## Department of Mathematics, University of Kentucky Analysis and PDE Seminar

Spring 2011

Partially supported by the Enrichment Fund of the College of Arts and Sciences and by the Department of Mathematics.

Date SPEAKER, TITLE and ABSTRACT Host
January 18
No Seminar Scheduled

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January 25 Professor David Adams, University of Kentucky
TITLE: Calderon-Zygmund operators and Wolff potentials
ABSTRACT: This is a preliminary report of work done recently by my self and V. Eiderman (who was here on our faculty last year 2009-2010, but who now resides at the Univ. Wisconsin, Madison). It involves a small extension of the potential theory associated with certain capacities of the Calderon-Zygmumd type for signed-Riesz operators on non-homogeneous spaces. As an application, we extend the known relations between s-Riesz capacities 0 < s < d = dimension of the underlining Euclidean space, and the capacities in Nonlinear Potential Theory, to the case s = 0.
Katy Ott
February 1 Dr. Leo Marazzi, University of Kentucky
TITLE: Absence of resonances near critical line for CC manifolds
ABSTRACT: We find a resonance free region polynomially close to the critical line on conformally compact manifolds with polyhomogeneous metric.
Peter Hislop
February 8 Professor Ciprian Demeter, Indiana University Bloomington
CANCELED DUE TO WEATHER
TITLE: Progress on the HRT conjecture
ABSTRACT: The HRT conjecture asserts that the time-frequency translates of any nontrivial function in L^2(R) are linearly independent. Prior to our work, the only result on HRT of a reasonably general nature was Linnell's proof in the case when the translates belong to a lattice. I will briefly describe an alternative argument to Linnell's (joint work with Zubin Gautam), inspired by the theory of random Schrodinger operators. Then I will explore a number theoretical approach to the HRT conjecture, for some special 4 point configurations.
Katy Ott
February 15 Dr. Vita Borovyk, University of Cincinnati
TITLE: Dispersive Estimates in Classical and Quantum Harmonic Lattice Systems
ABSTRACT: We consider infinite-volume classical (and quantum) harmonic lattice systems and study the decay of certain time-evolved observables in the large-time regime. The decay rate depends on the supports of observables and the lattice space dimension.
Peter Perry
February 22 Professor Manuel Maestre, University of Valencia and Kent State University
TITLE: Recent developments on Bohr Radii
ABSTRACT: PDF file
Larry Harris
February 24 (Thursday) Special Seminar, Thursday, February 24 at 11:00 a.m.
Professor Vasily Vasyunin, Steklov Institute and the University of Cincinnati
TITLE: How the Monge--Amp\ere equation helps in proving the John—Nirenberg inequality
ABSTRACT: I would like to present, using the famous John--Nirenberg inequality as an example, a rather new technique of obtaining integral estimates in analysis, the so-called Bellman function method. Originated in the stochastic control theory, it has become a hands-on and effective tool in proving various functional inequalities. The estimate being proved is recast as the solution of an extremal problem. Solving such problems exactly -- a step that often depends on solving Monge--Amp\ere equations on problem-specific Euclidean domains -- produces inequalities with sharp constants. The presentation will not require any special knowledge; in particular, one does not need to know what the John--Nirenberg inequality is. The main prerequisite is the Lebesgue integration theory.
Jim Brennan
February 24 (Thursday) van Winter Memorial Lecture
Thursday, February 24 at 4:00 pm in Chemistry-Physics Building, Room 155
Professor Gunther Uhlmann , University of California Irvine
TITLE: Cloaking and Transformation Optics
ABSTRACT: We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. Maxwell’s equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount some of the history of the subject and discuss some of the issues involved. Co-sponsored by the Departments of Mathematics and Physics and Astronomy and the van Winter Memorial Endowment.
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February 25 (Friday) Special Seminar, Friday, February 25 at 4:00 p.m., POT 745
Professor Gunther Uhlmann , University of California Irvine
TITLE: Inverse Problems in Geometry
Peter Hislop
March 1 Mr. Ryan Walker, University of Kentucky
TITLE: Domains, Ranges and Uniqueness in Photoacoustic Tomography
ABSTRACT: Photoacoustic tomography (PAT) is an emerging modality of medical imaging which produces high resolution, high contrast tomographic images without exposing patients to dangerous ionizing radiation. In a PAT scan, laser pulses are fired at a segment of the patient's body. The tissue in the segment absorbs some of this pulsed energy causing a rapid expansion and contraction of tissue that propagates a pressure wave through the segment. Detectors arranged around the patient measure this pressure wave and, because energy absorption depends on material composition, it is possible to construct an image of the internal structure of the segment. In this talk, I will present a wave equation model for PAT and exhibit a reconstruction formula in terms of a spherical mean operator. I will then investigate some interesting aspects of the geometry, domain, and range of the spherical mean operator which are relevant to the implementation of PAT.
Russell Brown
March 8 Professor Samangi Munasinghe, Western Kentucky University
TITLE: Geometric sufficient conditions for compactness of the $\overline{partial}$-Neumann and complex Green operators
ABSTRACT: The $\overline{partial}$-Neumann and complex Green operators are inverse operators of the complex Laplacian and the Kohn Laplacian respectively. In this talk we will discuss geometric sufficient conditions for compactness of the two operators. These conditions are formulated in terms of certain short time flows from weakly pseudoconvex points to strongly pseduoconvex points. This is joint work with Emil J. Straube.
Russell Brown
March 15
NO SEMINAR SCHEDULED -- Spring Break

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March 22
NO SEMINAR SCHEDULED

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March 29 Professor Antonio Sa Barreto, Purdue University
TITLE: The asymptotic behavior of the wave equation on hyperbolic manifolds.
ABSTRACT: We study the asymptotic behavior of the wave equation on hyperbolic manifolds.
Leo Marazzi
April 5 Professor Robert Buckingham , University of Cincinnati
TITLE: Dynamics of the semiclassical sine-Gordon equation
ABSTRACT: The small-dispersion or semiclassical sine-Gordon equation models magnetic flux propagation in long Josephson junctions. For a broad class of pure impulse initial data, we show the leading-order solution is given by modulated elliptic functions describing superluminal kink trains and breather trains at small times. This justifies the formal results obtained from Whitham averaging. We also describe the solution at transition points between the kink and breather trains in terms of Painleve functions. This is joint work with Peter Miller.
Peter Perry
April 12 Professor Fritz Gesztesy, University of Missouri
TITLE: Spectral Theory and Weyl Asymptotics for Perturbed Krein Laplacians
ABSTRACT: We study spectral properties for the Krein-von Neumann extension of the perturbed Laplacian $-\Delta+V$ defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a certain class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In this context we establish the Weyl asymptotic formula for the non-zero eigenvalues of the Krein-von Neumann extension, listed in increasing order according to their multiplicities. We prove this formula by showing that the pertrubed Krein Laplacian is spectrally equivalent to the buckling of a clamped plate problem. If time permits, we also intend to show the unitary equivalence of the inverse of the Krein-von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator in a Hilbert space to an abstract buckling problem operator. This establishes the Krein-von Neumann extension as a natural object in elasticity theory. This is based on various collaborations with Mark Ashbaugh (University of Missouri), Marius Mitrea (University of Missouris), Roman Shterenberg (University of Alabama at Birmingham), and Gerald Teschl (University of Vienna, Austria).
Peter Hislop
April 19 Professor Luis Vega, Universidad del País Vasco
TITLE: A new approach to Hardy uncertainty principle
ABSTRACT: In the lecture I will give a new proof, done in collaboration with L. Escauriaza, C. Kenig, and G. Ponce, of the well known result of G.H. Hardy about the rigidity on the simultaneous decay of a function and its Fourier transfom. This rigidity is a consequence of the uncertainty principle. In the case of Hardy the decay is measured on terms of gaussians and his proof uses in a fundamental way the properties of entire functions. Our approach is based on log-convexity properties of solutions of Schrodinger equations that have a gaussian decay. This gives us some flexibility, so that we are able to extend Hardy's result to some other settings where the use of the theory of analytic functions doesn't seem to be useful.
Russell Brown
April 21 (Thursday) Professor Alexei Rybkin , University of Alaska
TITLE: TBA
ABSTRACT: TBA
Peter Perry
April 21 (Thursday) Hayden-Howard Lecture
Professor Luis Vega, Universidad del País Vasco
TITLE: Vortex filaments and some dispersive geometric partial differential equations.
ABSTRACT: I will present work about self-similar solutions of the binormal flow. This is a flow in three dimensions of curves that move in the direction of their binormal with a speed proportional to their curvature. The flow was obtained in 1906 by Da Rios as an approximation of the evolution of a vortex filament. I will first justify the validity and limitations of this approximation to the Euler equations. Then, I will characterize the self-similar solutions (joint work with S. Gutiérrez) from a geometric point of view. Finally I will give some results obtained with V. Banica about the stability of these solutions. These results are based on the connection of this equation with the one-dimensional cubic non-linear Schrödinger equation. As a byproduct of our analysis we obtain some new scenarios of dispersive breakdown.
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April 26 Dr. Liang Song, Zhongshan University and University of Kentucky
TITLE: VMO spaces associated with operators
ABSTRACT: Classical Hardy spaces, BMO spaces and VMO spaces are associated with the Laplacian operator. Recently, motivated by the Kato conjecture, people begin to study spaces (such as Hardy and BMO spaces) associated with two kinds of operators. In this lecture, we will introduce VMO spaces associated with two kinds of operators. Firstly, let $L$ be the infinitesimal generator of an analytic semingroup on $L^2(R^n)$ with suitable upper bounds on its heat kernels, and $L$ has a bounded holomorphic functional calculus on $L^2(R^n)$. We introduce and develop a new function space VMO_L associated with the operator $L$. We also prove that a Hardy space $H^1_L$, which introduced by Auscher, Duong and McIntosh, is the dual of our new $VMO_{L^\ast}$, in with $L^{\ast}$ is the adjoint operator of $L$. Secondly, consider the second order diverge form elliptic operator $L$ with complex bounded coefficients. We also introduce a function space $VMO_L$ associated with $L$. Moreover, we prove that the Hardy space $H^1_L$, introduced by Hofmann and Mayboroda, is the dual of our $VMO_{L^{\ast}}$, in which $L^{\ast}$ is the adjoint operator of $L$. The main tools are theory of tent spaces, functional calculus and Gaffney estimates.
Zhongwei Shen
May 3 Ms. Megan Gier, University of Kentucky
Master's Exam
10:00 a.m., POT 745
TITLE: Cases of Equality in the Riesz Rearrangement Inequality
Peter Hislop
May 3 Professor Shijing Ding, School of Mathematical Sciences, South China Normal
2:00 -- 3:00 p.m., POT 745
TITLE: Incompressible Limit of the Compressible Hydrodynamic Flow of Liquid Crystals
ABSTRACT: This talk is concerned with the incompressible limit of the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions in $R^N, (N=2,3)$. We prove the local (and global) strong solution of the compressible system converges to the local (and global) strong solution of the incompressible system. Moreover, we get the convergence rates in some sense.
Changyou Wang
May 6
(Friday)
Mr. Murat Akman, University of Kentucky
Master's Exam
11:00 a.m., POT 745
TITLE: John Lewis's Proof of Little Picard's Theorem and Rickman's Theorem
Peter Hislop