FPAS-2001-10-19

First Prairie Analysis Seminar
October 19-20, 2001 

Kansas State University - Manhattan, Kansas

 

Featured Speaker: 

 Prof. Fang-Hua Lin,  Courant Institute, NYU. 

   "These two lectures will address some analytical and topological issues concerning Sobolev mappings between  
 manifolds.We shall discuss local and global topological obstructions for smooth maps to be strongly dense or to  
 be weakly sequentially dense in Sobolev mapping spaces,and the topology of Sobolev mappings.We shall also  
 outline a generalized varifold theory for Sobolev mappings and their applications in variational problems." 
 

 


 Invited Speakers: 

 Prof. Robert Hardt, Rice University 

  Size Minimization and Approximating Problems 

(Thierry De Pauw (Orsay) and Robert Hardt (Rice))

"A  k  dimensional rectifiable current is given by an oriented  k  
 dimensional rectifiable set  M  together with a positive integer-valued density function  D  .  The  mass  of the  
 current is then simply the integral of  D  over  M  (with respect to  k  dimensional Hausdorff measure).  In 1960  
 Federer and Fleming proved the existence of a rectifiable current of least mass for a given boundary.  For  q  in  
 [0,1] ,   the  q-mass of the current is the integral of  D^q  over  M .  
 The case  q = 0  corresponds to  size , introduced by Almgren as a way of using currents to model soap films.  
 We will discuss the existence of a rectifiable current of least  q-mass for a given boundary.  For that purpose we  
 make use of  scans  which are certain functions arising as limits of slices of rectifiable currents." 
 

 Prof. Yisong Yang , Poltytechnic Unversity and IAS (Princeton):

   Nonlinear Problems in Born-Infeld Theory

   "In 1933, M. Born and L. Infeld developed a geometric theory of electromagnetism to accommodate a finite-energy  
 point electric charge modeling the electron. Recently, this theory has become one of the major focuses of theoretical  
 physicists due to its relevance in superstrings and supermembranes. Mathematically, the Born-Infeld theory presents  
 new challenges to analysts. In this talk, we study the Bernstein problems for minimal surface equations, the existence  
 of static Klein-Gordon waves in arbitrary dimensions, and the existence of cosmic strings, in the framework of the  
 Born-Infeld theory."


Schedule

Friday afternoon: 

One 1 hour lecture by Fang-Hua Lin, one 1 hour invited lecture, and several 25 minute contributed talks.

Saturday:

Morning: One 1 hour invited lecture and several 25 minute contributed talks.

Afternoon: One 1 hour lecture by Fang-Hua Lin and several 25 minute contributed talks. 

All talks will be in Cardwell Hall, room to be announced. 

Contributed Speakers and Titles of their talks: 

David Auckly (Kansas State University) 
Title: A framework for analysis on noncompact quasiconformal 4-manifolds.

Sun-Sig Byun (University of Iowa) 
Title: Geometric approach to W^1,p estimates.

Luca Capogna (University of Arkansas) 
Title: Wave maps with target in the Heisenberg group.

Thierry de Pauw (Universite de Paris-Sud, and Rice University) 
Title: Nearly flat almost monotone measures are big pieces of Lipschitz graphs. 

Brian Hollenbeck (Emporia State University) 
Title: Best Constants for Operators Involving the Hilbert Transform.

Yaozhong Hu (University of Kansas) 
Title: On Logaritmic Sobolev and some other inequalities.

Frank Jochmann (Universität Leipzig) 
Title: Asymptotic behavior of solutions to nonlinear polarization  models.

Lev Kapitanski (Kansas State University) 
Title: $S^3$ and $S^2$ nonlinear $\sigma$-models.

Gocha Lepsveridze (Telavi State University, Georgia) 
Title: The Rate of Growth of Integral Means from Orlicz Clases. 

Mircea Martin (Baker University) 
Title: Multidimensional Generalizations of Alexander's Inequality.

Gabriel Nagy (Kasas State University) 
Title: Idemptotents in finite AW* factors. 

Richard Rochberg (Washington University, St. Louis) 
Title: Hankel and Schrodinger Forms on Dyadic Trees.

Sharon Schaffer Vestal (Missouri Western State College) 
Title: Using Functional Analysis to relate a wavelet GMRA to a 
multiwavelet MRA.

Eric Weber (Texas A & M University) 
Title: Frame Representations of Groups and Sampling Theory.

Shihshu Walter Wei  (Oklahoma University) 
Title: On the structure of minimal submanifolds in nonpositively curved manifolds. 

Karen Yagdjian (Kansas State University) 
Title: Parametric resonance and the global solutions to nonlinear hyperbolic equations.


Abstracts: 


David Auckly  
Title: A framework for analysis on noncompact quasiconformal 4-manifolds.

Abstract: This talk will report on analytical parts of a project joint  
with V. Kapovitch. An Alexandrov metric is a special structure that is 
studied by differential geometers. The existance question for Alexandrov 
metrics leads to the need for a gauge theory that would be applicable to 
non-compact quasiconformal 4-manifolds. In order to define a gauge 
theoretic moduli space, one needs to have a function space that 
compactly embeds into the continous functions, and is well defined when 
one considers the overlap maps. In 4 dimensions, the Sobolev space W^4_1, 
does not embed into the continous functions. On the other hand, the 
spaces, W^p_1 are not preserved by quasiconformal overlap maps for p>4. 
We define a weighted modified Sobolev space sitting between W^4_1 and each 
W^p_1. As an application of the weighted, modified Sobolev space, we will 
prove that any solution to a certain elliptic partial differential 
equation that grows slower than a given exponential function, must in 
fact decay exponentially. 

Sun-Sig Byun  
Title: Geometric approach to W^1,p estimates.

Abstract: We consider a Neumann problem for divergence form elliptic 
equations with discontinuous coefficients. We will prove a W^1,p 
esimate by a geometric approach which employs varified Vitali's 
covering lemma, Hardy-Littlewood maximal function and compactness 
method. Our methods can be easily extended to parabolic 
equations. 

Luca Capogna  
Title: Wave maps with target in the Heisenberg group

Abstract: This is a joint project with Jalal Shatah (Courant/NYU) 
concerning  well posedness of the Cauchy problem for  wave maps with 
target in the Heisenberg group. Such maps are solution of systems of 
quasilinear wave equations and satisfy differential constraints in the  
form of transport equations with rough coefficients. We can show local 
well-posedness for a large class of initial data. 

Thierry de Pauw 
Title: Nearly flat almost monotone measures are big pieces of Lipschitz graphs.

Abstract: Mass minimizing integral currents and stationary varifolds have strong 
regularity properties. For instance their support contains an open dense 
set which is an embedded smooth submanifold of the ambient space. The 
question asked in this talk is to what extent these regularity properties 
depend only upon some monotonicity property. We show that monotonicity 
implies partial regularity of the supports. 

Brian Hollenbeck  
Title: Best Constants for Operators Involving the Hilbert Transform.

Abstract: We calculate sharp constants in inequalities of the form, $\|Sf\|_{L^p} \le 
C_p\|f\|_{L^p}$, $1< p < \infty$, where $f$ is a complex-valued function and 
$S$ is an operator involving the Hilbert transform, $H$, and the 
identity operator, $I$.  This is equivalent to finding the $L^p$-norm of 
$S$. 
In particular, we calculate the norm of $aI + bH$, where $a, b \in \R$. 
We also prove $\|(I + iH)f\|_{L^p} \le 2 \csc \pip \|f\|_{L^p}$ for 
$1 < p < \infty$.  This immediately 
gives the norm of the Riesz projection to be $\csc \pip$, solving a conjecture 
made by Gohberg and Krupnik in 1968 and a problem posed by Pe\l czy\'nski 
in 1985. 

Yaozhong Hu  
Title: On Logaritmic Sobolev and some other inequalities.

Abstract: In this talk, I will present the Nelson's  
hypercontractivity inequality, Logarithmic 
Sobolev inequality and the motivation 
for this inequalities. I will also present  
a more general inequality which includes  
logarithmic Sobolev inequality, Poincare 
inequality as its particular cases. 
A simple proof of this general inequality 
will also be given.  

Some related inequalities such as Meyer's inequality, 
Interpolation inequality, correlation inequality 
will also presented. The correlation conjecture  
will also be presented.   

Frank Jochmann  
Title: Asymptotic behavior of solutions to nonlinear polarization  models.

Abstract: This talk is concerned with the Maxwell-Bloch system and the anharmonic 
oscillator model  describing the electromagnetic field 
in polarizable media. The main subject is the asymptotic behavior of the 
solutions to these models, in particular decay properties  and 
covergence to stationary states. 

Lev Kapitanski  
Title: $S^3$ and $S^2$ nonlinear $\sigma$-models.

Abstract: The Skyrme model (1961) was one of the first attempts to describe  
elementary particles as localized in space solutions of nonlinear PDEs.  
The fields take their values in SU(2)=S^3 and stabilize at spatial infinity. 
Thus, the configuration space splits into different sectors (homotopy classes)  
with a constant integer topological charge (the degree) in each sector.  
Faddeev's model (1975) was designed to provide additional internal structure  
(knottedness) to the localized solutions. The fields take their values  
in the two-dimensional sphere and the topological charge is the Hopf invariant. 
I will discuss some old and new results for these models. 

Gocha Lepsveridze  
Title: The Rate of Growth of Integral Means from Orlicz Clases.

Abstract:  Let $f\in L({\bf R}^n )$ be any function.  
For every $x\in {\bf  R}^n$ 
we consider integral means $1/|I|\int_{I} f$, where 
$I$ is an $n$ dimensional interval 
in ${\bf R}^n$. We  obtain the certain  weak type 
maximal inequalities from which 
are derived some exact  estimates on growth order of  
these means for functions from  
Orlicz classes  $L\Phi(L)({\bf R}^2)$ .   In 
particular, 
for any function  $f\in L\log_+^{\beta}({\bf 
R}^2),0<\beta<1$  the expression 
$\frac{1}{|I|}\int_I 
f/\log^{1-\beta}\Big(\frac{1}{M(I)}\Big)$ tends to $0$ 
as 
$\diam(I)\to 0$, where $M(I)$ denotes the length of 
the biggest side of the interval $I$. 
Estimates of growth order of Multiple  Fourier series 
are implied. 

Mircea Martin 
Title: Multidimensional Generalizations of Alexander's Inequality. 

Abstract: Alexander's inequality says that if 
$\Omega$ is a compact set in the complex plane and 
$C(\Omega)$ is the Banach algebra of complex-valued 
continuous functions on $\Omega$, then 
$$ {\rm dist}_{C(\Omega)}[\bar{z},R(\Omega)]\leq\left[\frac{1}{\pi}{\rm 
area}(\Omega)\right]^{1/2}, $$ 
where $\bar{z}$ is the complex conjugate coordinate 
function, and $R(\Omega)$ stands for the uniform closure in 
$C(\Omega)$ of rational functions that are analytic on open 
neighborhoods of $\Omega$. 
This inequality with many interesting applications is 
just a quantitative form of the classical Hartogs-Rosenthal 
theorem: Whenever ${\rm area}(\Omega)=0$, it follows that 
$R(\Omega)$ is a subalgebra of $C(\Omega)$ that contains $z$ 
and $\bar{z}$, and by the Stone-Weierstrass theorem one gets 
$R(\Omega)=C(\Omega)$. 
We will present several proper multidimensional 
generalizations of Alexander's inequality in the framework 
of Clifford analysis. The compact space $\Omega$ is now a 
subset of $\mathbb{R}^{m+1}$, $m\geq1$, and instead of 
$C(\Omega)$ we take the Banach algebra 
$C(\Omega,\frak{A}_m)$ of $\frak{A}_m$-valued continuous 
functions on $\Omega$, where $\frak{A}_m$ is the Clifford algebra 
with $m$ generators. The analog of $R(\Omega)$, denoted by 
$R(\Omega,\frak{A}_m)$, is defined as the uniform closure in 
$C(\Omega,\frak{A}_m)$ of functions Clifford-analytic on 
open neighborhoods of $\Omega$. 
As a direct generalization of Alexander's inequality, we 
will show that $$ {\rm dist}_{C(\Omega,\frak{A}_m)}[\bar{x},R(\Omega,\frak{A}_m)]\leq 
A_m[{\rm vol}(\Omega)]^{1/(m+1)}, $$ 
where $\bar{x}$ is the Clifford conjugate of the identity 
function on $\mathbb{R}^{m+1}$, $A_m$ is a universal 
constant that only depends on $m$, and ${\rm vol}(\Omega)$ 
is the Lebersgue measure of $\Omega$ in $\mathbb{R}^{m+1}$. 
Actually, this result will be derived from a more general 
inequality that estimates the distance in 
$C(\Omega,\frak{A}_m)$ from an 
arbitrary smooth $\frak{A}_m$-valued function to 
$R(\Omega,\frak{A}_m)$. In particular, that general 
inequality will imply 
$R(\Omega,\frak{A}_m)=C(\Omega,\frak{A}_m)$, whenever 
${\rm vol}(\Omega)=0$, so, once more, we end up with a 
quantitative form of the Hartogs-Rosenthal theorem, but now 
this theorem is in the setting of Clifford analysis. 

Gabriel Nagy  
Title: Idemptotents in finite AW* factors

Abstract: The talk addresses a problem posed by Kaplansky in the 1950's, 
which conjectures that an AW* factor is a von Neumann algebra. 
In connection with this question, we prove that the quasitrace of an  
idempotent in an AW* factor of type II_1 is equal to the dimension  
function of its left (or right) support. Based on this result, we discuss  
some linear algebraic reformulations of Kaplansky's conjecture.  

Richard Rochberg  
Title: Hankel and Schrodinger Forms on Dyadic Trees

Abstract:  Hankel forms on dyadic trees can be viewed as discrete models 
for Hankel forms on the Dirichlet space or as discrete models for 
Schrodinger forms.  I will describe how these discrete models are related 
to the classical questions and will describe boundedness criteria for the 
discrete forms. The criteria for those forms to be in Schatten-Von 
Neumann classes is not known. 

Sharon Schaffer Vestal  
Title: Using Functional Analysis to relate a wavelet GMRA to a multiwavelet MRA

Abstract: It is well-known that wavelets have an associated subspace structure  
calleda multiresolution analysis (MRA).  There are other wavelets, minimally 
supported frequency (MSF) wavelets, which are associated with a generalized 
multiresolution analysis (GMRA).  We will present a theorem that links the 
two structures and give examples illustrating this relationship. 

Eric Weber 

Title: Frame Representations of Groups and Sampling Theory.

Abstract:  We consider unitary representations of Abelian groups that give 
rise to a frame sequence.  By analyzing the group we can get information 
regarding the corresponding analysis operator; in particular, we have a 
way of "parametrizing" the range.  Such information is significant for 
multiplexing schemes.  We then demonstrate how this can be applied to 
sampling theory. 
 

Shihshu Walter Wei  
Title: On the structure of minimal submanifolds in nonpositively curved manifolds.

Abstract: We provide a topological obstruction for a complete submanifold 
with a specific uniform bound involving Ricci curvature to be minimally immersed 
in any complete simply-connected manifold of nonpositive sectional curvature. 
We prove that such minimal submanifolds of dimension greater 
than two have only one topological end.  The proof uses the Liouville 
theorem for bounded harmonic functions on minimal submanifolds of this sort 
due to Yau, and also adapts a technique of Cao-Shen-Zhu to show the 
existence of nonconstant bounded harmonic functions based on the Sobolev inequality of Hoffman-Spruck.  This extends the work 
of Yau. The same phenomena occur in a wider class of $n$-submanifolds with bounded mean 
curvature in an $L^n$ sense. By improving the techniques in Cao-Shen-Zhu, one can obtain the 
topological conclusion in the intrinsic settings. These generalize and 
unify the structure theorems in the extrinsic settings. 
 

Karen Yagdjian 
Title: Parametric resonance and global solutions to nonlinear hyperbolic equations.

Abstract: We show  how  parametric resonance can affect global existence of  
solutions to the Cauchy problem for  nonlinear hyperbolic equations.  Namely  
we give some examples of nonlinear hyperbolic equations and systems such that  
for arbitrary small smooth initial data, and for arbitrary large space dimension  
there are blowing up solutions.

 

 The Prairie Analysis Seminar is a joint project of the Department of Mathematics of  Kansas State University and the  

 Department of Mathematics of The University of Kansas. 

 Organizers:

 Marianne Korten 
 Estela Gavosto
 Charles Moore
 Rodolfo Torres

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