. Gérard and dating back to W. Junker), where this is achieved by diagonalizing the wave equation in terms of elliptic pseudodifferential operators and solving the Cauchy problem with possible smooth remainders. I will then indicate possible ways of replacing Cauchy data by scattering data and comment on how this relates to global constructions of Feynman propagators.
03/09	Jesse Gell-Redman (JHU)	Propagators and semilinear wave equations on Minkowski space.	e discuss inhomogeneous semilinear equations, e.g. those of the form $\Box u + \lambda u^p = f$, on Minkowski space and perturbations thereof. Our approach is to use geometric microlocal analysis to first construct an inverse for the wave operator - meaning we construct Banach spaces between which the wave operator is an invertible map - and then to prove estimates for the non-linear term in these Banach so that the equation under consideration can be solved by Poincaire series.
03/11 at 3pm SPECIAL DATE AND LOCATION	Hamid Hezari (UC Irvine) In Hackerman 320	L^p noms of eigenfunctions on negatively curved manifolds.	In a joint work with Gabriel Rivi\`ere, we improve Sogge's L^p bounds by a power of logarithm on negatively curved manifolds. Our proof is based on a new quantum ergodicity property of eigenfunctions which holds on balls of shrinking radius.
03/23	Peter Hintz (Stanford U)	Quasilinear wave equations on black hole spacetimes.	I will discuss the global small data solvability of quasilinear wave equations on geometric classes of spacetimes, which include asymptotically de Sitter and Kerr-de Sitter spacetimes. We obtain our results by showing the global invertibility of the underlying linear operator, which in the quasilinear setting has non-smooth coefficients, on suitable $L^2$-based function spaces. The linear framework is based on Melrose's b-analysis, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. Joint work with Andras Vasy.
03/30	Dean Baskin (Texas A&M)	High frequency estimates for the Helmholtz equation with application to boundary integral equations.	I will discuss joint work with Euan Spence and Jared Wunsch on high frequency estimates for the Helmholtz equation in exterior domains as well as for interior impedance problems. Motivation for these estimates comes from certain problems in numerical analysis.
04/06	Junping Shi (William and Mary)	Exact Multiplicity and Uniqueness of Positive Solutions of Nonlinear Elliptic Equations and Systems: Old and New Results.	The steady state of reaction-diffusion equations/systems and the standing waves of Schr\"odinger type systems all satisfy nonlinear elliptic equations/systems. Various methods can be applied to prove the existence and multiplicity of positive solutions of nonlinear elliptic equations/systems, but usually the uniqueness or the exact multiplicity of positive solutions is difficult to prove. Some classical methods of proving uniqueness and exact multiplicity of positive solutions for semilinear elliptic equations on ball domains will be reviewed. Then we show some recent results extending these old results in several ways: (i) exact multiplicity of positive solutions for quasilinear equations; (ii) uniqueness of positive solution for semilinear systems. In particular, some general oscillatory properties are established for the linearized system in the one-dimensional case if the systems are cooperative. Using these properties, the uniqueness of positive solutions can be proved for systems which are cooperative, superlinear, weakly sublinear and with a variational structure.
04/10 at 4pm SPECIAL DATE AND LOCATION	Xiaolong Han (ANU) In Krieger 413	Bilinear eigenfunction estimates.	Lp bilinear eigenfunction estimates concern proving the bound for the Lp norm of the product of two L2-normalized eigenfunctions. The general theme in bilinear theory is to use the lower eigenfrequency to control the product. In 2005 Burq-Gerard-Tzvetkov proved the L2 bilinear eigenfunction estimate in dimension two. In collaboration with Z. Guo and M. Tacy, we recently generalized it to Lp bilinear estimates in the full range p>=2 on n-dimensional manifolds for all n>=2. We further show that all the estimates are sharp by constructing various pairs of eigenfunctions that saturate the bounds. In this talk, I will report on these progress on bilinear theory and its application to nonlinear Schrodinger equations on some compact manifolds.
04/13	Elias Stein SPECIAL LOCATION (Princeton U) In Krieger 308	How to think about the Cauchy-Szego projection. (First talk of the Kempf Memorial Lectures)	A classical theorem of M.Riesz states in effect that for the unit disc the Cauchy-Szego projection is a bounded operator on L^p, for p finite and >1. The purpose of this talk is to explain the background and the ideas needed for the extension of that result to deduomains in several complex variables, in the setting where minimal assumptions of smoothness of the domain are required. The result described is joint work with L. Lanzani.
04/20	Shadi Tahvildar-Zadeh (Rutgers)	The Tiger and the Magic Hoop: The Dirac Particle and the Kerr-Newman Spacetime.	In 1965 Ezra (Ted) Newman and a team of graduate students taking his General Relativity course discovered what is now known as the Kerr-Newman metric. It was the first rotating electrovacuum solution of Einstein-Maxwell system to be found, i.e. a stationary axisymmtric spacetime with mass, charge, and angular momentum, thought to represent the vacuum outside a rotating, charged object. In this joint work with Michael Kiessling, we study Dirac's wave equation for a point electron in the topologically nontrivial, maximal-analytically extended Kerr-Newman spacetime, in the zero-gravity limit. Here, "zero-gravity" means G --> 0, where G is Newton's constant of universal gravitation. The following results are obtained: The formal Dirac Hamiltonian on the static spacelike slices is essentially self-adjoint; the spectrum of the self-adjoint extension is symmetric about zero, featuring a continuous spectrum with a gap about zero that, under two smallness conditions, contains a symmetric point spectrum. I will explain how this is connected with a new quantum-mechanical interpretation of the Dirac equation, proposed by us, in which the electron and the positron are not distinct individual particles, but merely two "topological-spin" states of a single, more fundamental particle. (Note: a physics background is not necessary in order to understand this talk!)
04/27	Jared Speck (MIT)	Small-Data Shock Formation in Solutions to $3D$ Quasilinear Wave Equations.	I will describe my recent monograph on the formation of shocks, starting from small smooth initial conditions, in solutions to two classes of quasilinear wave equations in $3$ spatial dimensions. The main result states that within the two classes, a \emph{necessary and sufficient} condition for small-data shock formation is the failure of S. Klainerman's classic null condition. The monograph provides a detailed geometric description of the dynamics from $t=0$ until the first shock. I will highlight some of the main ideas behind the proof including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong. An important precursor is the remarkable result of D. Christodoulou, who in 2007 proved an analogous shock-formation result for irrotational regions of small-data solutions to the relativistic Euler equations. The result was later extended by Christodoulou-Miao to apply to the non-relativistic Euler equations. Our work recovers both of these results as special cases and generalizes previous small-data singularity-formation results of F. John, S. Alinhac, and many others.

        Archive of Analysis Seminar:

        Fall 2014  Spring 2014  Fall 2013