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    2014-2015 Analysis Seminar

    Analysis / PDE Seminar

    Department of Mathematics
    University of North Carolina

    Wednesdays, 4:00-4:50pm, Phillips 381

    Next Talk:

    February 25th, 4:00-4:50pm, Phillips 381.
    Michael Taylor, UNC
    TITLE: Euler equation on a rotating sphere.

    ABSTRACT: Pictures of Jupiter show prominent horizontal bands. This phenomenon seems to be connected to the fact that the planet rotates fast. We discuss a model of fluid flow on a sphere, the 2D Euler equation modified by a Coriolis force. We investigate time averages of velocity fields, seen to approximate zonal velocity fields for large rotation number. These results expand on work of B. Cheng and A. Mahalov (2013). The main advance is to produce estimates valid globally in time. Also, time permitting, we might discuss some stability issues.

    Upcoming Events:

    Dates to keep in mind for UNC Analysis Mini-Schools: Gunther Uhlmann (Washington) - March 4-6, 205 (tentative); Alex Ionescu (Princeton) - April 8-10, 2015; Steve Zelditch (Northwestern) - Summer 2015.



    Spring 2015:

    February 4th, 4:00-4:50pm, Phillips 381.
    Xiaolong Han, ANU
    TITLE: Small scale quantum ergodicity in negatively curved manifolds.

    ABSTRACT: Quantum ergodicity studies the quantized counterpart of a classical dynamical system that is ergodic. If the geodesic flow is ergodic on the cosphere bundle of a compact manifold, then the quantum ergodic eigenfunctions tend equidistributive asymptotically in a fixed region. In a negatively curved manifold, the geodesic flow is chaotic and displays exponential decay of correlation. Using this property, we prove that the asymptotic equidistribution of quantum ergodic eigenfunctions can be improved to regions of small scales (balls logarithmically shrinking to points). I will also talk about the background on dynamical system, ergodicity, and chaos, in classical and quantum regimes, respectively.
    February 11th, 4:00-4:50pm, Phillips 381.
    Lillian Pierce, Duke
    TITLE: Bringing the Carleson operator out of Flatland.

    ABSTRACT: Must the Fourier series of an L^2 function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative by studying a particular type of maximal singular integral operator that has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey several of its generalizations, and then describe new joint work with Po-Lam Yung (Chinese University of Hong Kong) that introduces curved structure to the setting of polynomial Carleson operators.
    February 18th, 4:00-4:50pm, Phillips 381.
    Ioan Bejenaru, UCSD
    TITLE: Global well-posedness for the Cubic Dirac equation in the critical space.

    ABSTRACT: We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^(1/2)(\R^2)$. The theory we develop is the Klein-Gordon counterpart of the Wave Maps / Schroedinger Maps theory. This is joint work with Sebastian Herr.
    Dylan Muckerman, UNC

    March 17th and 18th, 4:00-4:50pm, Phillips 381.
    Jared Wunsch, Northwestern - Tuesday Talk is Part of the new UNC Student Colloquium Series

    March 19th, 2:00-2:50pm, Phillips 334 - Special Numerical Analysis Seminar.
    Ludwig Gauckler, TU Berlin
    TITLE: Stability of plane waves in the nonlinear Schr\"odinger equation: analysis and numerics.

    ABSTRACT: The cubic nonlinear Schr\"odinger equation on the torus has solutions that are plane waves. In the talk, the stability of these solutions and the stability of their numerical approximation will be discussed. In the first part of the talk, the stability of plane waves in the exact solution is discussed. We show orbital stability of plane waves over long times under generic perturbations of the initial data that are small in a high-order Sobolev norm. In the second part of the talk, a very popular numerical method for the nonlinear Schr\"odinger equation is considered, the split-step Fourier method. This method combines a Fourier spectral method in space with a splitting integrator in time. We will pursue the question whether the stability of plane waves in the exact solution transfers to this numerical discretization.
    April 15th, 4:00-4:50pm, Phillips 381.
    Lorena Bociu, NCSU

    April 16th, 4:00-4:50pm, Phillips 332.
    Dmitry Pelinovsky, McMaster

    April 22nd, 4:00-4:50pm, Phillips 332.
    Fang Wang, Shanghai Jiaotong University


    Upcoming Events:

    Dates to keep in mind for UNC Analysis Mini-Schools: Gunther Uhlmann (Washington) - March 4-6, 205 (tentative); Alex Ionescu (Princeton) - April 8-10, 2015; Steve Zelditch (Northwestern) - Summer 2015.



    Fall 2014:

    September 3rd, 4:00-4:50pm, Phillips 385.
    Colton Willig, UNC - Oral Examination

    September 5th, 4:00-4:50pm, Phillips 332 - Joint with Applied Seminar.
    Panos Kevrekidis, UMass Amherst
    TITLE: Existence, Stability and Dynamics of Solitary Waves and Vortices in Bose-Einstein Condensates: From Theory to Experiments.

    ABSTRACT: In this talk, we will present an overview of some of our recent theoretical, numerical and experimental efforts concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one- and higher-dimensional settings within Bose-Einstein condensates. We will discuss how this ultracold setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting possibilities for the future.
    October 8th, 4:00-4:50pm, Phillips 385.
    Matt Blair, University of New Mexico.
    TITLE: Strichartz and localized energy estimates for the wave equation in strictly concave domains.

    ABSTRACT: In this talk, we will introduce a family of energy estimates for solutions to the wave equation in domains with a strictly concave boundary satisfying homogeneous boundary conditions. They can be thought of as refinements of the classical local energy and local smoothing estimates. The estimates show that for frequency localized solutions, taking the square integral of the solution over small frequency-dependent collars of the boundary results in a stronger gain in regularity than would be expected for collars of a uniform size. We will also discuss the consequences for the development of Strichartz estimates with subcritical exponents in such domains, in particular providing an avenue for treating Neumann conditions.
    October 22nd, 4:00-4:50pm, Phillips 385.
    Jacob Bernstein, Johns Hopkins
    TITLE: A sharp lower bound on the entropy of closed hypersurfaces up to dimension six.

    ABSTRACT: The entropy of a hypersurface is defined to be the supremum of the Gaussian surface area of all translates and scalings of the hypersurface and it measures the geometric complexity of the hypersurface. It is always greater than one, with equality only for hyperplanes. I will discuss how to use weak mean curvature flows to show that, in low dimensions, the entropy of closed — i.e., compact and without boundary — hypersurfaces is bounded below by the entropy of the round sphere with equality only for the round spheres; proving a conjecture of Colding-Ilmanen-Minicozzi-White. This is joint work with Lu Wang.
    October 29th, 4:00-4:50pm, Phillips 385.
    Benjamin Harrop-Griffiths, UC-Berkeley
    TITLE: The lifespan of small solutions to the KP-I.

    ABSTRACT: We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets. This is joint work with Mihaela Ifrim and Daniel Tataru.
    November 5th, 4:00-4:50pm, Phillips 385.
    Chris Judge, Indiana
    TITLE: Laplace eigenfunctions on triangles.

    ABSTRACT: I will discuss joint work with Luc Hillairet concerning eigenfunctions of the Laplacian in both the Euclidean and hyperbolic settings. We show, for example, that the generic triangle has simple spectrum, and the generic noncompact hyperbolic triangle has no no nonconstant Neumann eigenfunctions. The methods synthesize quasimode analysis and analytic perturbation theory.
    November 12th, 4:00-4:50pm, Phillips 385.
    Mihaela Ifrim, UC-Berkeley
    TITLE: Water waves in holomorphic coordinates.

    November 19th, 4:00-4:50pm, Phillips 385.
    Michael Hitrik, UCLA
    TITLE: Invariant Lagrangian Tori and Spectra for Non-Selfadjoint Operators.

    ABSTRACT: We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. First, I would like to discuss an asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant tori for the classical flow. I shall then turn to spectral contributions coming from rational invariant tori, where a complete asymptotic description of individual eigenvalues is available, in suitable windows in the complex spectral plane. This is joint work with Johannes Sj\"ostrand.


    Spring 2014:

    February 19th, 4:00-4:50pm, Phillips 381.
    Benoit Pausader, Princeton
    TITLE: Asymptotic behavior for the nonlinear Schrodinger equation with partially periodic data.

    ABSTRACT: We consider the NLS equation on quotients of R^d, especially RxT^2. The question is to explore the asymptotic behavior of solutions in a more ``compact'' setting. We show how the scattering theory in the quintic case (the equivalent of the mass-critical case) is affected by the ``smaller'' volume and how, in the cubic case the asymptotic behavior is strongly modified by the presence of a secondary dynamics in logarithmic time. In the case of R or RxT (completely integrable case), this secondary dynamics can be explicitely integrated and only causes a phase correction. In the case of RxT^d, d>1, this dynamics is more complicated and leads to new regimes. In particular, in this case, one can find global solutions which start arbitrarily small in H^s and grow unboundedly with time.
    February 26th, 4:00-4:50pm, Phillips 381.
    Mayukh Mukherjee, UNC
    TITLE: Travelling waves in non-Euclidean settings.

    ABSTRACT: We study solutions of the form $v(t, x) = e^{i\lambda t}u(g(t)x)$, where $g(t)$ represents a one-parameter family of isometries, to nonlinear Schr{\"o}dinger and Klein-Gordon equations on Riemannian manifolds, both compact and non-compact ones. The emphasis will be on the NLKG. Here $g(t)$ is generated by a Killing field $X$ and the case of interest is when $X$ has length $\leq 1$, which leads to hypoelliptic operators with loss of at least one derivative. In the compact case, we will establish existence of travelling wave solutions via ``energy'' minimisation methods and prove that at least compact isotropic manifolds have genuinely travelling waves. We establish certain sharp estimates on low dimensional spheres that give sharp versions of results by Taylor and carry out the subelliptic analysis for NLKG on spheres of higher dimensions utilizing their homogeneous coset space properties. These subelliptic phenomena have no parallel in the setting of flat spaces. If time permits, we will make comments on related phenomena on complete noncompact manifolds which have a certain radial symmetry using concentration-compactness type arguments.
    March 27th, 2:00-2:50pm, Phillips 332. (Special Time and Place Due to Brauer Lectures)
    Jesse Gell-Redman, Toronto
    TITLE: Index formulas on singular spaces.

    ABSTRACT: Riemannian stratified spaces arise in many contexts, notably as the compactifications of moduli spaces, and on the K\"ahler-Einstein manifolds studied recently by Donaldson, Tian, and many other. We discuss various extensions of index formulas (such as the Chern-Gauss-Bonnet formula) to these spaces, work which was initiated by Cheeger in the 70s. The main object of study are naturally arising elliptic operators (e.g. the Hodge-deRham operator, the Dirac operator) and their mapping properties on these singular spaces. This is joint work with Pierre Albin at UIUC.
    April 2nd, 4:00-4:50pm, Phillips 381.
    Yakov Shlapentokh-Rothman, MIT
    TITLE: Boundedness and decay for the wave equation on sub-extremal Kerr spacetimes.

    ABSTRACT: After reviewing some background and the history of the problem, we will discuss our recent proof (joint with Mihalis Dafermos and Igor Rodnianski) of boundedness and decay statements for the wave equation on Kerr spacetimes in the full sub-extremal range |a| < M.
    April 3rd and 4th, Times and Places on Mini-School Web-Site
    Maciej Zworski, Berkeley
    TITLE: Dynamical zeta functions and an introduction to microlocal analysis.

    ABSTRACT: Tbd.
    April 9th, 4:00-4:50pm, Phillips 381 (Honor's Thesis Defense).
    Shreyas Tikare, UNC
    TITLE: Localized Energy Estimates for Wave Equations on (1+4)-dimensional Myers-Perry Space-Times.

    ABSTRACT: A robust measure of decay and dispersion for the wave equation is provided by the localized energy estimates, which have been essential in proving other types of dispersive estimates on black hole backgrounds such as Strichartz estimates. We study localized energy estimates on (1+4)-dimensional Myers-Perry space-times, which represent a family of rotating, axially symmetric, asymptotically flat black holes with spherical horizon topology and generalize the well-known Kerr space-times to higher dimensions. The main idea is to use the multiplier method. Unfortunately, due to the complicated nature of trapping, no differential operator provides us with a positive local energy norm. Once it is determined that all trapped geodesics lie on surfaces of constant r, we adapt the method of Tataru and Tohaneanu, which perturbs off the Schwarzschild case by instead commuting with an appropriate pseudodifferential operator to generate a positive commutator near the trapped set. This describes joint work with Parul Laul, Jason Metcalfe, and Mihai Tohaneanu.
    April 10th, 4:00-4:50pm, Phillips 332 (Colloquium).
    Alessio Figalli, Texas
    TITLE: Stability results for the Brunn-Minkowski inequality.

    ABSTRACT: Given a Borel A in R^n of positive measure, one can consider its semisum S=(A+A)/2. It is clear that S contains A, and it is not difficult to prove that they have the same measure if and only if A is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one may consider the semisum of two different sets A and B, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a set with finite unions of intervals to translate the problem to the integers Z. In this discrete setting the question becomes a well-studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk, which is intended for a general audience, I will review some results in the one-dimensional discrete setting and show how to answer to the problem in arbitrary dimension.
    April 24th, 4:00-4:50pm, Phillips 332 (Colloquium).
    Sylvia Serfaty, Courant/Paris 6
    TITLE: Questions of crystallization in Coulomb systems.

    ABSTRACT: We are interested in systems of points with Coulomb interaction. An instance is the classical Coulomb gas, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns, named Abrikosov lattices in physics. In joint works with Etienne Sandier and with Nicolas Rougerie, we studied both systems and derived a "Coulombian renormalized energy". I will present it, examine the question of its minimization and its link with the Abrikosov lattice and weighted Fekete points. I will describe its relation with the statistical mechanics models mentioned above and show how it leads to expecting crystallisation in the low temperature limit.


    Fall 2013:

    August 28th, 4:00-4:50pm, Phillips 381.
    Nick Moore, NYU - Joint Applied/Analysis Seminar

    TITLE: Hybrid analytical/numerical models for fluid-structure problems.

    ABSTRACT: Models that combine analytical and numerical techniques often reap the benefits of both, and I will discuss the use of such models for two fluid-structure problems. First, inspired by natural examples such as landform evolution, I will discuss the erosion of solid bodies by flowing fluids. Table-top experiments with soft-clay bodies eroding in flowing water show the formation of sharp corners and facets, contrary to the notion that erosion is a smoothing process. We develop a model in which an outer flow is coupled to a boundary layer flow which shears away solid material. This model allows us to rationalize the experimental measurements and extend our understanding of the process. Ultimately, we find that the eroding body converges to a terminal form characterized by nearly uniform shear stress, and then shrinks self-similarly in time. Second, I will discuss the motion of bodies through viscoelastic fluids. These fluids store and release elastic energy, leading to characteristically unsteady motion of immersed bodies. As an example, a body settling under gravity experiences an overshoot, in which its speed temporarily exceeds the terminal value. We develop a hybrid analytical/numerical method, valid in the so-called "weak-coupling" limit, that accurately captures the velocity overshoot. Unlike many traditional methods, our method allows efficient and stable computations when the viscoelastic relaxation timescale is long. I will briefly discuss work currently underway in which we apply the weak-coupling method to other viscoelastic-fluid-structure problems.

    Nick is visiting us a speaker in the Cha Cha Days Finale conference. The full schedule is available here.

    September 11th, 4:00-4:50pm, Phillips 381.
    Michael Taylor, UNC
    TITLE: Toeplitz operators on uniformly rectifiable domains.

    ABSTRACT: The class of uniformly rectifiable domains is essentially the maximal class of domains for which there is a viable theory of singular integral operators. There is a growing body of work on applications of such a theory to problems in PDE on such domains. In this talk, we discuss Toeplitz operators, which arise as follows. Let M be a compact Riemannian manifold, U a uniformly rectifiable domain in M, and D a first order elliptic differential operator on M. There is a natural projection P of L^2(bU) onto the space of boundary values of functions on U annihilated by D. If F is a continuous, matrix-valued function on the boundary bU, we define the Toeplitz operator T_F u=PFu. We show such operators are Fredholm if F(x) is invertible for all x in bU, and discuss properties of the index. We take more generally F to be bounded and of vanishing mean oscillation, and obtain simultaneously extensions to higher dimensions and to domains with rough boundary of index results of Brezis-Nirenberg. We also discuss a cobordism argument, effective in comparing our index problem to one involving a smoothly bounded domain. This is a description of work with Irina and Marius Mitrea.
    September 18th, 4:00-4:50pm, Phillips 381.
    David Webb, UNC - Oral Examination in fulfillment of Ph.D. requirements

    October 2nd, 4:00-4:50pm, Phillips 381.
    Kevin Zumbrun, Indiana - Lecture 1 of the First UNC PDE Mini-School Series 2013/2014
    TITLE: Modulation and stability of periodic traveling waves, and the link through Whitham's modulation equations to hyperbolic conservation laws and shock fronts.

    ABSTRACT: See the Mini-School web-site for full details.
    October 23rd, 4:00-4:50pm, Phillips 381.
    Andrew Lawrie, Berkeley

    TITLE: Stable soliton resolution for exterior wave maps.
    ABSTRACT: We consider one-equivariant wave maps in $3$ space dimensions exterior to the unit ball which take values in the unit $3$-sphere and with a Dirichlet boundary condition at $r=1$. We establish relaxation of such a wave map with finite energy to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizon, Chmaj, Maliborski who proposed this problem as a model in which to study stable soliton resolution and who observed this asymptotic behavior numerically. This is joint work with Carlos Kenig and Wilhelm Schlag.

    November 6th, 4:00-4:50pm, Phillips 381.
    Yaiza Canzani, Harvard
    TITLE: Expectation of the sup-norms of high frequency random waves.

    ABSTRACT: In this talk I'll present some recent results on the sup-norms of random approximate eigenfunctions of the Laplace operator on a compact aperiodic Riemannian manifold. We prove that if $f_{\lambda}$ is chosen uniformly at random from the space of L^2-normalized linear combinations of Laplace eigenfunctions of frequencies in $(\lambda, \lambda+1]$, then the expected value of $\|f_\lambda\|_\infty$ grows at most like $C \sqrt{\log \lambda}$ as $\lambda \to \infty$, where $C$ is an explicit constant that depends only on the dimension and volume of the manifold. In addition, we obtain concentration of the $L^\infty$-norm around its mean and study the analogous problems for Gaussian random waves on the manifold. This is joint work with Boris Hanin. 
    November 13th, 4:00-4:50pm, Phillips 381.
    Fabricio Macia, Madrid
    TITLE: Dispersion and observability for completely integrable Schrödinger flows.

    ABSTRACT: I will present some results on weak dispersion and unique continuation (observability) for linear Schrödinger equations that are obtained as the quantization of a completely integrable Hamiltonian system. The model case corresponds to the linear Schrödinger equation (with a potential) on the flat torus. Our results are obtained through a detailed analysis of semiclassical measures corresponding to sequences of solutions, which is performed using a two-microlocal approach. This is a joint work with Nalini Anantharaman and Clotilde Fermanian-Kammerer.
    November 20th, 4:00-4:50pm, Phillips 381.
    Natasa Pavlovic, Texas
    TITLE: Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti.

    ABSTRACT: The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) or nonlinear Hartree (NLH) equations, from many body quantum dynamics is central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP. That was achieved in seminal papers of Erd\"{o}s-Schlein-Yau. Recently, with T. Chen, C. Hainzl and R. Seiringer we obtained a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in ${\mathbb{R}}^3$. One of the main tools in our analysis is the quantum de Finetti theorem.

    In the talk, we will present a brief review of the derivation of NLS via the GP, describing the context in which the new uniqueness result appears, and will then focus on the uniqueness result itself.

    December 4th, 4:00-4:50pm, Phillips 381.
    Anir Zenginoglu, Cal Tech
    TITLE: Infinity on a grid shell: wave equations on unbounded domains.

    ABSTRACT: It is commonly stated that hyperbolic equations cannot be solved numerically on unbounded domains without the introduction of an artificial outer boundary.  I will show that a method called hyperboloidal compactification provides the unbounded domain solution resolving all problems associated with the truncation of the computational domain. I will present a few recent applications including a study of the focussing cubic wave equation, the numerical calculation of black hole Green functions, and the definition of quasinormal modes for black hole perturbations.


    Spring 2013:

    February 13th, 4:00-4:50pm, Phillips 381.
    Jianfeng Lu, Duke
    TITLE: Nonexistence Results for Energy Functionals with Competing Attracting and Repelling Interactions.

    ABSTRACT: Energy functionals with competing attracting and repelling interactions arise frequently in applications. In this talk, we will consider two related variational models of this kind: the Thomas-Fermi-Dirac-von Weiszacker theory and the nonlocal isoperimetric functional. In particular, we will focus on some recent results in nonexistence of minimizers with large number of electrons (resp. large volume) for these constrained minimization problems. (joint work with Felix Otto).
    February 20th, 4:00-4:50pm, Phillips 381.
    Graham H. Cox, UNC

    TITLE: Essential spectrum of the 2D Euler equation in the absence of long periodic orbits.  

    ABSTRACT: We consider the linearized Euler equation for a 2D incompressible fluid. It is known that the $L^2$ essential spectrum is purely imaginary, and in fact coincides with the imaginary axis when the flow has arbitrarily long periodic orbits. This has been shown using sequences of approximate eigenfunctions supported near a carefully chosen streamline of the flow.

    In this talk we present a new calculation of the essential spectrum, using ideas from geometric measure theory to give a direct integral decomposition of the domain and hence reduce the computation to a family of 1D spectral problems. This method has the advantage of being applicable when the flow does not have arbitrarily long orbits, in which case we find a finite number of spectral gaps on the imaginary axis.

    February 27th, 4:00-4:50pm, Phillips 381.
    Lydia Bieri, Michigan
    TITLE: From the Analysis of Einstein-Maxwell Spacetimes in General Relativity to Gravitational Radiation.  

    ABSTRACT: A major goal of mathematical General Relativity (GR) and astrophysics is to precisely describe and finally observe gravitational radiation, one of the predictions of GR. In order to do so, one has to study the null asymptotical limits of the spacetimes for typical sources such as binary neutron stars and binary black hole mergers. D. Christodoulou showed that every gravitational-wave burst has a nonlinear memory, displacing test masses permanently. In joint work with P. Chen and S.-T. Yau we investigated the Einstein-Maxwell (EM) equations in GR and proved that the electromagnetic field contributes at highest order to the memory effect. In this talk, we discuss the null asymptotics for spacetimes solving EM equations, compute the radiated energy and derive limits at null infinity and compare them with the Einstein vacuum (EV) case. The physical insights are based on geometric-analytic investigations of the solution spacetimes. I will also discuss some very recent results in the situation of binary neutron star mergers.
    March 20th, 4:00-4:50pm, Phillips 381.
    Semyon Dyatlov, Berkeley
    TITLE: Resonances for normally hyperbolic trapped sets.

    ABSTRACT: Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.
    March 27th, 4:00-4:50pm, Phillips 381.
    Christine Breiner, Columbia
    TITLE: A gluing construction for constant mean curvature surfaces.

    ABSTRACT: Constant mean curvature (CMC) surfaces are critical points to the area functional with an enclosed volume constraint. Classic examples include the round sphere and a one parameter family of rotationally invariant surfaces discovered in 1841 by Delaunay. In this talk I outline a generalized gluing method we develop that produces infinitely many new examples of embedded CMC surfaces of finite topology. In particular, I explain two key steps of the gluing construction. First, I will demonstrate how we solve a modified global linearized problem on a candidate surface in the presence of possible obstructions. Second, I will show how perturbations of this surface can be used to induce such a modification, which allows us to solve the problem via a fixed point theorem. This work is joint with Nicos Kapouleas.
    April 17th, 4:00-4:50pm, Phillips 381.
    Tanya Christiansen, Missouri
    TITLE: Distribution of resonances for Schrödinger operators

    ABSTRACT: Physically, a resonance may correspond to a decaying wave. This is in contrast with an eigenvalue of a selfadjoint operator, which in many models corresponds to a periodic wave. Mathematically, resonances are analogs of discrete spectral data on certain noncompact domains or manifolds. Their behavior is less well understood than that of eigenvalues of selfadjoint operators, leaving many questions unanswered. We give an introduction to resonances and the problem of understanding their distribution, concentrating on the case of Schrödinger operators on R^d.

    Fall 2012:

    September 5th, 4:00-4:50pm, Phillips 381.
    Jeremy Marzuola, UNC
    TITLE: Behavior of a toy model dynamical system related to the cubic, defocusing nonlinear Schrödinger equation on the torus.  

    ABSTRACT: With Jim Colliander, Tadahiro Oh and Gideon Simpson, we explore some solutions that arise from a dynamical systems model introduced by Colliander-Keel-Staffilani-Takaoka-Tao (2010) to study the interaction of resonant frequencies in the cubic, defocussing NLS equation on the torus. This dynamical system gives a rich class of solutions, some of which we can prove exist, some of which we study probabilistically and some of which we show exist in approximative models, including a discrete Burger's equation. The goal is to present solutions that have cascades towards large frequency. We will discuss how the toy model fits into the study of the NLS equation and what information can be gained towards the idea of "weak turbulence." If given time, we will discuss some of the perturbative techniques being pursued in an ongoing work with Sebastian Herr and Gideon Simpson towards reconciling the toy model with in particular the Burger's equation model.
    September 19th, 4:00-4:50pm, Phillips 381.
    Robby Marangell, Sydney
    TITLE: The spectrum of periodic wavetrains of the nonlinear Klein-Gordon equation 

    ABSTRACT: Consider the nonlinear periodic Klien-Gordon equation $u_tt − u_xx+V ′(u) = 0$, where u is a scalar valued function of x and t, and the potential V(u) is C2 and periodic. Periodic traveling waves in the problem are organized by the wave speed c, and the energy E over a period T. Linearizing about a periodic traveling wave leads to a spectral problem which is 1) a quadratic pencil in the spectral parameter, and 2) not self adjoint. The spectrum can be completely characterized for large values of the spectral parameter through homotopy/asymptotic arguments and energy bounds, and for very small values of the spectral parameter through a Taylor expansion of the monodromy matrix near the origin. For values in between the very large and very small, a Hill’s equation is introduced, and by exploiting the known Hill’s equation theory as well as some geometry, the Floquet spectrum of the Hill’s equation can be related to the temporal spectrum of the linearized periodic Klein-Gordon operator.
    September 26th, 4:00-4:50pm, Phillips 381.
    Michael Taylor, UNC
    TITLE: Fractional diffusion (and diffusion-reaction) equations.  

    ABSTRACT: We consider variants of reaction-diffusion equations \pa u/\pa t = Delta u +f(u), both when Delta is replaced by a fractional power of the Laplace operator and when \pa/\pa t is replaced by a fractional time derivative. One tool will be an extension of the Duhamel formula. A number of interesting special functions arise in the analysis.
    October 10th, 4:00-4:50pm, Phillips 381.
    Jason Metcalfe, UNC
    TITLE: On the Strauss conjecture for the wave equation on black hole backgrounds.  

    ABSTRACT: We will discuss progress on resolving an analog of the Strauss conjecture for the wave equation in the presence of geometry. In recent years, such geometry has included exterior domains, nontrapping and asymptotically Euclidean metrics, and hyperbolic space. A recent result, joint with H. Lindblad, C. Sogge, M. Tohaneanu, and C. Wang, explores the same in the presence of trapping. In particular, our result permits black hole metrics such as Schwarzschild and Kerr with small angular momentum.
    October 24th, 4:00-4:50pm, Phillips 381.
    Jared Speck, MIT
    TITLE: On Big Bang Spacetimes.

    ABSTRACT: The purpose of this talk is to provide a preliminary report on some results that I recently obtained in collaboration with Igor Rodnianski. The results concern small perturbations of the well-known Friedmann-Lemaˆıtre-Robertson-Walker solution to the Einstein-stiff fluid system (stiff FLRW). This solution is a special case of a family of spatially uniform solutions that arise in cosmology. It models a stiff fluid evolving in a spacetime that expands as t → ∞ and that collapses as t ↓ 0. In particular, the stiff FLRW solution has a “Big Bang” singularity at t = 0. To study the perturbed solutions, we place “initial” data on a Cauchy hypersurface Σ1 = {t = 1} that are close to the stiff FLRW data as measured by a Sobolev norm. No symmetry assumptions are made on the data. We then study the global behavior of the perturbed solution in the collapsing direction. We first show that the spacetime region of interest can be foliated by a family of spacelike Cauchy hypersurfaces Σt, t ∈ (0,1], of constant mean curvature −t^(−1/3). We then analyze the behavior of the solution as t ↓ 0 and provide a detailed description of its asymptotics. The main conclusion is that the perturbed solution remains globally close to the stiff FLRW solution and has approximately monotonic behavior. In particular, the perturbed solution also has a Big Bang singularity at t = 0. More precisely, at the singularity, various curvature invariants uniformly blow-up and the volume of Σt collapses to 0. These blow-up results demonstrate the validity of the Strong Cosmic Censorship conjecture for the past half of the perturbed spacetimes. From the point of view of analysis, our main results can be viewed as a proof of stable blow-up for an open set of solutions to a highly nonlinear elliptic-hyperbolic system. The most important aspect of our analysis is the identification of a new energy almost-monotonicity inequality that holds for the solutions under consideration.
    October 31st, 4:00-4:50pm, Phillips 381.
    Hans Christianson, UNC
    TITLE: A Brief Introduction to Eigenfunctions.  

    ABSTRACT: Eigenfunctions for the Laplace-Beltrami operator are some of the most familiar objects in mathematical analysis and geometry. As an analyst, one tries to gain an understanding of what eigenfunctions look like, especially how geometry influences qualitative properties of eigenfunctions. In this talk, I will introduce some basic ideas of scarring, non-concentration, quantum ergodicity, and approximate solutions.
    CANCELLED!! - November 14th, 4:00-4:50pm, Phillips 381.
    Peter Perry, Kentucky
    TITLE: Inverse Scattering and Dispersive Nonlinear PDE in Two Space Dimensions.  

    ABSTRACT: The inverse scattering method has been applied with great success to study integrable systems such as the Korteweg-de Vries (KdV) and cubic nonlinear Schrodinger (NLS) equations in 1+1 (one space and one time) dimensions. The Riemann-Hilbert method used there has become a powerful tool both for studying large-time asymptotic behavior of solutions to such integrable PDE's, and for studying problems in random matrix theory and the theory of orthogonal polynomials. In the 1980's, Fokas-Ablowitz and Beals-Coifman developed a method analogous to the Riemann-Hilbert method, the so called $\overline{\partial}$ ("d-bar") method, which has applications, in principle at least, to dispersive nonlinear equations in two space dimensions as well as to problems involving normal matrix distributions and orthgoonal polynomials in the plane. In this talk we'll discuss recent progress in $\overline{\partial}$ ("d-bar") methods as applied to Davey-Stewartson and Novikov-Veselov equations, two completely integrable, dispersive nonlinear PDE's in 2+1 dimensions which respectively generalize the NLS and KdV equations. ,We'll also assess prospects for further progress in integrable systems in two space dimensions.
    December 5th, 4:00-4:50pm, Phillips 381.
    Brett Kotschwar, Arizona State
    TITLE: Unique continuation for the Ricci flow, with and without analyticity.  

    ABSTRACT: Certain fundamental geometric properties of the Ricci flow equation, including the preservation of the isometry and reduced holonomy groups, reduce to statements of unique continuation for the Ricci flow and other associated weakly parabolic systems. I will discuss two approaches to such unique continuation problems -- the first, a general approach, based on the embedding of the problem into a prolonged system of mixed differential inequalities amenable to Carleman-type estimates, and the second, a more direct approach, via estimates on the derivatives of the curvature tensor establishing the (interior) time-analyticity of solutions to the Ricci flow.

    Spring 2012:

    January 18th, 4:00-4:50pm, Phillips 381
    Hans Christianson, UNC
    TITLE: Local smoothing with loss for the Schrödinger equation.  

    ABSTRACT: It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L^2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. In this talk we will recall how this is proved and then discuss several simple geometric settings where the result no longer holds, which is a consequence of trapped geodesics, e.g. those which do not escape to infinity. My goal is for the talk to be very accessible, beginning with integrations by parts in $\reals^2$, and moving on from there. This is mostly joint work with J. Wunsch (Northwestern).
    January 25th, 4:00-4:50pm, Phillips 381
    Peter Hislop, Kentucky
    TITLE: Eigenvalues and resonances for finite-area surfaces hyperbolic near infinity.  

    ABSTRACT: It is known that Laplace-Beltrami operators for certain non-compact, finite-area hyperbolic surfaces have an infinite number of eigenvalues embedded in the continuous spectrum. This is expected to be a rare occurrence. In 1983, Y. Colin de Verdiere proved that compactly supported conformal perturbations of the hyperbolic metric generically have no embedded eigenvalues for general, compactly supported perturbations of the metric. We also prove that the Laplace-Beltrami operator for these perturbed metrics have an infinite number of resonances using the Phillips-Sarnac integral. Related results have been obtained by Wolpert using the S-matrix and by Parnovski using trace formulae. This is joint work with L. Marazzi and P. Perry.
    February 10th, 2:00-2:50pm, Phillips 332.
    Pierre Albin, UIUC
    TITLE: The signature operator on stratified pseudomanifolds.   

    ABSTRACT: The signature operator of a Riemannian metric is an important tool for studying topological questions with analytic machinery. Though well-understood for smooth metrics on compact manifolds, there are many open questions when the metric is allowed to have singularities. I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified pseudomanifolds and some of its topological applications.
    February 14th, 4:00-4:50pm, Phillips 367. Joint Applied/Analysis Seminar.
    Kay Kirkpatrick, UIUC
    TITLE: Bose-Einstein condensation and quantum many-body systems.   

    ABSTRACT: Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body systems, a step towards large deviations for Bose-Einstein condensation.
    February 24th, 2:00-2:50pm, Phillips 332.
    Wilhelm Schlag, Chicago
    TITLE: Wave maps exterior to a ball.  

    ABSTRACT: We present a model in which equivariant wave maps are modified by excising a ball around zero in Minkowski space. This has the effect of turning a supercritical equation into a subcritical one (in 1+3 dimensions). The goal here is to understand the problem of relaxation to a ground state harmonic map, and we shall present partial progress in this direction. The proof hinges on the Kenig-Merle concentration compactness argument. Joint work with Andrew Lawrie at Chicago.
    March 21st, 4:00-4:50pm, Phillips 381. Joint Applied/Analysis Seminar.
    Jonathan Weare, Chicago
    TITLE: Coarse graining results in crystal surface relaxation.  

    ABSTRACT: I will present resent results on the coarse graining of two models of crystal surface relaxation. First I discuss joint work with Hala Al Hajj Shehadeh and Robert V. Kohn on an ODE model of a 1D monotone crystal surface. We prove that the slope of a fnite size crystal in this setting converges (in the long time limit) to a similarity solution. We also give an informal derivation of a fully non-linear fourth order PDE (large crystal) limit of the ODE's as well as analogues of our similarity results in the continuum. Next I'll discuss current work with Jeremy Marzuola investigating certain scaling limits of a general family of Kinetic Monte Carlo models of crystal surface relaxation. We informally derive two fully non-linear fourth order PDE in two different scaling limits. Both PDE's are similar to (but not exactly the same as) PDE's that have been proposed as large scale limits for the models in question. Our aim is to clarify how each arrises as well as to establish the limits rigorously.
    March 28th, 4:00-4:50pm, Phillips 381
    Herbert Koch, Bonn
    TITLE: The Miura map and estimates for KdV in H^{-1}.  

    ABSTRACT: I will present uniform in time estimates for solutions to KdV in H^{-1} in terms of the corresponding norm of the initial data, stability and asymptotic stability of solitons in H^{-1}. The proof relies on the Galilean invariance of KdV, the Miura map near the kink, and a study of mKdV near the kink in L2.
    April 3rd, 4:00-4:50pm, Phillips 367.
    Anna Mazzucato, Penn State
    TITLE: Boundary layer analysis for certain non-linear fluid flows.  

    ABSTRACT: We study certain classes of Taylor-Couette flows in circular pipes. Due to the symmetry of the problem, the equations of motion reduce to a weakly non-lnear systems in two space dimensions. We prove optimal convergence rates in viscosity of Navier-Stokes solutions to Euler solutions using effective equations for flow correctors, under some compatibility conditions between the initial and boundary data. This is joint work with Daozhi Han, Dongjuan Niu, and Xiaoming Wang.
    April 4th, 4:00-4:50pm, Phillips 381. Joint Applied/Analysis Seminar.
    Alex Barnett, Dartmouth
    TITLE: Fast computation of high frequency Dirichlet eigenmodes via the spectral flow of the interior Neumann-to-Dirichlet map.  

    ABSTRACT: We present and analyze a new method for numerical computation of the spectrum and eigenfunctions of a planar star-shaped domain with Dirichlet boundary condition. The method is 'fast' since it is computes a cluster of eigenfunctions (numbering of order the square-root of the eigenvalue) in the time usually taken to compute a single one. In practice, with 400 wavelengths across the domain, and relative error 1e-10, this speed-up is around 1e3. It is related to the little-understood 'scaling method', but, in constrast, has a rigorous error analysis and allows higher-order accuracy. We will include some applications to quantum chaos.
    April 13th, 3:00-3:50pm, Phillips 332. Special Analysis Seminar.
    Kevin Zumbrun, Indiana

    TITLE: Viscous roll waves and stability of periodic Kuramoto-Sivashinsky-Korteweg de Vries waves in the KdV limit.

    ABSTRACT: An interesting phenomenon of inclined thin film flow is that asymptotic behavior appears to be dominated by arrays of approximate solitary waves, or ``roll waves,'' despite that individual solitary waves are (quite clearly) unstable. Here we discuss some recent work shedding light on this phenomenon, both heuristically and analytically. In particular, we i) describe a recent result establishing equivalence (up to nondegeneracy conditions) of spectral and nonlinear stability of periodic arrays, and ii) focusing on a canonical KS-KdV \to KdV limit describing weakly unstable small-amplitude flow, we verify by a singular perturbation analysis that, up to the precision of certain elliptic integral computations, spectral stability hold in this limit precisely for a certain band of periods [X_L,X_R]. The former result resolves the 35-year open problem, pointed out almost immediately after the introduction of the model, of nonlinear stability of numerically-observed bands of spectrally stable periodic Kuramoto-Sivashinsky and KS-KdV waves. The latter has interest both as a perturbed integrable system problem which permits a complete solution, particularly in the key low- and high-frequency regimes, where we find it necessary to introduce a number of new tools in order to carry out the analysis, and as a canonical physical problem analagous to the Burgers limit for small-amplitude shock waves studied by Kawashima-Matsumura-Nishihara, Goodman, Liu, and many others in the closely related setting of compressible gas dynamics.

    Reference 1

    Reference 2

    Reference 3

    Reference 4

    April 18th, 2:00-2:50pm, Phillips 332.
    Toan Nguyen, Brown
    TITLE: On the stability of boundary layers in the Navier-Stokes equations.  

    ABSTRACT: In fluid dynamics, a classical problem is to understand the dynamics of viscous fluid flows past solid bodies, especially in the regime of very high Reynolds numbers (or very small viscosity). Boundary layers are typically formed in a thin layer near the boundary. In this talk, I shall review recent developments on stability of boundary layers (for compressible and incompressible fluids). In particular, I will present some main ingredients of a stability analysis of suction/blowing boundary layers. I will also discuss various ill-posedness results obtained for the classical Prandtl boundary layers and the relevance of boundary-layer expansions in the inviscid limit of the Navier-Stokes equations.

    Fall 2011:

    August 31st, 4:00-4:50pm, Phillips 381
    Jason Metcalfe, UNC
    TITLE: Local well-posedness for quasilinear Schrödinger equations.  

    ABSTRACT: I will speak on a recent joint study with J. Marzuola and D. Tataru which proves low regularity local well-posedness for quasilinear Schrödinger equations. Similar results were previously proved by Kenig, Ponce, and Vega in much higher regularity spaces using an artificial viscosity method. Our techniques, and in particular the spaces in which we work, are motivated by those used by Bejenaru and Tataru for semilinear equations.
    September 7th, 4:00-4:50pm, Phillips 381
    Nathan Totz, Duke
    TITLE: A rigorous justification of the modulation approximation to the 2D full water wave problem.  

    ABSTRACT: In this joint work with Sijue Wu (U. Mich.), we consider solutions to the 2D inviscid infinite depth water wave problem neglecting surface tension which are to leading order wave packets of the form $\alpha + \epsilon B(\epsilon \alpha, \epsilon t, \epsilon^2 t)e^{i(k\alpha + \omega t)}$ for $k > 0$. Multiscale calculations formally suggest that such solutions have modulations $B$ that evolve on slow time scales according to a focusing cubic NLS equation. Justifying this rigorously is a real problem, since standard existence results do not yield solutions which exist for long enough to see the NLS dynamics. Nonetheless, given initial data within $O(\epsilon^{3/2})$ of such a wave packet in $L^2$ Sobolev space, we show that there exists a unique solution to the water wave problem which remains within $O(\epsilon^{3/2})$ to the formal approximation for times of order $O(\epsilon^{-2})$. This is done by using a version of the evolution equations for the water wave problem developed by Sijue Wu with no quadratic nonlinearity.
    September 23rd, Time and Place TBD
    Magdalena Czubak, SUNY Binghamton
    TITLE: On some properties of the Navier-Stokes equation on the hyperbolic space.  

    ABSTRACT: Contrary to what is known in the Euclidean case, finite energy and finite dissipation solutions to the Navier-Stokes equation on a two dimensional hyperbolic space are nonunique. We review the nonuniqueness result and discuss possible ways to arrive at uniqueness of solutions in the hyperbolic setting. This is based on joint works with Chi Hin Chan and Pawel Konieczny.
    September 23rd, Joint Applied/Analysis Seminar, 4:00-4:50pm, Phillips 332
    Mikael Rechtsman, Technion
    TITLE: Nonlinear optics of microstructures: recent results and open problems.  

    ABSTRACT: Experimental nonlinear optics has motivated quite a number of deep (and multidisciplinary) mathematical questions over the years, particularly in the context of nonlinear Schrodinger equations. In this talk, I will give a number of examples of past and current problems in nonlinear optics, with the goal of bridging the gap between mathematical analysis and the laboratory. Examples will include past results such as stable spatial solitons in saturable materials, discrete solitons, and more current ones that surround Anderson localization of light, and disorder-enhanced transport in quasicrystals (and more, time permitting).
    September 28th, 4:00-4:50pm, Phillips 381
    Carla Cederbaum, Duke
    TITLE: The Newtonian Limit of Geometrostatics.  

    ABSTRACT: Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.
    October 5th, 4:00-4:50pm, Phillips 381 Ha Pham, Stanford TITLE: A simple diffractive boundary value problem on an asymptotically anti-de Sitter space.  

    ABSTRACT: We will discuss the diffractive problem (the existence of `shadow region') for a case of asympotically anti-de Sitter (AdS) spaces whose boundary is not totally geodesic. Asymptotically AdS spaces are Lorentzian manifolds modeled on actual AdS space at infinity. The main result establishes conor- mality (relative to a certain weighted L2-space) when one is close enough to the boundary for the forward solution. The approach adopted is motivated from a conformally related problem by Friedlander. The difficulties for our problem start with the fact that our operator does not a priori have a well-understood solution in terms of Airy functions as in Friedlander's case; and in fact we have to construct these from scratch. With translation invariance in the boundary variables (assumed for our model operator), most major technical difficulties reduce to studying and constructing a global resolvent for a semiclassical ODE on R+, which at one end is a b-operator while having a scattering behavior at infinity. We use different techniques near 0 and infinity to analyze the local problems. Near infinity we use local resolvent bounds, while near zero we build a local semiclassical parametrix. The two local resolvent/parametrix estimates are then patched by a construction of Datchev-Vasy to get a global resolvent with a polynomial bound in h.
    October 10th, 1:00-1:50pm, Phillips 324C. Amanda French, UNC TITLE: TBD.  

    October 12th, 4:00-4:50pm, Phillips 381
    Gustav Holzegel, Princeton
    TITLE: Stability problems for asymptotically AdS black holes.  

    ABSTRACT: I will present some recent results regarding the stability of asymptotically anti de Sitter black holes. After introducing the relevant spacetime geometries, I will turn to a discussion of the massive wave equation on Kerr-AdS spacetimes and finally to the spherically symmetric coupled Einstein Klein-Gordon system. We prove that the latter system is well-posed and moreover, that the AdS-Schwarzschild metric is asymptotically stable within this model. The results on the coupled problem are joint work with Jacques Smulevici.
    October 17th, 2:00-2:50pm, Phillips 301 Betsy Stovall, UCLA

    TITLE: On finite time blowup solutions to certain nonlinear Klein--Gordon equations.

    ABSTRACT: We consider the focusing nonlinear Klein--Gordon equation

    u_{tt} - \Delta u + m^2 u = |u|^p u
    u(0) \in H^1, \quad u_t(0) \in L^2,
    for $m \in [0,1]$ and $\frac4d < p < \frac4{d-2}$, with $d \geq 2$.  In the case when the solution $u$ blows up in finite time, we study the behavior of various norms of $u(t)$ and $u_t(t)$ as $t$ approaches the blowup time.  This is joint work with Rowan Killip and Monica Vi\c{s}an.

    October 26th, 4:00-5:00pm, Phillips 381 Alex Ionescu, Princeton TITLE: The energy-critical defocusing NLS in periodic settings.  

    ABSTRACT: I will discuss some recent work, joint with B. Pausader, on constructing global solutions of defocusing energy-critical nonlinear Schrodinger equations in periodic and semiperiodic settings.

    October 27th, 4:00-4:50pm, Phillips 332 - NOTE: SPECIAL TIME AND PLACE AS THIS IS A COLLOQUIUM SPEAKER

    Boris Khesin, Toronto Title:  Optimal Transport and Geodesics on Diffeomorphism Groups

    ABSTRACT:  In 1965 Vladimir Arnold described the hydrodynamical Euler equation as an equation of geodesics on the group of volume-preserving diffeomorphisms with respect to the right-invariant L^2 metric. In the talk we describe the corresponding extrinsic geometry by regarding the group of volume-preserving diffeomorphisms as a Riemannian submanifold in the group of all diffeomorphisms. Various applications of the latter approach include a relation to L^2 and H^1 optimal mass transport problems, a non-holonomic version of the Moser theorem, and integrable PDEs with several space variables.
    November 2nd, 4:00-4:50pm, Phillips 381
    Austin Ford, Northwestern
    TITLE: Coisotropic regularity and second microlocalization.  

    ABSTRACT: I will discuss the notion of coisotropic regularity, a generalization of Hörmander's Lagrangian regularity where the homogeneous Lagrangian submanifold of the cotangent bundle is replaced by a homogeneous coisotropic. Fixing the coisotropic $\Gamma$, I will then discuss a "second microlocal" pseudodifferential calculus associated to $\Gamma$ which tests for coisotropic regularity in a way analogous to how the usual pseudodifferential calculus tests for smoothness. Finally, I will mention how such a calculus can shed light on wave evolution on singular spaces.
    November 3rd, 4:00-4:50pm, Phillips 332 - NOTE: SPECIAL TIME AND PLACE AS THIS IS A COLLOQUIUM SPEAKER Ken McLaughlin, Arizona

    TITLE: Oscillatory Phenomenon in a Scaling Limit for the Periodic Linear Schrödinger Equation

    ABSTRACT:  In the 1830s Henry Fox Talbot discovered a self-imaging phenomenon in coherent illumination of a periodic diffraction grating. Since then, studies of the Talbot effect (as it is called) have themselves recurred frequently. One such incarnation was initiated by Michael Berry; eventually some interesting fractal dimension results were proven by subsequent researchers. I will explain some of these connections, and some recent developments concerning oscillatory phenomena reminiscent of Gibbs' phenomenon. In passing I will explain some possible misconceptions concerning the discovery of Gibbs' phenomenon. The main mathematical connection is that the Fourier series solution is quite directly related to exponential sums appearing in classical analytic number theory. Joint work with Nigel Pitt.

    November 4th, Joint Applied/Analysis Seminar, 4:00-4:50pm, Phillips 332 Ken McLaughlin, Arizona

    TITLE:  The Partition Function of Random Matrix Theory, its Behavior as the Matrix Size Grows, and a Number of Applications

    ABSTRACT: We will begin with a basic introduction to random matrices. I will explain how the partition function emerged as the magical elixir of solvable 2 dimensional quantum gravity, without a proof concerning its asymptotic behavior. Then a number of applications of the proof (in 2003, joint work with Nick Ercolani) will be presented. In particular I will explain some very recent large deviation results for fundamental eigenvalue statistics.

    November 16th, 4:00-4:50pm, Phillips 381 Matt Hernandez, UNC TITLE: Resonant leading term geometric optics expansions with boundary layers for quasilinear hyperbolic boundary problems.  

    ABSTRACT: We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems. Recent work (2010) by J.-F. Coulombel, O. Gues, and M. Williams treated the case where the boundary frequency lies in the hyperbolic region. In this case the authors constructed expansions in terms of almost-periodic functions which are highly oscillatory wavetrains with real phases. We treat generic boundary frequencies by incorporating in our expansions wavetrains with real phases, and, in addition, boundary layers which arise as functions with nonreal phases. The technique of simultaneous Picard iteration is employed in both studies, but dealing with nonreal phases forces us to adapt to functions which are not almost-periodic and introduces difficulties such as approximately solving complex transport equations. A major task of our study is to clearly establish the interactions and dependencies between the wavetrains and the boundary layers.
    November 30th, 4:00-4:50pm, Phillips 381
    David Borthwick, Emory
    TITLE: Spectral theory for infinite-volume hyperbolic manifolds.  

    ABSTRACT: This talk will focus on the spectral theory of the Laplacian on infinite-volume manifolds with hyperbolic ends. For such spaces, the discrete spectral data are the resonances, which can be defined in analogy to eigenvalues as poles of an analytic continuation of the resolvent. Many of the usual spectral theory tools have been adapted to this setting, with the resonance set taking the role of the spectrum. There is, for example, a trace formula expressing the wave trace as a sum over the resonance set. After introducing spectral theory in this setting, we'll discuss the particular problem of understanding the distribution of resonances. We'll present some recent results on resonance counting for hyperbolic surfaces.
    November 30th, 4:00-4:50pm, Phillips 381
    Spyros Alexakis, Toronto
    TITLE: Loss of compactness and bubbling for complete minimal surfaces in hyperbolic space.  

    ABSTRACT: We consider the Willmore energy on the space of complete minimal surfaces in H^3 and study the possible loss of compactness in the space of such surfaces with energy bounded above. This question has been extensively studied for various energy functionnals for closed manifolds. The first such study was that of Sacks and Uhlenbeck for harmonic maps. The key tools to study the loss of compactness in that case are epsilon-regularity and removability of singularities; the loss of compactness can then occur due to bubbling at a finite number of points where energy concentrates. We find the analogous results in our setting of complete surfaces. These are the first results in this direction for surfaces with a free boundary. joint with R. Mazzeo.

    Spring 2011:

    January 19th, 4:00-4:50pm, Phillips 381
    Mark Williams, UNC
    TITLE: Resonant leading term geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems.  

    ABSTRACT: Simultaneous Picard iteration is a powerful technique introduced by Joly, Metivier, and Rauch in the early 1990s for justifying leading term geometric optics expansions in nonlinear hyperbolic initial value problems. We explain how this method can be extended to apply to boundary problems, including the free boundary problem for oscillatory multidimensional Euler shocks. (joint work with J.-F. Coulombel and O. Gues).
    January 26th, 4:00-4:50pm, Phillips 381
    Igor Rodnianski, Princeton
    TITLE: On formation of trapped surfaces in General Relativity.  

    ABSTRACT: The talk will describe the ingredients of the recent results on formation of trapped surfaces. We will review the background material, including the incompleteness theorem of Penrose, heuristic mechanism, the short-pulse method and the evolutionary control introduced and exploited by Christodoulou in his breakthrough work. We will also explain recent extensions, amplifications and generalizations of the result as well as future directions.
    January 27th, 4:00-4:50pm, Phillips 332 - NOTE: SPECIAL TIME AND PLACE AS THIS IS A COLLOQUIUM SPEAKER
    Nicolas Burq, Orsay
    TITLE: Random data for wave equations: From Paley and Zygmund to dispersive PDE's.  

    ABSTRACT: Link to Colloquium Page.
    February 9th, 4:00-4:50pm, Phillips 381
    Kiril Datchev, MIT
    TITLE: Wave decay on asymptotically hyperbolic manifolds.  

    ABSTRACT: We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates, and deduce as a corollary local exponential decay of solutions to the wave equation in a variety of hyperbolic geometric settings. This project is joint work with Andras Vasy.
    February 16th, 4:00-4:50pm, Phillips 381 - Special Talk for Oral Examination Requirement
    John Helms, UNC

    February 17th, 4:00-4:50pm, Phillips 332 - NOTE: SPECIAL TIME AND PLACE AS THIS IS A COLLOQUIUM SPEAKER
    Michael Weinstein, Columbia
    TITLE: Dynamics of nonlinear dispersive systems: Analysis and Applications  

    ABSTRACT: This talk will overview results and open problems on the dynamics of coherent structures for certain nonlinear dispersive PDEs, a class of infinite dimensional Hamiltonian systems. Many such systems have spatially localized solutions, describing coherent structures (hydrodynamic, electrodynamic, quantum...) such as soliton pulses or vortices, with remarkable stability properties. The general PDE dynamics can be viewed in terms of the nonlinear interaction of such coherent structures with linear dispersive waves. The infinite-time behavior is the subject of nonlinear scattering. Intermediate but very long-time transients, e.g. metastable states, play an important role in the analysis and a central role in applications. A detailed understanding involves ideas from dynamical systems (Hamiltonian theory of normal forms,...) and scattering theory (wave operators, non-self adjoint spectral theory,...), variational and harmonic analysis. We will consider these questions in the context of the nonlinear Schroedinger - Gross Pitaevskii equation, a class of PDEs having wide applications to classical and quantum systems. Applications to the control of soliton-like states in nonlinear optical and quantum systems will be discussed.
    Michael Weinstein, Columbia
    TITLE: Dynamics of nonlinear dispersive systems: Analysis and Applications - Part II  

    ABSTRACT: This talk will overview results and open problems on the dynamics of coherent structures for certain nonlinear dispersive PDEs, a class of infinite dimensional Hamiltonian systems. Many such systems have spatially localized solutions, describing coherent structures (hydrodynamic, electrodynamic, quantum...) such as soliton pulses or vortices, with remarkable stability properties. The general PDE dynamics can be viewed in terms of the nonlinear interaction of such coherent structures with linear dispersive waves. The infinite-time behavior is the subject of nonlinear scattering. Intermediate but very long-time transients, e.g. metastable states, play an important role in the analysis and a central role in applications. A detailed understanding involves ideas from dynamical systems (Hamiltonian theory of normal forms,...) and scattering theory (wave operators, non-self adjoint spectral theory,...), variational and harmonic analysis. We will consider these questions in the context of the nonlinear Schroedinger - Gross Pitaevskii equation, a class of PDEs having wide applications to classical and quantum systems. Applications to the control of soliton-like states in nonlinear optical and quantum systems will be discussed.
    February 23rd, 4:00-4:50pm, Phillips 381
    Nikos Tzirakis, UIUC
    TITLE: High frequency perturbation of cnoidal waves.  

    ABSTRACT: In this talk I will consider the Korteweg-de Vries (KdV) equation with periodic boundary conditions. There is a family of well known solitary wave solutions known as cnoidal waves. We study high frequency perturbations of stationary cnoidal waves. It is shown that for finite time the cnoidal wave can be recovered, as the high frequency perturbation evolves almost linearly. The perturbation is not necessarily small and it is assumed to be finite in $L^2$ (energy) norm. This is joint work with B.Erdogan and V. Zharnitsky.
    March 2nd, 4:00-4:50pm, Phillips 381
    Chris Sogge, Johns Hopkins
    TITLE: On the $L^p$ norms and dispersive properties of eigenfunctions  

    ABSTRACT: We provide a necessary and sufficient condition that $L^p$-norms, $2< p < 6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $\lambda$. The condition that ensures this is that their $L^2$ norms over $O(\lambda^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $\lambda$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). We also discuss connections between our results and Quantum Ergodicity. Time permitting, we shall also discuss related joint work with S. Zelditch on estimates for manifolds with nonpositive curvature.
    March 23rd, 4:00-4:50pm, Phillips 381
    Marius Mitrea, Missouri
    TITLE: How to express smoothness in geometrical terms, and a sharp version of the Hopf-Oleinik boundary point principle.  

    ABSTRACT: Suppose a surface has the property that, at every point, it can be pinched in between the two rounded components of a fixed hour-glass shape, appropriately re-positioned. How regular does this condition make the given surface? We shall address this issue and then proceed to use this type of geometrical characterization of smoothness in order to prove a sharp version of the celebrated Hopf-Oleinik boundary point principle for 2-nd order nondivergence form differential operators.
    March 25th, 9:00-9:50am, Special Joint Applied/Analysis Seminar, Room 328 Phillips
    Dirk Hundertmark, UIUC
    TITLE: Mathematical Challenges from Non-linear Fiber Optics.  

    ABSTRACT: We describe some recent rigorous work on soliton-like pulses in dispersion managed optical fiber channels: Dispersion management in glass fibers refers to the engineering of an optical fiber channel with alternating spans of positive (normal) and negative (anomalous) dispersion fiber (periodic or otherwise) in order to achieve greater stability, bandwidth etc of optical information transfer. This technology has lad to a hundred-fold increase in bandwidth in long-haul optical transmission lines over intercontinental distances and it is widely used commercially nowadays. The simplest mathematical model describing pulses in a glass-fiber cable is the scalar one-dimensional nonlinear Schrödinger equation with cubic nonlinearity. For dispersion managed glass fiber cables, the coefficient of dispersion is a function of distance (e.g., periodic) along the fiber waveguide. To model dispersion managed fiber channels, one also averages over one period, yielding the Gabitov-Turitsyn equation, which is a non-local version of the non-linear Schrödinger equation. It is well known that with constant negative (i.e., anomalous) dispersion, there are soliton-like localized solutions and, not much of a surprise, for dispersion managed systems if the dispersion is, on the average, anomalous, then there are again stable solitons. However, in physical experiments, as well as numerical studies, it has long been observed that one gets soliton-like localized solutions even for average dispersion equal to zero (!) dispersion. This was a surprise, both physically and mathematically, because the conventional wisdom had been that solitons emerge from a combination of non-trivial linear dispersion and nonlinearity. Something more subtle is going on in the zero average dispersion case, which is also the most important case from an applications point of view. Rigorous results on soliton-like pulses for the Gabitov-Turitsyn equation, the so-called dispersion management solitons, have been rare, which is mainly due to its non-locality, which makes it hard to study. Rigorous results for zero average dispersion are even rarer, since this case is a singular limit. This is quite in contrast to the enormous amount of experimental, numerical and theoretical work (if one searches for dispersion management on Google scholar, one gets roughly 552,000 hits). We will discuss recent work on the decay and regularity properties of dispersion management solitons. Our results include a simple proof of existence of solutions of the dispersion management equation under mild conditions on the dispersion profile, which includes all physically relevant cases, regularity of weak solutions, and most recently a proof of exponential decay of dispersion management solitons, which confirms the theoretically and experimentally seen fact that dispersion management solitons are very well-localized.
    March 30th, 2:00-3:00pm, 301 Phillips ... NOTE: special time and place in order not to conflict with Brauer Lectures
    Monica Visan, UCLA and Texas
    TITLE: The energy-supercritical nonlinear wave equation.  

    ABSTRACT: I will report on joint work with Rowan Killip toward the global well-posedness and scattering conjecture for the energy-supercritical nonlinear wave equation in three space dimensions.
    April 1st, 3:00-4:00pm, 332 Phillips
    Don Hadwin, University of New Hampshire
    TITLE: When is an operator S an analytic function of an operator T?.  

    ABSTRACT: Here.
    April 4th, 11:00-11:50am, Phillips 301
    Parul Laul, UNC
    TITLE: Localized energy estimates for wave equations on high dimensional Schwarzschild space-times.  

    ABSTRACT: Localized energy estimates for the wave equation on Minkowski and (1+3)-dimensional Schwarzschild space-times have had various applications; for example, in the proof of Price's Law. We discuss a similar localized energy estimate for the homogeneous wave equation \Box \phi=0 on the (1+n)-dimensional hyperspherical Schwarzschild manifold. Note, this talk is to defend a dissertation written for completion of the Ph.D. in Mathematics at the University of North Carolina.
    April 6th, 4:00-4:50pm, Phillips 381
    Laurent Thomann, Nantes
    TITLE: KAM for the NLS with harmonic potential.  

    ABSTRACT: We show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. This is a joint work with B. Grébert.
    April 13th, 4:00-4:50pm, Phillips 381
    Jonathan Mattingly, Duke
    TITLE: Hypoelipticity, Smoothing and ergodicity for Stochastic Partial Differential Equations.  

    ABSTRACT: I will discuss the idea of a hypoelliptic diffusion on a function space. Such infinite dimensional diffusions are generated by stochastically forced PDEs. I will begin by giving a probabilist perspective of Hörmander's "sum of squares theorem" and show how similar ideas can capture the way randomness moves through a SPDE through nonlinear interactions. I will end by commenting how these results can be used to prove an ergodic theorem.
    April 20th, 4:00-4:50pm, Phillips 381 - CANCELLED!!!!
    John Toth, McGill - CANCELLED!!!!

    April 27th, 4:00-4:50pm, Phillips 381
    Braxton Osting, Columbia
    TITLE: Spectral Optimization Problems for Engineering Long-Lifetime States.  

    ABSTRACT: Consider a system governed by the wave equation with wave speed $c(x)$, where $c(x)$ is inhomogeneous within a bounded region $\Omega \subset \mathbb R^d$, constant in $\mathbb R^d \setminus \Omega$, and is bounded point-wise from both above and below everywhere. The initial value problem with localized data admits an expansion in terms of the resonant states of the system. The resonant states decay exponentially at a rate given by the imaginary part of the corresponding resonance and thus the resonance with smallest imaginary part dominates at large time scales. We pose the material design problem: Find a wave speed $c_opt$ within an admissible class which has a Helmholtz resonance with minimal imaginary part. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. We also show that the optimal structure is 'bang-bang' in the sense that if point-wise bounds are imposed on $c(x)$, then the optimal wave-speed, $c_opt$, will achieve these bounds almost everywhere. I'll present several optimal structures obtained using gradient-based optimization methods. Applications of this design problem include using micro- and nano-scale photonic crystal devices to control light. I'll also discuss a related design problem for the forced Schroedinger Equation where Fermi's Golden Rule governs the lifetime of a meta-stable state.

    Fall 2010:

    September 1, 4:00-4:50pm, Phillips 381
    Michael Taylor, UNC
    TITLE: Scattering of nonlinear waves on 3D hyperbolic space.  

    ABSTRACT: We discuss the existence and completeness of wave operators for small amplitude solutions to semilinear wave equations on 3D hyperbolic space. We consider nonlinearities of the form a|u|^b, with emphasis on the case 5/3 <= b <= 3. This is part of a project with Jason Metcalfe.
    September 8, 4:00-4:50pm, Phillips 381
    Dean Baskin, Northwestern
    TITLE: Wave equations on asymptotic de Sitter spaces.  

    ABSTRACT: Asymptotically de Sitter spaces are asymptotic solutions of the vacuum Einstein equations with positive cosmological constant and can be thought of as a Lorentzian analogue of asymptotically hyperbolic spaces. In this talk, I will construct the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces and use this construction to obtain Strichartz estimates.
    September 22, 4:00-4:50pm, Phillips 381
    Parul Laul, UNC
    TITLE: Localized energy estimates for wave equations on high dimensional Schwarzschild space-times.  

    ABSTRACT: Localized energy estimates for the wave equation on Minkowski and (1+3)-dimensional Schwarzschild space-times have had various applications; for example, in the proof of Price's Law. We discuss a similar localized energy estimate for the homogeneous wave equation \Box \phi=0 on the (1+n)-dimensional hyperspherical Schwarzschild manifold.
    September 29, 2:00-2:50pm, Phillips 367 -- NOTE: SPECIAL TIME AND PLACE DUE TO GERGEN LECTURES AT DUKE!!!
    Vedran Sohinger, MIT
    TITLE: Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations.  

    ABSTRACT: In this talk, we study the growth of Sobolev norms of solutions to 1D and 2D Nonlinear Schrodinger Equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate of the low-to-high frequency cascade. We present a frequency decomposition method which allows us to obtain polynomial bounds in the case of the 1D Hartree equation with sufficiently regular convolution potential, and the cubic NLS on the plane, and which allows us to bound the growth of fractional Sobolev norms of the Cubic NLS on the real line.
    October 6, 4:00-4:50pm, Phillips 381
    Gideon Simpson, Toronto
    TITLE: A Survey of Spectral Properties and the Role of Numerics in Analysis.  

    ABSTRACT: I will discuss several results on so-called spectral properties, estimates on the coercivity of auxiliary bilinear forms of the nonlinear Schrodinger equation (NLS). These properties have been instrumental in studying NLS in two ways: proving the stability of the log-log blow up of NLS and ruling out nonzero imaginary eigenvalues of the linearized operator. I will review successes in this direction, and mention obstacles in showing this property generically, Additionally, the proof of this property currently relies on solving a small number of one dimensional ODEs, raising the question of how one should view numerically assisted proofs. Work presented in this talk is in collaboration with J.L. Marzuola and I. Zwiers.
    October 13, 4:00-4:50pm, Phillips 381
    Matt Blair, UNM
    TITLE: Strichartz Estimates for the Schrödinger Equation in Exterior Domains.  

    ABSTRACT: We consider the problem of proving scale-invariant Strichartz inequalities for the Schrödinger equation in exterior domains. One of the main difficulties here is that the boundary conditions affect the flow of energy, which complicates estimates. We will discuss recent progress in developing inequalities for these boundary value problems, with emphasis on a joint work with H. Smith and C. Sogge.
    October 27, 4:00-4:50pm, Phillips 381
    Mihai Tohaneanu, Purdue University
    TITLE: Pointwise decay on nonstationary spacetimes.  

    ABSTRACT: Let $u$ be a solution to the equation $\Box_g u = 0$ where $g$ is some (nonstationary) Lorentzian metric and $\Box_g$ its associated d'Alembertian. If we assume a priori that certain local energy norms for $u$ and its higher derivatives hold, we can prove that $u$ decays pointwise like $t^{-3-}$. As an application, we can prove the aforementioned decay on Kerr spacetimes and some perturbations. This is joint work with Jason Metcalfe and Daniel Tataru.
    November 3, 4:00-4:50pm, Phillips 381
    Chengbo Wang, Johns Hopkins
    TITLE: Strauss conjecture for nontrapping obstacles.  

    ABSTRACT: In this talk, we discuss our recent work on the 2-dimensional Strauss conjecture for nontrapping obstacles. This is a joint work with H. Smith and C. Sogge. Recently, Hidano, Metcalfe, Smith, Sogge and Zhou proved the Strauss conjecture for nontrapping obstacles when the spatial dimension n equals 3 and 4. Their method is to prove abstract Strichartz estimates, including the |x|-weighted Strichartz estimates. In the Minkowski spacetime, the |x|-weighted Strichartz estimates (also from my work with Fang) can be utilized to prove the Strauss conjecture with n = 2, 3, 4. In this work, we remedy the difficulty for n=2 by proving the generalized Strichartz estimates of the type L^q_t L^r_{|x|} L^2_\theta. The corresponding problem for n>4 are still open.
    November 8, 4:00-4:50pm, Phillips 381 -- NOTE: SPECIAL TIME AND PLACE!!!
    Fabrice Planchon, Nice
    TITLE: Revisiting endpoint Serrin's criterion for the Navier-Stokes equation through profile decompositions.  

    ABSTRACT: Recently, Escauriaza-Seregin-Sverak proved that for weak solutions to the incompressible Navier-Stokes equation, uniform (in time) boundedness of the spatial L^3 norm prevents blow-up. We revisit this result in the framework of L^3 mild solutions (à la Kato), following the Kenig-Merle roadmap for dispersive equations (extraction of a minimal blow-up solution through suitable profile decompositions, compactness at blow-up time and finally, like in E-S-S, backward uniquess which precludes existence of such a critical solution). This is joint work with Isabelle Gallagher and Gabriel Koch.
    November 10, 4:00-4:50pm, Phillips 381
    Melissa Tacy, IAS
    TITLE: Semiclassical eigenfunction estimates.  

    ABSTRACT: Concentration phenomena for Laplacian eigenfunctions can be studied by obtaining estimates for their $L^{p}$ growth. By considering eigenfunctions as quasimodes (approximate eigenfunctions) within the semiclassical framework we can extend such estimates to a more general class of semiclassical operators. This talk will focus on $L^{p}$ estimates for quasimodes restricted to hypersurfaces and the links between such estimates and properties of classical flow.
    November 17, 4:00-4:50pm, Phillips 381
    Jonathan Luk, Princeton
    TITLE: The null condition and global existence for wave equations on a slowly rotating Kerr spacetime.  

    ABSTRACT: Motivated by the problem of the nonlinear stability of Kerr spacetimes, we study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region $\{r\leq \frac{t}{4}\}$.
    December 1, 4:00-4:50pm, Phillips 381
    Jared Wunsch, Northwestern
    TITLE: Spreading of quasimodes on integrable systems.  

    ABSTRACT: Consider a sequence of eigenfunctions of the Laplace operator on a manifold with completely integrable geodesic flow. You might suppose that the eigenfunctions could concentrate completely upon a single closed orbit, but this turns out not (usually) to be the case. I will describe how the study of local Lagrangian regularity, and its refinement, the second-microlocal calculus, lead to this result. This is partly joint work with A. Vasy.


    Past Semesters:

    Spring 2010

    Tues., 1/19/10, Special Lecture, PH332, 4:00
    Pierre Albin, IAS and Courant Institute
    TITLE:  Ricci flow and the determinant of the Laplacian on non-compact

    ABSTRACT:  The determinant of the Laplacian is an important invariant of closed surfaces and has connections to the dynamics of geodesics, Ricci flow, and physics.  Its definition is somewhat intricate as the Laplacian has infinitely many eigenvalues.  I'll explain how to extend the determinant of the Laplacian to non-compact surfaces where one has to deal with additional difficulties like continuous spectrum and divergence of the trace of the heat kernel.  On surfaces (even non-compact) this determinant has a simple variation when the metric varies conformally.  I'll explain how to use Ricci flow to see that the largest value of the determinant occurs at constant curvature metrics.  This is joint work with Clara Aldana and Frederic Rochon.
    Wed., 1/20/10, Ergodic Theory Seminar, PH385, 4:00
    Richard Oberlin, UCLA
    TITLE: A variation norm Carleson Theorem

    ABSTRACT: The Carleson-Hunt theorem shows that every p-integrable function f on the circle, 1<p<infinity, the Fourier series of f converges to f almost everywhere.  We give an extention of this theorem which provides quantitative information about the rate of convergence and some applications.  Joint work with A. Seeger, T. Tao, C. Thiele, and J. Wright
    Fri., 1/22/10, Special Lecture, PH383, 3:00
    Frederic Rochon, University of Toronto
    TITLE:  A cohomological formula for the index of fibred cusp operators

    ABSTRACT:  Fibred cusp operators are defined on a non-compact manifold with asymptotic behavior specified by a fibration at infinity.  They arise naturally in various geometric contexts and in scattering theory.  For such operators to be Fredholm, ellipticity is not enough.  An invertibility condition at infinity must also be satisfied, making the computation of the index a subtle matter.  In this talk, after reviewing the basic properties and features of fibred cusp operators, we will describe how the index of such operators can naturally be obtained by integrating a certain cohomology class, effectively providing a generalization of the Atiyah-Singer index theorem to this context.
    Mon., 1/25/10, Special Lecture, PH332, 4:00
    Jeremy Marzuola, Columbia University
    TITLE:  Eigenfunction concentration for polygonal billiards

    ABSTRACT:  In this joint work with Andrew Hassell and Luc Hillairet, we extend the results on concentration of eigenfunctions of the Laplacian on billiards as proved earlier by the speaker. There, the geometric and control theoretic methods developed in the works of Burq-Zworski to study eigenfunctions for billiards with rectangular components were applied to study eigenfunction concentration on billiards with slits.  Here, we take an arbitrary polygonal billiard and show that eigenfunction mass cannot concentrate away from the vertices.
    Wed., 1/27/10, Special Lecture, PH332, 4:00
    Jeffrey Streets, Princeton University
    TITLE : Geometric evolution equations and four-dimensional geometry

    ABSTRACT: After giving some background on geometric evolution equations, I will discuss some unique aspects of four-dimensional geometry, and then introduce new equations aimed at exploiting and understanding these special structures. Specifically I will discuss a coupling of the Ricci and Yang-Mills flows, and a fourth-order parabolic equation arising as the gradient flow of a quadratic curvature functional.  Finally I will discuss a natural geometric flow on complex, possibly non-K\"ahler manifolds which generalizes the K\"ahler-Ricci flow, and the potential applications to understanding the topology of complex surfaces.
    Mon., 2/1/10, Special Lecture, PH332, 4:00
    Hans Christianson, MIT
    TITLE : The water-waves problem with surface tension.

    ABSTRACT:  In this talk, I will discuss recent work with V.M. Hur and G. Staffilani on the water-waves  problem with surface tension.  I will first introduce the problem, and give some history, and some ideas of the formulation we use.  I will explain  how this formulation allows the use of Fourier analysis techniques to gain some qualitative knowledge about the solution.  I will describe our main results, which are a family of mixed time-space $L^p L^q$ estimates called Strichartz estimates.
    Wed., 2/3/10, Special Lecture, PH332, 4:00
    Richard Oberlin, UCLA
    TITLE: Variations on the Kakeya problem

    ABSTRACT: A subset of R^d is called a Kakeya set if it contains a translate of every line segment of unit length. The Kakeya problem asks "how small (in terms of Hausdorff dimension) can Kakeya sets be?" We will discuss related questions. Includes joint work with Jordan Ellenberg and Terence Tao.
    Thurs. 2/4/10, Special Time/Date, 2:00, PH301
    Richard Aron, Kent State University
    TITLE: Cluster set theorems for H^\infty functions.

    Wed. 2/17/10
    John Pfaltzgraff, UNC
    TITLE: Loewner Theory and Schwarzians in C^n.

    ABSTRACT:  Loewner theory and Schwarzians in higher dimensions.
    Recent results generalizing the one variable Schwarzian derivative and Loewner differential equation theory to holomorphic mappings of the complex n-ball into n-space will be described. In higher dimensions one encounters a set of Schwarzians that are Moebius invariant. Loewner chains are constructed and used to derive “Schwarzian” univalence and quasiconformal extension criteria for holomorphic mappings of the n-ball into complex n-space. These generalize the classical single variable results of Nehari, Ahlfors, Becker, Pommerenke, C. Epstein and others.  The talk will begin with a summary of some single variable history and results to motivate the new several variables work.
    Wed., 2/24/10
    Galia Dafni, Concordia University
    TITLE: On a "classical" inequality and connections to Hardy spaces

    ABSTRACT: The inequality $$\|g -g_Q\|_2 \leq C\sum_{i=1}^n\Big\|\frac{ \partial g}{\partial
    x_i}\Big\|_{H^{-1}(Q)},$$ where $Q$ is a cube in $\mathbb{R}^n$, $g_Q$ is the mean of $g$ over $Q$, and $H^{-1}(Q)$ denotes the dual of the Sobolev space $H^1_0(Q)$, was used by Coifman, Lions, Meyer and Semmes in their proof of the decomposition of the real Hardy space
    $\mathcal{H}^1(\mathbb{R}^n)$ into ``div-curl" quantities.  It is a homogeneous special case of an inequality for Sobolev spaces on a  Lipschitz domain in $\mathbb{R}^n$, due to Ne\v{c}as.  We
    show how to use this inequality to obtain div-curl decompositions for local Hardy spaces
    on domains, as well as its connection to the problem of solving $\mbox{div } \vec  V =
    f$.  We discuss how this latter problem is related to a maximal characterization of Hardy-Sobolev spaces on Riemannian manifolds, which are also identified with the Haj{\l}asz Sobolev space.
    Wed., 3/3/10,
    Thesis Defense
    Nathan Pennington, UNC

    Wed., 3/24/10
    Michael Taylor, UNC
    TITLE: Introductory Stochastic Analysis

    ABSTRACT: We show how the Wiener process (Brownian motion) gives rise to a probability measure on path space and establish some basic results on continuity of the process.  We also
    define Wiener's stochastic integral and see how it works in some basic classes of stochastic differential equations. Familiarity with measure theory and some basic Hilbert space theory will provide sufficient prerequisites.  Some material on Brownian motion can be found on my web page (along with other material related to talks in the Analysis/PDE seminar).
    Wed. 3/31/10, Honors Thesis Defense
    Jacob Perry, UNC
    TITLE: Solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary

    ABSTRACT: I'll be discussing techniques for proving global existence for quasilinear wave equations in waveguides.  In particular, I'll highlight the development  of the localized energy estimate for the perturbed wave equation, and how it can be applied to solve quasilinear waves.
    Wed., 4/14/10
    Ivana Alexandrova, Eastern Carolina University
    TITLE:  Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation

    ABSTRACT: We consider the problem of quantum resonances in magnetic scattering by two solenoidal fields at large separation in two dimensions, and we study how a trajectory oscillating between the two fields gives rise to resonances near the real axis when the distance between the two centers of the fields goes to infinity. We give a sharp lower bound on resonance widths in terms of backward amplitudes calculated explicitly for scattering by each solenoidal field. The study is based on a new type of complex scaling method. As an application, we also discuss the relation to semiclassical resonances in scattering by two solenoidal fields.  This is joint work with Hideo Tamura.
    Wed., 4/21/10
    Robby Marangell, UNC
    TITLE: Applications of the Maslov index to stability in nonlinear waves

    ABSTRACT: In the first part of the talk I will outline the use of the Maslov index to determine the instability of standing wave solutions to Nonlinear Schrodinger equations with inhomogeneous potentials. The second part of the talk will focus on how the Maslov index can be used in determining the stability and instability of kink-wave solutions to the sine-Gordon equation. If time, I will try to discuss the applications for this work, as well as unifying both techniques under a single theory. 

    Fall 2009:

    Wed., 9/23/09
    Nathan Pennington, UNC
    TITLE:  Solutions to LANS equations in integrable in time spaces

    Mon., 9/28/09 (GMA Visions)
    Parul Laul, UNC
    TITLE:  Local Energy Estimates of the Wave Equation
    Wed., 9/30/09
    Nathan Pennington, UNC TITLE:  Solutions to LANS equations in integrable in time spaces (Continued)

    Wed., 10/7/09
    Michael Taylor, UNC
    TITLE:  Flat 2D tori with sparse spectra

    ABSTRACT: Flat 2D tori have as the spectra of their Laplace operators the negatives of square-lengths of elements of lattices in 2D Euclidean space.  One example is the 2D integer lattice.  In such a case, all the square length are integers, but most positive integers are not square lengths.  We have "sparse spectrum."  The talk will deal with the construction of other flat 2D tori with sparse spectra.

    There will be a little Fourier analysis, a little algebra, and a computer program.  Some time before the talk, Taylor's web page will have a pdf file on this topic, which can be downloaded.
    Wed., 10/14/09
    Michael Taylor, UNC
    TITLE: Linear and nonlinear waves on hyperbolic space

    ABSTRACT: We give a sequence of two talks.  We begin by producing formulas for solutions to the linear wave equation on 3D hyperbolic space.  These formulas are then used together with some harmonic analysis to establish "dispersive estimates" and "Strichartz estimates" on solutions to wave equations.  We then show how this leads to results on global existence for a class of nonlinear wave equations. This is a report on joint work with Jason Metcalfe.
    Wed., 10/21/09
    Michael Taylor, UNC
    TITLE: Linear and nonlinear waves on hyperbolic space

    ABSTRACT: We give a sequence of two talks.  We begin by producing formulas for solutions to the linear wave equation on 3D hyperbolic space.  These formulas are then used together with some harmonic analysis to establish "dispersive estimates" and "Strichartz estimates" on solutions to wave equations.  We then show how this leads to results on global existence for a class of nonlinear wave equations. This is a report on joint work with Jason Metcalfe.

    Wed., 11/4/09
    Joseph Cima, UNC
    TITLE: Introduction to H^1

    ABSTRACT: There will be a brief introduction to the analytic theory on the disc.Then we will define the "real" Fefferman and Stein Hardy space(s) on R^n. Much of this material is taken from Stein's book, Harmonic Analysis (Princeton Press- which I have checked out of the library!), Chapters 3,4.

    I will give a few of the shorter proofs and just   quote other pertinent results. Mainly how can one tell if an L^1 function can be in the Hardy 1 space, definitions of atoms, definition of  BMO ,the topological dual etc.

    This talk is suitable for graduate students.  A second lecture will be given two weeks later.
    Wed., 11/11/09
    Ian Zwiers, University of Toronto
    TITLE: Blowup of the Cubic Focusing Nonlinear Schrodinger Equation on a Ring

    ABSTRACT: We prove there exist solutions to the three-dimensional cubic focusing nonlinear Schrodinger equation that blowup on a circle, in the sense of L2 concentration on a ring, bounded H1 norm outside any surrounding toroid, and growth of the global H1 norm with the log-log rate.
    In this talk we will described the bootstrapping scheme of the proof. Namely that, for sufficient decay outside a surrounding toroid, the dynamic near the circle of concentration is essentially two-dimensional. Then, for appropriate data, the robust two-dimensional log-log blowup regime occurs. Merle & Raphael's precise description of this singular behaviour allows us to prove an unusual persistance of regularity away from the circle of concentration, which re-establishes the sufficient decay outside the surrounding toroid.

    Wed., 11/18/09
    Joseph Cima, UNC
    TITLE: Introduction to H^1

    ABSTRACT:  A proof of a homogeneous "div-curl" lemma and again how  can one get functions in the Hardy 1 space. (Paper  of Meyer, Semmes, Coiffman, See a chapter in M.Taylor's  book on tools used in PDE.

    If time allows I will give a proof of   a result of Jones , Zinsmeister, Iwaniec,  Bonami and an interesting bounded operator mapping BMO into a space called the "Hardy-Orliz" space.
    Wed., 12/2/09
    Jeff Jauregui, Duke University

    Spring 2009:

    Wed., 1/28/09
    Benjamin Dodson, UNC
    TITLE: Global existence for some nonlinear Schrodinger equations

    Wed., 2/4/09
    Benjamin Dodson, UNC
    TITLE: Global existence for some nonlinear Schrodinger equations (Continued)
    Fri. 1/30/09 through Sun. 2/1/09
    Carolina Meeting on Harmonic Analysis and PDE
    Wed., 2/25/09
    Monica Visan, University of Chicago
    TITLE: Nonlinear Schrodinger equations at critical regularity

    ABSTRACT: We introduce the nonlinear Schrodinger equation (NLS) and define criticality.  We then survey the history of the two most studied NLS at critical regularity, namely, the mass- and energy-critical NLS.  We review some of the techniques developed to solve these problems and describe recent progress on the energy-critical and energy-supercritical NLS.  This includes joint work with Rowan Killip, Terence Tao, and Xiaoyi Zhang.
    Wed., 3/4/09
    John Anderson, Holy Cross
    TITLE: $H^1$-BMO duality and Hardy-Orlicz spaces

    ABSTRACT: In 1971 Charles Fefferman identified the dual of the Hardy space $H^1$ with the space BMO of functions of bounded mean oscillation.  The dual pairing is somewhat subtle, because the product of an $H^1$ function and a BMO function need not be integrable.  In a recent paper Bonami, Iwaniec, Jones and Zinsmeister (BIJZ) identify the product of an $H^1$ function and an analytic BMO function (acting as a distribution) with a function in a certain Hardy-Orlicz space.  I will give an exposition of this and other results in the BIJZ paper.
    Wed., 3/25/09
    Benjamin Dodson, UNC
    Dissertation Defense

    TITLE: The Pinsky phenomenon and indefinite signature Schrodinger equation

    ABSTRACT: The thesis examines the focusing phenomenon and Gibbs phenomenon for the linear Schrodinger equation of indefinite signature.  Local and global existence results for the nonlinear equation are also obtained.
    Thurs., 3/26/09 (Colloquium)
    Steve Hofmann, University of Missouri
    TITLE: Local $Tb$ Theorems and Applications in PDE

    ABSTRACT: Singular integral operators ("SIOs") arise often in complex analysis and in the theory of partial differential equations.  For example, the prototypical SIOs are the Hilbert Transform, which relates the real and imaginary parts of the boundary values of an analytic function in the upper half plane and, in higher dimensions, the Riesz Transforms, which relate the normal and tangential components of the boundary values of the gradient of a harmonic function in the half space.  The terminology "singular integral" refers to the fact that the Schwarz kernels of these integral operators possess a singularity which just fails to be integrable, and therefore the operators must be defined in some limiting or "principal value" ("p.v.") sense.  The aforementioned prototypical SIOs are of convolution type, so their boundedness on L^2 (which is the funadmental desired property of such operators) may be verified via Plancherel's Theorem (that is, one may exploit the fact that convolution operators are "diagonalized" by the Fourier transform).

    On the other hand, there are many other important examples of SIOs, arising, e.g., in the theory of variable coefficient elliptic PDE, and in the theory of analytic functions in domains with non-smooth boundaries, which are not of convolution type, and for these, Plancherel's Theorem is not available as a tool to establish L^2 boundedness.  In this talk, we shall present a survey of progress on the development of criteria to verify the L^2 boundedness of non-convolution SIOs, and we shall discuss some applications.

    Wed., 4/15/09
    Yuri Latushkin, University of Missouri TITLE: An index theorem, the spectral flow, and the spectral shift function for relatively trace class perturbations

    ABSTRACT: This is a joint work with Fritz Gesztesy, Konstantin Makarov, Fedor Sukochev, and Yuri Tomilov. Under relatively trace class assumptions, we calculate the Fredholm index of the operator d/dt+A(t) via Krein's spectral shift function of the operators A(+infinity) and A(-infinity). The main technical result is that the difference of the corresponding Morse projections is of trace class; this is proved using double operator integrals.
    Wed., 4/22/09
    Walter Strauss, Brown University
    TITLE: Stability Criteria in a Collisionless Plasma

    ABSTRACT: Stability of a state in a physical system refers to the asymptotic behavior of the nearby states. For a collisionless plasma (solar wind, hot fusion,...) that is modeled by the relativistic Vlasov-Maxwell system, many equilibria are stable but many others are unstable.  In this talk, presenting joint work with Zhiwu Lin, I will consider axisymmetric equilibria of the form f(e, p) that are decreasing in the particle energy e and also depend on the particle angular momentum p. Then a necessary and sufficient condition for linear stability is the positivity of a certain linear operator L^0. This operator L^0 is much less complicated than the generator of the full linearized system. It has an interesting non-local term that can definitely affect its positivity.  There is a similar reduction in the simpler case of 1.5 dimensional symmetry. For the important example of a purely magnetic equilibrium, explicit conditions for the linear/nonlinear stability/instability are obtained.
    Thurs., 4/23/09 (Colloquium)
    Walter Strauss, Brown University
    TITLE: Steady Rotational Water Waves

    ABSTRACT: Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,...)  with an arbitrary vorticity function. Consider such a wave traveling at a constant speed over a flat bed.  Using local and global bifurcation theory and topological degree, one can prove that there exist many such waves of large amplitude.  I will outline the existence proof, joint with Adrian Constantin, and also exhibit some recent computations, joint with Joy Ko, of the waves using numerical continuation.  The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid.

    Fall 2008:

    Wed., 9/3/08
    Benjamin Dodson, UNC
    TITLE:  Nonlinear Perturbations of the Minkowski Schrodinger equation

    Wed., 9/17/08
    Becca Thomases, UC Davis
    TTILE:  Mixing Transitions and Oscillations in Low-Reynolds Number Viscoelastic Fluids

    ABSTRACT:  In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow.  For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing.  The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics.  To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations.  Next we study this and related models numerically for low-Reynolds number flows in two dimensions.  For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state.  The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices.
    Wed., 9/24/08
    Nathan Pennington, UNC
    TITLE: Derivation of the Lagrangian averaged Navier-Stokes equation and local existence with small initial data

    Wed., 10/22/08
    Mihai Tohaneanu, UC Berkeley
    TITLE:  Local energy decay on Schwarzchild and Kerr backgrounds

    ABSTRACT:  Understanding the decay of linear waves is crucial in dealing with the problem of stability of the Kerr space time. I will talk about one way to measure this decay, namely local energy estimates, from which one can deduce many other useful estimates (uniform energy bounds, pointwise bounds, Strichartz estimates etc). This is joint work with Jeremy Marzuola, Jason Metcalfe and Daniel Tataru (for Schwarzschild) and Daniel Tataru (for Kerr).
    Wed., 10/29/08
    Paul Kessenich, University of Michigan
    TITLE:  Global Existence for a 3D Incompressible Isotropic Viscoelastic Material

    ABSTRACT:   Incompressible viscoelastic materials can be studied using Oldroyd-B model which views the material as an elastic polymer immersed in a Newtonian fluid.  Previous global existence results for this model have used parabolic methods in which the size of the initial disturbance is inevitably dependent on the Newtonian viscosity in the Oldroyd-B equations.  This talk will focus on the use of hyperbolic methods to prove global existence for a slightly more general set of equations so that the smallness of the initial data is not restricted by the viscosity parameter.  As the viscosity goes to zero, a previously proven global existence result for incompressible isotropic elastodynamics can be recovered.
    Wed., 11/5/08
    Yaguang Wang, Northwestern
    TITLE:  Zero Viscosity Limit for Incompressible Navier-Stokes Equations with Navier Boundary Conditions

    ABSTRACT:  In this talk, we shall study the zero viscosity limit and behavior of boundary layers in a viscous incompressible flow with the Navier-friction boundary condition. For different relation of the slip lengths to the viscosity, we obtain several criteria on the convergence from the velocity of viscous fluids to that of in-viscous fluids in energy space when the viscosity tends to zero. The asymptotic behavior of boundary layers is derived by using multi-scale analysis. This is a joint work with Zhouping Xin.
    Wed., 11/12/08
    Tadeusz Iwaniec, Syracuse University TITLE: Estimates of Jacobian determinates - Hardy & Littlewood meet with Fefferman & Stein in maximal inequalities

    ABSTRACT: This lecture features new types of maximal functions and commutators of singular integrals, linear and nonlinear.  The topics are motivated by the study of nonlinear differential expresssions which arise naturally in geometric function theory, nonconvex calculus of variations and nonlinear elasticity.  In particular, estimates of the Jacobian determinant in the Hardy space by means of subdeterminants involves an interpolation between maximal averages over the balls (Hardy maximal function) and spherical maximal averages (Fefferman-Stein maximal operator).
    Thurs., 11/13/08 (Colloquium), 4:00, Phillips 332
    Tadeusz Iwaniec, Syracuse University TITLE:  New Prospects of Quaiconformal Geometry, an Invitation to n-Harmonic Hyperelasticity

    ABSTRACT:  Quasiconformal geometry and nonlinear elasticity theory share common problems of compelling mathematical interest.  Both theories are governed by specific energy integrals together with their associated Lagrange-Euler equations.  The mappings of particular interest (elastic deformations) are the ones with smallest energy.  In this talk I will present our latest advances and the results concerning existence, global injectivity (the principle of non-penetration of matter) and boundary behavior of deformations of smallest n-harmonic energy, a prototype of the planer Dirichlet problem.  This brings us to a polyconvex variational integral for the inverse deformations.  Distinctly, the inverse deformations are the mappings that minimize the L1 - average of the distortion function, among all homeomorphisms between two domains (very much reminiscent of the Teichmuller extremal problems).  Then the total energy of a mapping and its inverse becomes of sufficient interest to call for closer examination.  The great virtue of all our energy integrals is that they are invariant under conformal change of variables.  For this reason we call this theory Quasiconformal Hyperelasticity.  Our approach offers significantly larger class of mappings than the quasiconformal geometry.  Interplay between nonlinear analysis and topology is critical in our approach.  In particular, the underlying integration of rather special nonlinear differential expresssions (these are the free Lagrangians defined on homeomorphisms in a given homotopy class) becomes truly a work of art.

    Report is joint with Jani Onninen.  This talk will be accessible to graduate students in analysis.
    Wed., 11/19/08
    Nick Costanzino, Penn State University TITLE:  Front propagation is discontinuous heterogeneous media

    ABSTRACT:  We investigate the behaviour of multidimensional pulsating fronts in discontinuous heterogeneous media. In addition to extending the existence result of Berestycki-Hamel-Roque to the case of discontinuous coefficients, we show the dependence of the front speed on both bulk and local properties of the medium.  This is joint work with A. Novikov (Penn State) and L. Heltai (SISSA).

    Spring 2008:

    Wed., 1/16/08
    Marius Mitrea, University of Missouri
    TITLE:  Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains

    ABSTRACT:  We establish the well-posedness of the inhomogeneous Dirichlet problem for the bi-Laplacian  in arbitrary three-dimensional Lipschitz domains, with data from Besov-Triebel-Lizorkin spaces, for the optimal range of indices (smoothness and integrability). The main novel contribution is to allow for certain non-locally convex spaces to be considered, and to establish integralrepresentations for the solution.
    Thurs., 1/17/08
    4:00pm, Phillips 332
    Marius Mitrea, University of Missouri
    TITLE: Singular Integral Operators and Boundary Value Problems under Sharp Geometric Measure Theoretic Assumptions

    ABSTRACT:  It has long been recognized that there are subtle connections between the boundedness  of singular integral operators (SIO) and the geometric measure-theoretic properties of sets. A fundamental result in this direction is the boundedness of SIO with reasonable kernels on uniformly rectifiable surfaces. These are Ahlfors regular surfaces (i.e., behave like a (n-1)-dimensional object at all scales), and contain ``big pieces of Lipschitz images of (n-1)-dimensional sets'' in a uniform fashion. This earlier work has interfaced tightly with geometric measure theory, but until now it has not been systematically applied to problems in PDE.

    The aim of this talk is to explore the role that SIO may play in the treatment of boundary  value problems under sharp geometric measure theoretic assumptions on the underlying domain. In particular, I will describe some recent joint work with S. Hofmann and M. Taylor in which we forge new links between the analysis of SIO on uniformly rectifiable surfaces and problems in PDE, most notably boundary problems for the Laplace operator and other second order elliptic operators, including systems.
    Wed., 1/23/08
    Benjamin Dodson, UNC
    TTILE:  Pinsky Phenomena and the Minkowski Schrodinger Equation

    ABSTRACT:  In $\mathbf{R}^{n}$, the linear Schrodinger equation $\frac{\partial u}{\partial t} - i \Delta u = 0$, $u(0,x) = \chi_{B(0;1)}$ exhibits some interesting behavior. But now change the metric on $\mathbf{R}^{n}$ to a more general metric of signature (p,q). The equation $\frac{\partial u}{\partial t} - i \Delta_{p,q} u = 0$ has significantly altered convergence phenomena. In particular there is a generalization of the Pinsky phenomenon.
    Wed., 2/13/08
    Nathan Pennington, UNC
    TITLE: The Lagrangian averaged Navier-Stokes equation with small initial data

    ABSTRACT: The Lagrangian averaged Navier Stokes equation is a PDE that models the averaged motion of an ideal incompressible fluid filtering over spatial scales smaller than $\alpha$.  The equation is used to reproduce the large-scale averaged motion of the Navier-Stokes equations without the use of artificial viscosity or dissipation.  We will briefly discuss the derivation of the PDE and will show the existence of a short time solution $u$ to the Lagrangian averaged Navier Stokes equation $\partial_t u + (u\,\cdot\,\nabla)u+\text{div}\,\tau^\alpha = -\text{grad}\,p+\nu\Delta u$ with initial data $u(0)=u_0$ in the homogeneous Lebesque space $\dot{L}_{s,p}(\mathbb{R}^m)$ for $p\ge m$ and for $s>0$.
    Wed., 2/20/08

    Thurs., 2/28/08 (Special day), Time: 2-3
    Sarah Raynor, Wake Forest University
    TITLE: Neumann Fixed Boundary Regularity for an Elliptic Free Boundary Problem

    ABSTRACT: We examine the regularity properties of solutions to an elliptic free boundary problem near a Neumann fixed boundary.  Consider a nonnegative function u, defined variationally, which is harmonic where it is positive and satisfies a gradient jump condition weakly along the free boundary (the boundary of the set where u is positive).  Our main result is that u is Lipschitz continuous.  Additionally, we prove various basic properties of such a minimizer near a portion of the fixed boundary on which Neumann conditions hold weakly. Our results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains, which have independent interest.
    Wed., 3/19/08
    Yaguang Wang, Shanghai Jiaotong University and Northwestern University TITLE:   Existence and Stability of Compressible Current-Vortex Sheets in 3-D MHD

    Thurs., 3/20/08
    4:00pm, Phillips 332
    Christopher Sogge, Johns Hopkins University
    TITLE: How Focusing and Dispersion Influence the Solution of Certain Linear and Nonllinear Equations

    ABSTRACT: In this survey, I shall present several estimates whose strength depends on whether certain types of dispersion is present and certain types of focusing is absent.  These include size estimates for eigen functions, and Strichartz estimates both in manifolds without boundary and manifolds with boundary.  I shall also go over some applications to the theory of nonlinear hyperbolic equations.
    Wed., 3/26/08

    Thurs., 3/27/08
    4:00pm, Phillips 332
    Victor Guillemin, MIT
    TITLE: Classical and quantum Birkhoff canonical forms in one dimension
    Wed., 4/2/08
    Ramona Anton, Université Paris Sud, XI and Johns Hopkins University TITLE:  Non-linear Schr\"odinger equations on domains with boundary

    ABSTRACT:  We are interested in proving global existence results in the energy space for the semi-linear Schr\"odinger equation on domains of dimension 2 or 3. The main ingredients are generalized Strichartz inequalities adapted to the domains, which have some loss of derivatives. We present the results and the strategy for three types of domains.
    Wed., 4/16/08
    Michael Goldberg, Johns Hopkins University
    TITLE: The Schr\"odinger Equation with a Non-Smooth Magnetic Potential

    ABSTRACT: We prove Strichartz estimates for the absolutely continuous evolution of a Schr\"odinger operator $H = (i\nabla + A)^2 + V$ in ${\bf R}^n$, $n \ge 3$.  Both the magnetic and electric potentials are time-independent and have polynomial pointwise decay.  The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity.  This condition improves upon previous results that require half a derivative of smoothness or more.
    Wed., 4/23/08
    Gigliola Staffilani, MIT TITLE: Weak turbulence for periodic 2D NLS
    Thurs., 4/24/08
    4:00pm, Phillips 332
    Gigliola Staffilani, MIT
    TITLE: "The nonlinear Schrodinger Equations: the old and the new"

    4/28 - 4/30/08

    Brauer Lectures
    Charles Fefferman, Princeton University
    TITLE: Interpolation and extension of functions

    More info

    Fall 2007:

    Wed., 10/10/07
    Michael Shearer, NCSU
    TITLE:   Particle-size segregation in granular flow: a conservation law in two space dimensions.

    ABSTRACT:  Kinetic sieving is the process by which large particles rise in granular avalanches, while smaller particles fall. Recent models of this effect reduce to a scalar conservation law in two dimensions and time, but with non-constant coefficients, reflecting the shear needed to induce segregation.  Various topics of significance to applications are considered using the theory and constructions of scalar hyperbolic equations: steady solutions in which the direction of flow is time-like, leading to a sharp estimate of how long a chute should be to guarantee full segregation; breaking of interfaces, forming an evolving lens-shaped mixture zone; and the connection to recent experiments of Daniels on shear flow, for which the model is adjusted to account for nonuniform shear, with the consequent loss of constant solutions.
    Wed., 10/24/07
    Xiao-Biao Lin, NCSU
    TITLE: Gearhart-Pr¨uss Theorem and linear stability for Riemann solutions of conservation laws

    ABSTRACT: We first review the Hille-Yosida Theory, Paley-Wiener Theorem and Gearhart-Pr¨uss Theorem on the asymptotic behavior of semigroups. We then consider the spectral and linear stability of of the Riemann solutions with multiple Lax shocks for systems of conservation laws u_T  + f(u)_X= 0. Using the self-similar change of
    variables x = X/T , t = ln(T), Riemann solutions become stationary to the system u_t +(Df(u)-xI)u_x = 0. In the
    space of O((1 + |x|)^(-m)) functions, we show that if the real part of \lambda is greater than -m, then  \lambda is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, there are resonance values where the determinant can be arbitrarily small but nonzero. A C_0 semigroup is constructed and using the Gearhart-Pr¨uss Theorem, we show that the solutions are of O(e^{\gamma t}) if \gamma is greater than the largest real parts of the eigenvalues and the resonance values.  We present examples where Riemann solutions have two or three Lax-shocks. We will
    discuss how the linear stability can be used to determine the nonlinear stability of Riemann solution with shocks.
    Mon., 10/29/07
    (GMA Visions)
    4:00pm, Phillips 367
    Jason Metcalfe, UNC
    TITLE: Local energy estimates for wave equations on Schwarzschild black hole backgrounds

    ABSTRACT: One measure of dispersion for wave equations which is known to be fairly robust is the localized energy estimates.  We will briefly discuss these estimates in Minkowski space-time (i.e. flat space).  We shall then discuss the wave equation on Schwarzschild black hole backgrounds and some recent attempts at proving analogous
    estimates in this setting.  At the end, some related open problems will be outlined.
    Wed., 10/31/07
    Mark Williams, UNC
    TITLE: ODEs with fast transitions and spherical viscous shocks.

    ABSTRACT:  We discuss a new technique for constructing solutions to two-point ODEs that exhibit fast transitions in the interior of the domain.  The technique is applied to the construction of spherical viscous shocks.
    Wed., 11/7/07
    Michael Lacey, Georgia Tech.
    TITLE: Irregularities of Distribution and Related Questions

    ABSTRACT:  The subject of Irregularities of Distribution concerns identification of optimal rates of convergence to uniform distribution. This classical topic is mostly understood in average case analysis. Certain endpoint estimates remain stubbornly resistant, despite decades of research on the topic. These questions are in turn related to arising in Harmonic Analysis, Probability Theory, and Approximation Theory. Given $ N$ points $ P_N$ in the unit cube, define the Discrepancy Function by $$ D_N (x) = |P_N \cap [0,x)| - N |[0,x)| $$ where $x$ is the rectangle with antipodal verticies at the origin and at $x$ in the unit cube. The importance of this function is highlighted by the classical Koksma-Hlawka inequality. We describe the average case analysis of $ D_N$, namely a universal lower bound on the $ L^2 $ norm of $ D_N$ due to Klaus Roth.  The endpoint lower bounds for $ D_N$ remain a mystery in dimensions three and higher.  We describe a new result, joint with Dmitriy Bilyk and Armen Vagharshakyan, on  the $ L ^{\infty}$  of the Discrepancy function, in all dimensions three and higher.
    Wed., 11/28/07
    Jeremy Marzuola, Columbia University
    TITLE:  Wave packet parametrices for evolution equations governed by PDO's with rough symbols

    ABSTRACT:  We prove existence of solutions to a generic class of dispersive equations under certain integrability conditions along the Hamilton flow of the leading order pseudodifferential operator governing the evolution of the solution.
    Wed., 12/5/07
    Jason Metcalfe, UNC
    TITLE: On 4D nonlinear wave equations in exterior domains

    ABSTRACT:  This talk is on a recent joint work with Y. Du, C. Sogge, and Y. Zhou.  Here, we discuss two results concerning 4D wave equations in exterior domains.  The first is a proof of an exterior domain analog of the 4D Strauss conjecture.  The second is an exterior domain analog of a result of Hormander concerning almost global existence for quasilinear wave equations with nonlinear dependence on the solution not just its derivatives.  The key to the proof is a combination of certain localized energy estimates with a certain Hardy-type inequality.

    For questions, concerns, or to recommend a speaker, please contact Hans Christianson or Jeremy Marzuola.

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