Titles/Abstracts for the 2007 Midwest Geometry Conference -- Speaker: Peter Perry
Title: Scattering Theory on Complex Manifold and CR-Invariants
Abstract:
This is a report on joint work with Peter Hislop and Siu-Hung Tang. Suppose that $X$ is a compact complex manifold of complex dimension $m=n 1$ and that the boundary $M$ of $X$ is strictly pseudoconvex. We consider scattering theory for a K\"ahler metric associated to the K\"ahler form $\omega_\varphi=-i\partial\overline{\partial}\log(-\varphi)$ where $\varphi$ is a defining function for $M$. For certain complex compact manifolds (including pseudoconvex domains in $\mathbb{C}^m$), we can construct a defining function which is a global approximate solution of the Monge-Amp\`ere equation so that the metric is an approximate K\"ahler-Einstein metric. In this case, CR-covariant differential operators on $M$ may be recovered as residues of the scattering operator and the CR-$Q$-curvature exhibited as a special value of the scattering operator acting on constant functions. Our results are an analogue in CR-geometry of Graham and Zworski's results relating scattering on asymptotically Poincar\'e-Einstein manifolds to conformally covariant diferential operators and the $Q$-curvature. This work was announced in C. R. Acad. Sci. Paris,
Ser. I \textbf{342} (2006), 651-654.
Speaker: s.n.hosseinimotlagh
Title: Analytical Solutions to Brajinskii’s Equations for Nuclear Fusion Plasma
Abstract:
Brajinskii’s equations are the fundamental relations governing the behavior of the plasma produced during a fusion reaction, especially ICF plasma. These equations contains six partial differential coupled together. In this paper we have tried to give analytical solutions to these equations using a one dimensional method. Laplace transform technique is the main tool to do that with an orbitrary boundary and initial conditions for some special cases.
Speaker: John Ryan
Title: Dirac Operators and Conformal structure
Abstract:
The purpose of this talk is to give a review of the interaction between Dirac type operators and the conformal group. The focus will be on setting up structures so that one can transpose tools from classical harmonic analysis in euclidean space to similar stuctures on some conformally flat spin manifolds. Automorphic forms in n real variables will be a key tool used.
Speaker: Ronald Fulp
Title: Infinite dimensional super Lie groups
Abstract:
A super Lie group is a group whose operations are $G^{\infty}$ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are $G^{\infty}$ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.
In this context, we prove that if ${\frak{h}}$ is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group ${\mathcal{G}},$ then ${\frak{h}}$ is the super Lie algebra of a sub-super Lie group of ${\mathcal{G}}.$ Additionally, we show that if ${\frak{g}}$ is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group ${\mathcal{G}}$ such that the even part of ${\frak{g}}$ is the even part of the super Lie algebra of ${\mathcal{G}}.$ In general, the module structure on ${\frak{g}}$ is required to obtain ${\mathcal{G}},$ but the ``structure constants" involving the odd part of ${\frak{g}}$ can not be recovered without further restrictions. We also show that if ${\mathcal{H}}$ is a closed sub-super Lie group of a super Lie group ${\mathcal{G}},$ then ${\mathcal{G}} \rightarrow {\mathcal{G}}/{\mathcal{H}}$ is a principal fiber bundle.
Finally, we show that if ${\frak{g}}$ is a graded Lie algebra over the complex numbers, then there is a super Lie group whose super Lie algebra is the Grassmann shell of ${\frak{g}}.$ We also briefly relate our theory to techniques used in the physics literature.
We emphasize that some of these theorems are known when the space of supernumbers is finitely generated in which case one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the $G^{\infty}$ category.
Speaker: Eero Saksman
Title: Quasiconformal flows and Q-curvature
Abstract:
The Jacobian problem for quasiconformal maps asks for a characterization of weights on R^n that are comparable to Jacobian determinants of quasiconformal homeomorphisms. The talk describes a joint work with M. Bonk and J. Heinonen in this area that uses the technique of quasiconformal flows. As an application of our results we show that a conformal deformation of R^4 is bi-Lipschitz equivalent to R^4 if it has sufficiently small total Q-curvature. This yields a natural generalization of a result by J. Fu in dimension 2.
Speaker: Genkai Zhang
Title: Residue and Diximier traces of Toeplitz operators
Abstract:
We compute the Residue trace and Diximier trace of
Toeplitz operators with symbols being pseudo-differential operators
on the boundary of a strongly
pseudoconvex domains. We find certain conformal invariant
L^n-norm on smooth functions on the unit ball of C^n
defined in terms of the boundary CR-operators.
Speaker: Andrew Waldron
Title: Supersymmetric Quantum Mechanics and SuperLichnerowicz Algebras
Abstract:
Supersymmetric quantum mechanics has a deep relation
to geometry. Famous examples are the relations
between Pontryagin classes and N=1 supersymmetry,
and Morse theory and N=2 supersymmetry. We present
spinning particle models with an osp(2p|Q) internal
symmetry group and show how they relate to geometry.
The simplest model reproduces Lichnerowicz's algebra
of differential operators on symmetric tensors.
The Lichnerowicz wave operator is central, and the
model is a deformation of the Jacobi group.
Higher models produce a superalgebra of operators on
arbitrary tensors and spinor-tensors on a manifold.
Speaker: Labbi Mohammed Larbi
Title: On Gauss-Bonnet-Weyl Curvatures
Abstract:
The symmetric functions $s_{2k}$ of the eigenvalues of the shape operator of a hypersurface of the Euclidean space are intrinsic quantities. Therefore, they can be generalized and defined for any
Riemannian manifold. The so obtained scalar invariants are called
Gauss-Bonnet-Weyl curvatures and are denoted by $h_{2k}$. Note that the curvature $h_{2k}$ coincides with the Gauss-Bonnet integrand in dimension $2k$, and in higher dimensions it appears naturally as an integrand in the well known Weyl's tube formula. \\
In this talk we study some properties of these invariants using the the exterior product of double forms. In particular, we prove that these curvatures, like the usual scalar curvature, satisfy a nice variational formula.\\
References: 1. Double forms, curvature structures and the (p,q)-curvatures, Transactions of the American Mathematical Society. 357, n10, 3971-3992 (2005).
2. Manifolds with positive second Gauss-Bonnet curvature, Pacific journal of Math. Vol. 227, No. 2, 2006.
3. On a variational formula for the Gauss-Bonnet-Weyl curvatures, The paper is available online at http://arxiv.org/abs/math.DG/0406548
Speaker: Xin Tang
Title: ON REPRESENTATIONS OF QUANTUM GROUPS $U_{q}(f(K,H))$
Abstract:
In this talk, we first explain how to use methods in spectral
theory to construct interesting families of irreducible weight representations for the quantum groups $U_{q}(f(K,H))$. Then we
will explain how to construct irreducible Whittaker representations (which are definitely not weight representations) of $U_{q}(f(K,H)$ through the Whittaker model for the center of $U_{q}(f(K,H))$. This is an ongoing work joint with Yunge Xu.
Speaker: Bi-invariant and non-invariant metrics on low-dimensional Lie groups
Title: Ryad Ghanam
Abstract:
In this talk we will discuss the existence of pseudo-Riemannian metrics on indecomposable solvable Lie groups in dimensions six and less that are compatible with the canonical bi-invariant connection. We will give necessary and sufficient conditions for a metric to be bi-invariant and consider also the integrability properties of non-invariant metrics.
Speaker: Sungwook Lee
Title: Maximal surfaces in a certain 3-dimensional homogeneous spacetime
Abstract:
In this talk, we present a generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauss map of maximal spacelike surfaces and its harmonicity are also discussed.
Speaker: Nahid Sultana
Title: constant mean curvature surfaces of revolution in the Schwarzschild space
Abstract:
We study the constant mean curvature (CMC) surfaces of revolution in the ambient space which is spherically symmetric and related to the general relativity. There is a strong relation between the general theory of relativity and differential geometry. In both of these fields, the notion of curvature is a basic concept and plays a central role. Spherically symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity, including the Schwarzschild solution. Therefore, we consider a 3-dimensional positive-definite slice of the Schwarzschild metric and define the Schwarzschild space $\mathcal M^3$. We describe the surfaces of revolution in $\mathcal M^3$, and constarct some minimal and closed non-minimal CMC surfaces of revolution.
Speaker: Klaus Kirsten
Title: Exotic phenomena for zeta functions and heat kernels on conical manifolds
Abstract:
We discuss exotic phenomena and a complete
classification of the meromorphic structure of zeta-functions
associated with self-adjoint extensions of Laplace-type operators
over conic manifolds. We show that in general these zeta-functions
have countably many logarithmic branch cuts on the non-positive
real axis. In addition, unusual locations of poles with
arbitrarily large multiplicity occur. It is shown, that this
behavior results in the corresponding small-t asymptotic expansion
of the heat kernel to have logarithmic terms to arbitrary positive
and negative powers.
Speaker: Peter J. Olver
Title: Differential Invariants of Equi-affine Surfaces
Abstract:
I will show that the algebra of equi-affine differential invariants of a generic surface in R^3 is entirely generated by the third order Pick invariant via invariant differentiation. The proof is based on the equivariant method of moving frames.
Speaker: A Rod Gover
Title: Non-local operators on conformally Einstein manifolds
Abstract:
Many conformal constructions simplify drammatically if the structure
is known to be conformally Einstein, or ``almost Einstein'' as in the
case of the Poincare-Einstein manifolds. The geometry behind this is
examined with a focus on specialising various constructions of
non-local operators and global invariants that were developed with Tom
Branson.
Speaker: Peter Gilkey
Title: The spectral geometry of the canonical Riemannian submersion of a compact Lie G
Abstract:
ABSTRACT: Let G be a compact Lie group which is equipped
with a bi-invariant Riemannian metric. Let m(x,y)=xy be the
multiplication operator. The associated fibration m:GxG->G is
a Riemannian submersion with totally geodesic fibers. The
associated spectral geometry of the submersion is studied.
Eigen functions on G pull back to eigen functions on GxG with
the same eigenvalue. Eigen p-forms for p>0 on the base pull
back to eigen p-forms on GxG with finite Fourier series;
there are examples where the number of eigenvalues in the
Fourier series of the pull back on GxG is arbitrarily large.
If w is a harmonic p-form on the base, necessary and
sufficient conditions are given to ensure the pull back of w
is harmonic on GxG.
Speaker: Ivan Avramidi
Title: Heat Kernel on Homogeneous Bundles over Symmetric Spaces
Abstract:
We study the heat kernel asymptotics of Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained formal result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel on two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$.
Speaker: Rodrigo Ristow Montes
Title: A Congruence Theorem for Minimal Surfaces in $S^5$ with Constant Contact Angle
Abstract:
We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations with constant contact angle satisfy a Laplacian equation of the Holomorphic angle. Also, we will give a characterization of flat minimal surfaces in $S^5$ with constant contact angle.
Speaker: Michael Eastwood
Title: Monogenic Functions in Conformal Geometry
Abstract:
This talk will discuss the construction of conformally invariant differential operators, in a way that includes the "monogenic functions," arising in Clifford analysis. Particular emphasis will be given to the link between invariance under a symmetry group on the one hand and differential geometric invariance on the other. The link between these two notions of invariance was a key feature of Tom Branson's work. The results in this talk will be well-known to those who know and this would have included Tom. The particular exposition, however, is joint work with John Ryan.
Speaker: Michiel van den Berg
Title: Heat content of a complete Riemannian manifold with singular initial conditions
Abstract:
Upper bounds are obtained for the heat content of an open set D with singular intial condition f on a complete Riemannian manifold provided (i) the Dirichlet-Laplace-Beltrami operator satisfies a strong Hardy inequality, and (ii) f satisfies an integrability condition.Precise asymptotic reuslts for the heat content are obtained for an open bounded and connected set in Euclidean space with C^2 boundary,and with initial condition f(x)= d(x)^-a ,0 < a < 2, where d(x) is the distance from x to the boundary of D.
Speaker: Stephen A. Fulling
Title: An Index Theorem for Quantum Graphs
Abstract:
A quantum (or metric) graph is a 1-complex equipped in the obvious way with a Schrodinger equation. Heat kernel expansions, etc., can be computed. Boundary conditions must be chosen at the vertices; the most natural choice is a certain generalization of the Neumann condition.
In that case the constant term in the heat-kernel expansion equals half the difference between the vertex number and the edge number. An index interpretation is achieved by comparing with another operator on the same graph with different boundary conditions. The construction is a generalization of Gilkey's treatment of Neumann and Dirichlet boundary conditions on the interval as the simplest case of the de Rham complex on a manifold with boundary. This work builds on research done with an undergraduate student, Justin Wilson, and members of the quantum graph research group at Texas A&M University.
Speaker: Jan Slovak
Title: Feffermann type inclusions of parabolic geometries
Abstract:
The lecture reports on all Feffermann type constructions for parabolic geometries where the underlying manifolds do not change.
The original Feffermann's construction produces an $S^1$--bundle over each manifold with an integrable CR--structure, equipped with a conformal structure. Recently, non-degenerate rank 2 vector distributions on 5-dimensional manifolds were studied by Nurowski and those associated with non-degenerate rank 3 distributions on 6-dimensional manifolds were discussed by Bryant. In both cases there are natural parabolic geometries associated with these distributions, which serve as an intermediate structure between the distribution and the conformal geometry.
Using classical results by A. Onischchik, we classify all possibilities of such inclusions of parabolic geometries. Apart of known examples, a new series of embeddings of 2--graded $C_\ell$ geometries into 1--graded $D_{\ell 1}$ geometries has been detected. These geometries correspond to the generic $\ell$--dimensional distributions of codimension $\frac12\ell(\ell-1)$ and the Bryant's example fits into this series with $\ell=3$.
Speaker: Prof. Dr. Kishore Marathe
Title: Topics in Physical Mathematics
Abstract:
In recent years the interaction between geometric topology and classical
and quantum field theories has attracted a great deal of attention from
both the mathematicians and physicists. We will discuss some topics
where this has led to new viewpoints as well as new results. They
include categorification of knot polynomials and a special case of
the gauge theory to string thery correspondence.
Speaker: Guanghao Hong
Title: A New Proof of Reifenberg's Topological Disk Theorem
Abstract:
This is a report on joint work with my advisor Lihe Wang. Reifenberg's topological disk theorem played a key role in his solution of higher dimensional Plateau problem. The theorem says roughtly that if a subset S of R^n is locally close to a m-dimensional plane at its every point and at every scale then S is locally Euclidean. The proof of theorem depends on construction of a series of approximating surfaces. Reifenberg built these surfaces locally, involving several averaging processes. This made his proof very complicated and messy. However, our new proof is based upon the global construction of the approximating surfaces, which is the key point. So our proof is rather direct and clean.
Speaker: Huilian Jia
Title: Regularity Theory in Orlicz Spaces for Elliptic Equations in Reifenberg Domains
Abstract:
In this paper, we establish the global estimates for divergence form elliptic equations in the Orlicz space $L^{\phi}(\Omega)$, where
$\Omega$ is a bounded Reifenberg domain in $\mathbb{R}^n$. Our
result generalizes the $W^{1,p}$ estimates.
Speaker: Oleg Svidesrkiy
Title: Contractions of certain Lie subalgebras of the Conformal algebra
Abstract:
Contractions of the Lie algebras d=u(2), f=u(1,1) to the oscillator algebra l and to abelian algebras are realized as the result of the adjoint action of SU(2,2). Similar contractions of (seven-dimensional) isometry Lie algebras iso(D) and iso(F) to iso(L), where D, F and L are corresponding Lie Lie groups endowed with bi-invariant metrics of Lorentzian signature, are determined.
Speaker: Gestur Olafsson
Title: The Paley-Wiener Theorem on Compact Symmetric Spaces and Applications
Abstract:
The classical Paley-Wiener theorem describes the size of the support of a smooth function on Eucledian space in term of the growth of the holomorphically extended Fourier transform of the function. We discuss the generalization of the Paley-Wiener Theorem for the symmetric spaces of the compact type. Note, that the Fourier transform is a priory only defined on a discrete subset. At the end we will discuss the special case where all the multiplicities are even and apply that to the shifted wave equation on compact symmetric spaces.
Part of our results is a joint work with Tom Branson and Angela Pasquale. Then general case is a joint work with Henrik Schlichtkrull.
Speaker: David Vogan
Title: Cutting and pasting group representations
Abstract:
The most fundamental homogeneous spaces for reductive groups are the
partial flag varieties G/P (and their various real forms). The most
fundamental representations for reductive groups are spaces of
sections of vector bundles on partial flag varieties (and their
various cohomological cousins).
More interesting representations turn up as subrepresentations of
these fundamental ones, often cut out as solutions of some
differential equations. I'll talk about a very general formalism for
taking representations apart into pieces related to flag varieties,
and how to use it to look for these interesting representations.
Speaker: Oksana Bihun
Title: Deformation Minimal Bending and Morphing of Riemannian Manifolds
Abstract:
Let $M$ and $N$ be compact Riemannian n-manifolds embedded in $R^{n 1}$ with the corresponding Riemannian metrics $g_M$ and $g_N$ and the volume forms $w_M$ and $w_N$ induced by the metrics. We consider the problem of optimization of the functionals $\Phi(h)=\int_M(|J(h)|-1)^2 w_M$ and
$\Psi(h)=\int_M\|h^*g_N-g_M\| w_M$ over the class of diffeomorphisms
$Diff(M,N)$, where $J(h)$ denotes the Jacobian of $h$. The above
functionals measure distortion of the manifold $M$ produced by the
diffeomorphism $h$. Results on existence and uniqueness of minima of the functionals $\Phi$ and $\Psi$ will be presented. A morph between manifolds $M$ and $N$ is a smooth homotopy between the latter manifolds. The problem of deformation minimal morphing of manifold $M$ to manifold $N$ will be discussed.
Speaker: Spyros Alexakis
Title: The Deser-Schwimmer conjecture and Branson's Q-curvature
Abstract:
I will present my work on the Deser-Schwimmer conjecture regarding the algebraic structure of global conformal invariants. This work provides an algebraic description of Tom Branson's Q-curvature. In view of work by Chang-Qing-Yang and Graham-Zworski, Q-curvature provides an understanding of re-normalized volume and conformal anomalies for conformally compact Einstein manifolds.
Speaker: Kengo Hirachi
Title: Q-curvature in CR geometry
Abstract:
For 3-dimensional CR manifolds, the Q-curvature appears in the logarithmic term in the Szego kernel. I will generalize this relation to higher dimensions by generalizing the notion of Q-curvature. This family of Q-curvatures is associated with non-linear CR invariant differential operators and is constructed by using the ambient metric.
Speaker: Igor Zelenko
Title: On Local Geometry of Nonholonomic Vector Distributions
Abstract:
The talk is based on the joint work with Boris Doubrov. First we
will describe a new rather effective procedure of symplectification
for the problem of local equivalence of nonholonomic vector
distributions. The starting point of this procedure is to lift a
distribution $D$ to a special odd-dimensional submanifold $W_D$ of
the cotangent bundle, foliated by the characteristic curves (the
abnormal extremals of the distribution $D$). In particular, if $D$
is a rank $2$ distribution then the submanifolds $W_D$ is nothing
but the annihilator of the square of $D$, while if $D$ is a distribution of odd rank it is the annihilator of $D$ itself. The dynamics of the lifting (to $W_D$) of the distribution $D$ along
the characteristic curves (of $W_D$) is described by certain
curves of flags of isotropic and coisotropic subspaces in a linear
symplectic space. So, the problem of equivalence of distributions
can be essentially reduced to the differential geometry of such
curves: symplectic invariants of these curves automatically produce
invariants of the distribution $D$ itself and the canonical frame
bundles, associated with such curves can be in many cases
effectively used for the construction of the canonical frames of the
distributions $D$ itself on certain fiber bundles over $W_D$. In
this way we succeeded to construct the canonical frames
for germs of rank 2 distributions in $\mathbb R^n$ with $n>5$ and of
rank 3 distributions in $\mathbb R^n$ with $n>6$ from certain
generic classes. The first case generalizes the classical work of E.
Cartan (1910) on rank $2$ distributions in $\mathbb R^5$. The second
case is also new: the only rank 3 distributions with functional invatiants, treated before, were rank $3$ distributions in $\mathbb R^5$ (Cartan, 1910) and in $\mathbb R^6$ (N. Tanaka school and independently R. Bryant in 70th). In all these cases the most symmetric models will be given as well.
Speaker: Ende Zhang
Title: ZhangEnde Space Theory (a new monograph, the preprint is on www.zhangende.net)
Abstract:
In geometry’s terms, this work is Flat ZhangEnde Manifold Geometry. This monograph consists of 16 chapters. In this new monograph, the author has developed about one hundred and eighty brand new definitions, notions, theorems, formulas, etc. including (1) the order of complex numbers; (2) positivity and negativity of complex numbers; (3) complex inequalities; (4) the complex axis; (5) ZhangEnde complex plane (which contains Euclidean plane); (6) n-dimensional ZhangEnde space Z^n (which is a complex space and contains Euclidean space as a subspace); (7) Solid Geometry In Z^3; (8) ZhangEnde plane coordinate system (which contains planar Cartesian coordinate system as a special case); (9) ZhangEnde space coordinate system (which contains the 3-dimensional Cartesian coordinate system as a special case); (10) geometric definitions of the six traditional complex trigonometric functions in ZhangEnde plane; (11) complex straight lines & equations; (12) complex circles & equations; (13) complex ellipses & equations; (14) complex hyperbolas & equations; (15) complex parabolas & equations; (16) complex triangles & their complex areas; (17) complex parallelograms; (18) The Theorem of Complex Cosines; (19) The Theorem of Complex Sines; (20) complex polar coordinates; (21) complex spheres - equations and their complex volumes; (22) complex spherical coordinates; (23) ZhangEnde inner product; (24) ZhangEnde defined angle between complex vectors; (25) ZhangEnde norm (which is generally a positive complex value); (26) ZhangEnde distance in Z^3 (which is generally a positive complex value); (27) ZhangEnde Plane Analytic Geometry; (28) ZhangEnde Space Analytic Geometry; (29) ZhangEnde arc differential (which is generally complex valued and contains Euclidean arc differential as a special case); (30) Complex tangent vector and normal plane to a complex curve in ZhangEnde space; (31) The Existence Theorem of Maximum and Minimum Values of Holomorphic Functions; (32) Differential Calculus & Integral Calculus of Complex Functions of One and Several Variables; (33) ZhangEnde Passage Existence Theorem; and so on and so forth. Among other things, this book contains current mathematics system's Euclidean Plane Geometry, Euclidean Solid Geometry, Trigonometry, Vector Analysis, Analytical Geometry, Calculus, Theory of Complex Functions, etc. as special cases. So this book may also be regarded as generalizations of them. || The preprint of this work is available on the website: www.zhangende.net
Speaker: Gary R. Jensen
Title: Isoparametric Hypersurfaces with Four Principal Curvatures
Abstract:
Let $M$ be an isoparametric hypersurface in the sphere
$S^n$ with four distinct principal curvatures.
M\"{u}nzner showed that the four principal curvatures
can have at most two distinct multiplicities $m_1, m_2$,
and Stolz showed that the pair $(m_1,m_2)$ must either be
$(2,2)$, $(4,5)$, or be equal to the multiplicities of an
isoparametric hypersurface of FKM-type, constructed by
Ferus, Karcher and M\"{u}nzner from orthogonal
representations of Clifford algebras. In this talk I will describe the following result obtained in joint work with Thomas E. Cecil and Quo-Shin Chi: If the multiplicities satisfy $m_2 \geq 2m_1 - 1$,
then the isoparametric hypersurface $M$ must be of
FKM-type. Together with known results of Takagi
for the case $m_1 = 1$, and Ozeki and Takeuchi for $m_1 = 2$,
this handles all possible pairs of multiplicities except
for 4 cases, for which the classification problem remains open.
Speaker: Steven Rosenberg
Title: Characteristic Classes on Loop Spaces
Abstract:
The free loop space LM of a Riemannian manifold M comes with a family of Riemannian metrics parametrized by a Sobolev space parameter. The curvature of these metrics takes values in pseudodifferential operators on a trivial bundle over the circle. There is a Chern-Weil theory of characteristic classes and Chern-Simons theory of secondary classes, using the Wodzicki residue in place of the usual trace. These classes are computable in theory, and in practice if your computer is fast enough. The characteristic classes on the tangent bundle TLM all vanish, and the Chern-Simons classes are apparently nonzero, modulo checking of computations. Unlike in finite dimensions, where the Chern-Simons classes are conformal invariants, on loop spaces all but the first Chern-Simons class are smooth invariants. So it would be very nice to have firm examples of nonzero Chern-Simons classes.
This is joint work with Yoshiaki Maeda and Fabian Torres-Ardila, and based on earlier work with Sylvie Paycha.
Speaker: Juha Pohjanpelto
Title: Pseudogroups, Moving Frames, and Differential Invariants
Abstract:
I will report on my ongoing joint work with Peter Olver
on developing systematic and constructive algorithms for
analyzing the structure of continuous pseudogroups and
for identifying various invariants for their action. In this
talk I will focus on the method of moving frames combined
with techniques from commutative algebra to discuss
Tresse-Kumpera type existence results for differential
invariants of a pseudogroup action and to describe methods
for analyzing the algebraic structure of such invariants.
Speaker: Omid Ghayour
Title: CONFORMAL CAPACITY ON FINSLER MANIFOLDS
Abstract:
With Sina Hedayatian.
Joint research on conformal capacity will be presented.
Speaker: Andrea Malchiodi
Title: Conformal metrics on four manifolds with constant Q-curvature
Abstract:
In 1983 S.Paneitz discovered a conformally invariant operator on four manifolds whose principal part is the squared Laplacian. Two years later Tom Branson introduced a related scalar quantity, the Q-curvature, which
in many ways enjoys analogous properties to the Gauss curvature in two dimensions.
In the spirit of the classical uniformization theorem, we investigate the problem of finding conformal metrics with constant Q-curvature.
Speaker: Adrian Butscher
Title: New Constructions of Stationary Submanifolds of the Sphere
Abstract:
If one searches for k-dimensional submanifolds with critical k-dimensional volume in a Riemannian manifold, then one is led towards elliptic partial differential equations involving the mean curvature vector of the submanifold. I will present new constructions of volume-critical submanifolds of the sphere in two contexts: hypersurfaces with constant mean curvature in spheres of any dimension; and Legendrian submanifolds in spheres of odd dimension that are stationary under variations preserving the contact structure. These are constructed by solving the associated elliptic PDE using singular perturbation theory. I will then highlight some of the analytic and geometric similarities between these two contexts.
Speaker: Bent Orsted
Title: Calculating spectra of intertwining operators
Abstract:
A classical problem in Lie theory is to solve differential equations
by knowing their symmetries. A more modern aspect is to study
representations of a Lie group in spaces of solutions to linear
differential equations admitting the group as symmetries. A particular problem is to calculate the spectrum of canonical
classes of differential or integral operators, the so-called
covariant or intertwining operators. In this talk, which is
based on joint work with Gestur Olafsson and Tom Branson, we shall
give methods to generate the spectrum in an intrinsic way of such
operators, and we shall give examples of applications to
representation theory. The technical setting will be in spaces of
sections of vector bundles over flag manifolds G/P, or over
open orbits here of symmetric subgroups.
Speaker: Lina Wu
Title: Homotopy Groups in p-Harmonic Theory
Abstract:
In this talk, we are interested in representing homotopy groups by
p-harmonic maps , with applications in minimal varieties.
Speaker: Yvette Kosmann-Schwarzbach, Ecole Polytechnique, Palaiseau, France
Title: Lie algebroids, Poisson geometry and modular classes
Abstract:
Lie algebroids unify differential geometry and algebra in the sense
that both the tangent bundle of a smooth manifold and finite-dimensional Lie algebras are examples of Lie algebroids, and they play an important
role in the geometry of Poisson manifolds.
The notion of modular class, a class in the Lichnerowicz-Poisson cohomology of a Poisson manifold, can be generalized to the case of Lie algebroids equipped with a bivector whose Schouten square vanishes and, in particular, the modular class can be computed for a coboundary Lie bialgebra, i.e., a Lie algebra equipped with a cobracket defined by a solution of the classical Yang-Baxter equation.
Speaker: Leo Rodriguez
Title: Black Hole Statistics via the Virasoro and Kac-Moody Algebra
Abstract:
We investigate the implications of the Cardy formula in the determination of black hole entropy. The Cardy formula directly relates the density of states to the central extensions of the algebras derived from conformal field theories. We consider algebras that include semi direct products of the Virasoro and Kac-Moody sectors. From these mergers we compute the shifted central charges and thus the entropy.
Speaker: Dmitriy Khots, dkhots@cox.net
Title: Geometrical Aspects of Observer's Mathematics
Abstract:
The work is a joint effort with Boris Khots. A new system of observer-dependent arithmetic on finite sets of finite decimal fractions is developed (see, for example, www.mathrelativity.com). In the new arithmetic setting, Euclidean, Lobachevskii and Riemannian geometries (in a sense of parallel lines) are examined. As a result, we prove that the Euclidean geometry can be applied only in a neighborhood of a fixed line, whereas Lobachevsky geometry can be established beyond this neighborhood. Riemannian geometry exists in a special space model. Further, a space is found that contains a dense subset of unreachable points – “Malevich Effect”.
Speaker: Andreas Juhl
Title: Q-curvature and holography
Abstract:
We present a formula for the Q-curvature of a Riemannian manifold
of even dimension in terms of data derived from the associated
Poincare-Einstein metric. The formula identifies Q as a sum of the holographic anomaly and explicit divergence terms which are determined
by the structure of harmonic functions for the Poincare-Einstein
metric. As a consequence, for a conformally flat metric Q splits as the sum of (a multiple of) the Pfaffian and an explict divergence term. We describe the relation of this formula to the so-called residue family.
The residue family is a one-paramter family of differential operators which is canonically associated to a Poincare-Einstein metric. This new
object satisfies a conformal transformation law and satisfies a system of recursive relations. The work is partly joint with R. Graham.
Speaker: Carlo Morpurgo
Title: Sharp Moser-Trudinger and Onofri-Beckner's inequalities on the CR sphere
Abstract:
The classical Moser-Trudinger inequality represents the imbedding of the Sobolev space W^{n/p,p} into an exponential class, where n is the dimension of the ambient space. The sharp form of this inequality is known for domains of R^n and for compact Riemannian manifolds. When p=2 another sharp form of the imbedding can be given via the so-called Onofri-Beckner type estimate, which was first derived by Onofri on S^2 and later discovered by Beckner in the general n-dimensional setting. The derivatives are controlled by the intertwining operator of order n, also known as the Paneitz operator, which makes the inequality invariant under the conformal group. In this talk I will present some natural versions of the above results on the CR sphere S^{2n 1}. This is joint work with Luigi Fontana.
Speaker: Oussama Hijazi
Title: Q-curvature in Spin Geometry
Abstract:
On a Spin manifold the fundamental Schrödinger-Lichnerowicz formula simply says that the square of the Dirac operator differs from the rough Laplacian by the scalar curvature pointwise multiplication. Based on the conformal covariance of the Dirac operator and the conformal scalar Laplacian (the Yamabe operator) this formula leads to a relation between fermionic and bosonic spectral invariants. The aim of the talk is to report on a joint work with Tom on some aspects of the corresponding role of the Q-curvature.
Speaker: carla farsi
Title: Orbifold characteristic Classes and Index Theory
Abstract:
In this talk I will present some results on orbifold characteristic classes, and their application to orbifold index theorey.
Speaker: Robin Graham
Title: A survey of Q-curvature
Abstract:
This talk will survey the development of our understanding of Q-curvature, beginning with Tom's definition in 1995 and ending
with a new formula for Q-curvature which is recent joint work with Andreas Juhl. The focus will be on Q-curvature for general metrics
in arbitrary even dimensions. The new formula is explicit in
terms of ingredients involving the Poincare metric in one higher dimension. The formula resolves an open problem by realizing Q
in the conformally flat case as a multiple of the Pfaffian plus a divergence.
Speaker: Andreas Cap
Title: A holonomy characterization of Fefferman spaces
Abstract:
I will outline how Fefferman spaces of CR manifolds can be locally characterized among conformal structures of the appropriate signature by the existence of a parallel orthogonal complex structure on the standard tractor bundle. Equivalently, this can be phrased as the fact that the a conformal structure for which the conformal holonomy is contained in U(p,q) is locally conformally isometric to a Fefferman space. This implies that the conformal holonomy Lie algebra is contained in su(p,q), so u(p,q) cannot occur as a conformal holonomy Lie algebra. Finally, I will show how to use these results and the machinery of BGG sequences to obtain a strengthening of Sparling's characterization of Fefferman spaces.
Speaker: Shihshu Walter Wei
Title: p-Harmonic geometry and related topics
Abstract:
We'll begin with a brief general view of p-harmonic goemetry. Then we'll discuss Some recent progress in p-harmonic geometry and related topics.
Speaker: Pham Trieu Duong
Title: The first BVP for parabolic systems in domains with singularities
Abstract:
The study of boundary value problems for systems of PDE in smooth domains was started with the well-known result of Agmon, Duglis and Nirenberg in 50s of last century. This theory says that the solution behaves smoothly relying on the smoothness of right-hand side,of the coefficients and on the regularity of boundary. In non-smooth domains, V. A. Kondratiev has initiated the reseach of the normal solvability and the asymptotical behavior of solution near conical points of boundary for elliptic systems. Following to this result, the russian mathematicians such as V. G. Mazya, B. A. Plamenievsky have achieved the analoguos results in more general Sobolev spaces. The similar theory for
non-stationary systems is very interesting that explores the method of study of the stationary systems in more abstract domains: dihedral domains, domains with cusps etc. In this moment, there is a lot of open questions for non-linear systems in non-smooth domain about the asymtotical behavior of solution.
This talk is devoted to the fisrt initial BVP for strongly parabolic systems in cylindrical domains of conical type. Some results on the unique solvability, the smoothness with respest to (x,t) and the
asymptotic expansion of generalized solution near conical point are obtained by the ellipticized method. The application in mathematical phisics is also brought to consider.
Speaker: Nguyen Manh Hung/ Hanoi University of Education/hungnmmath@hnue.edu.vn
Title: On the second BVP for strongly Schrodinger systems in conical cylinders
Abstract:
In this talk we consider the second nonhomogeneous initial boundary
value problem for strongly Schrodinger systems in cylinders with nonsmooth base.Some results on the unique solvability and on the smoothness with respect to time variable of generalized solution of this problem are given.
Speaker: Thomas Ivey
Title: Constructing Hopf Hypersurfaces in the Complex Hyperbolic Plane
Abstract:
Hopf hypersurfaces are real hypersurfaces in a complex
space form whose normal vector is related to a
principal direction by multiplication by the complex structure.
In complex projective space, Hopf hypersurfaces
are locally tubes over complex submanifolds,
and the same is true in complex hyperbolic space
when the principal curvature is large.
When the ambient space is the complex hyperbolic plane,
we have found a construction which gives all Hopf hypersurfaces
of small principal curvature, in terms of a surface
in a twistor space which is the product of two
3-dimensional spheres.
(The surface is the product of a pair of arbitrary contact curves.)
This construction also completes the classification
of pseudo-Einstein hypersurfaces.
This is joint work with Patrick J. Ryan of McMaster University.
Speaker: Lotfi Hermi
Title: On Riesz and Carleman Means of Eigenvalues
Abstract:
Riesz means are generalizations of the Weyl counting function
for eigenvalues. In this presentation, we prove a new monotonicity
principle for Riesz means of eigenvalues of the Dirichlet Laplacian, and discuss consequences for Riesz and Carleman means. This monotonicity principle complements the famous result of Aizenman-Lieb.
We also show that the Berezin-Li-Yau inequality arises, when its order
is greater than or equal to 2, from a class of universal inequalities
for these Riesz means and this new monotonicity result, as well as offer three alternative new routes to this inequality, in addition to the well-known proof by Ari Laptev (by himself, and with T. Weidl). We also prove the equivalence, again when the order of the Riesz mean is greater than or equal to 2, of the Berezin-Li-Yau inequality for the Riesz mean with the classical inequality of Mark Kac for the trace of the heat kernel.
Connections between various new and classical results are made via a
host of integral transforms such as the Laplace, Weyl, and
Riemann-Liouville fractional transforms, adding new tools to an already rich class of convexity and Legendre transform methods.
At the heart of some of these inequalities are pure commutator
techniques and sum rules. In the course of developing these inequalities we prove new Gaussian-type bounds for the spectral zeta function and conjecture new ones.
This is joint work with Professor Evans Harrell of Georgia Tech.
Speaker: Omid Ghayour and Sina Hedayatian
Title: Capacity on Finsler Spaces
Abstract:
Finsler space is the most natural and advanced generalization of Euclidean space, which has many applications in theoretical physics. The physical notion of capacity is the electrical capacity of a -dimensional conducting surface, which is defined as the ratio of a given positive charge on the conductor to the value of the potential on its surface.
The capacity of a set as a mathematical concept was introduced first by N. Wiener in 1924 and was subsequently developed by O. Forstman, C. J. de La Vallee Poussin and several other physicists and mathematicians in connection with the potential theory.
The concept of conformal capacity was introduced by Loewner and has been extensively developed for $R^n$. In particular it was used by G. D. Mostow to prove his famous theorem on rigidity of hyperbolic spaces. The concept of capacity on Riemannian geometry was introduced by J. Ferrand and developed in the joint work’s of M. Vuorinan and G. J. Martin.
We introduce the concept of capacity for Finsler spaces and prove that, it depends only on the conformal structure of $(M,g)$ , more precisely:
Theorem. Let $(M,g)$ be a connected non-compact Finsler manifold, then the capacity of a compact set on $M$ is a conformal invariant.
Speaker: Omid Ghayour and Sina Hedayatian
Title: Capacity on Finsler Spaces
Abstract:
Finsler space is the most natural and advanced generalization of Euclidean space, which has many applications in theoretical physics. The physical notion of capacity is the electrical capacity of a -dimensional conducting surface, which is defined as the ratio of a given positive charge on the conductor to the value of the potential on its surface.
The capacity of a set as a mathematical concept was introduced first by N. Wiener in 1924 and was subsequently developed by O. Forstman, C. J. de La Vallee Poussin and several other physicists and mathematicians in connection with the potential theory.
The concept of conformal capacity was introduced by Loewner and has been extensively developed for $R^n$. In particular it was used by G. D. Mostow to prove his famous theorem on rigidity of hyperbolic spaces. The concept of capacity on Riemannian geometry was introduced by J. Ferrand and developed in the joint work’s of M. Vuorinan and G. J. Martin.
We introduce the concept of capacity for Finsler spaces and prove that, it depends only on the conformal structure of $(M,g)$ , more precisely:
Theorem. Let $(M,g)$ be a connected non-compact Finsler manifold, then the capacity of a compact set on $M$ is a conformal invariant.
Speaker: Niels Martin Moeller
Title: Extremal zetas and determinants for Dirac operators and Branson's conjecture
Abstract:
In this talk I present new extremal results, also across conformal classes, of determinants and zeta functions of the square of the (Atiyah-Singer-)Dirac operator on a closed $m$-dimensional Riemannian spin manifold $M$. The main point is obtaining finite index of critical points, for the corresponding pseudodifferential Hessians. This involves the important problem of a general change of metrics of the Dirac operator. Dealing with this, we can compute the Hessian's leading symbol, using methods devised by Kate Okikiolu. In even dimensions the determinant's Hessian generally has a log-polyhomogeneous symbol and we state the needed results on hypoellipticity of symbols and lower semi-boundedness of spectrum. Also, a new viewpoint on the structure of the Hessian is provided by factoring out the (pseudodifferential) projections onto invariant directions. The analysis finally gives the pattern of essential extremum types as the dimension of $M$ varies. The pattern has a modulo 4 periodicity in dimension and together with results by K. Okikiolu it verifies, with finitely many exceptional directions, the modification to local, general variations of a conjecture by T. Branson, which stated originally in the global, conformal, round sphere case, that the determinant of the Atiyah-Singer-Dirac squared should have exactly opposite extremal behavior to that of the Yamabe operator.
Speaker: Josef Dorfmeister
Title: CMC-surfaces and loop groups
Abstract:
The discussion of surfaces of constant mean curvature (CMC) surfaces in R^3 has gained much momentum after the sensational work of Henry Wente. Similar to the minimal surface case, real analytic and more complex analytic approaches (generalized Weierstrass representation) coexist.
This talk will introduce the generalized Weierstrass representation for surfaces of constant mean curvature and illustrate it with concrete examples (and pictures). In particular, time permitting, a classification of CMC-trinoids of genus g = 0 will be presented.
Speaker: En-Bing Lin
Title: Multiscale Analysis of Symplectic Mechanics
Abstract:
We present the interplay between wavelet analysis and symplectic mechanics by studying the relationships between QMF systems, orbit methods and systems of Hamilton equations.
Speaker: Claude LeBrun
Title: On Conformally Kaehler, Einstein Manifolds
Abstract:
In this talk, I will describe recent joint work with X.X. Chen and
B. Weber which shows that any compact complex surface with
positive first Chern class admits an Einstein metric which is
conformally related to a Kaehler metric. The key new ingredient
is the existence of such a metric on the blow-up of the complex
projective plane at two distinct points.
Speaker: Ian Hawthorn
Title: Keeping it simple using $so(2,3)$.
Abstract:
The Lie group $SO(2,3)$ contracts to the Poincar\'e group and hence the two provide equally valid descriptions of local symmetry in a curved space. This general talk will look at the advantages of adopting this viewpoint consistently. We will see that both notation and interpretation tend to be simplified. In particular the Dirac equation arises naturally and is more easily understood. Considerations of curvature lead to a natural ten dimensional notational framework in which the equations of general relativity are simply expressed.
Speaker: Doojin Hong
Title: Translation to Bundle Operators
Abstract:
We give the explicit formulas for conformally invariant operators with leading term an m-th power of Laplacian on the product of spheres with the natural Lorentzian product metric by considering the corresponding object in the ambient space.
Speaker: Kate Okikiolu
Title: A negative mass theorem on the 2-torus
Abstract:
Consider the scalar Laplacian on a closed, compact surface M
with Riemannian metric g.
For a point p in M, the Robin mass m(p) is the value of the
Greens function G(p,q) at p=q after the logarithmic singularity
has been removed. When M is a 2-torus, we can find a metric
conformal to g having unit area, such that the Robin mass m(p) is
constant, and less than the value on the round 2-sphere of
unit area. This is a negative mass theorem on the 2-torus.
This result leads to the existence of sharp logarithmic HLS theorems and Onofri theorems on the 2-torus, and is also related to the positive mass theorem from relativity, and the extremal results for
the determinant of the Laplacian.
Speaker: Henrik Schlichtkrull
Title: Heat flow on Riemannian symmetric spaces
Abstract:
The application of harmonic analysis to analytic problems
on Riemannian manifolds was often in Tom's mind.
In this talk I will present an example of that.
The heat flow is the map that takes a function f in L^2(X)
to the function u(.,t) on X, where u(x,t) solves the heat
equation on X with initial condition u(x,0)=f(x). It was
observed by Segal and Bargmann, that when X=R^n, then the
image of L^2 by this map is a Hilbert space of holomorphic
functions on C^n.
I will discuss the possibility of extending this result to other Riemannian symmetric spaces X (by presenting results of
B.Hall, M.Stenzel, J.Mitchell, B.Kr"otz, G.Olafsson, R.Stanton
and others).
Speaker: Diego Moreira
Title: Least Supersolution Approach to Regularizing Fre Boundary Problems
Abstract:
In this talk, we discuss a free boundary problem obtained as a limit as
$\varepsilon \to 0$ to the following regularizing family of semilinear
equations $\Delta u = \beta_{\varepsilon}(u) F(\nabla u)$, where
$\beta_{\varepsilon}$ approximates the Dirac delta in the origin and $F$ is
a Lipschitz function bounded away from $0$ and infinity. The least
supersolution approach is used to construct solutions satisfying geometric
properties of the level surfaces that are uniform. This allows to prove that
the free boundary of the limit has the "right" weak geometry, in the measure
theoretical sense. By the construction of some barriers with curvature, the
classification of global profiles for the blow-up analysis is carried out
and the limit function is proven to be a viscosity and pointwise solution
(a.e) to a free boundary problem. Finally, the free boundary is proven to be
a $C^{1,\alpha}$ surface around $\mathcal{H}^{n-1}$ a.e. point.
Speaker: Bob Stanton
Title: Symplectic methods in Lie theory
Abstract:
I will report on joint work with Marcus Slupinski. For a finite dimensional simple Lie algebra, over a field of characteristic zero, endowed with a Heisenberg grading we give a symplectic description of the orbits in a naturally occuring prehomogeneous vector space. We then construct a tower of Lie subalgebras related to certain nilpotent adjoint orbits which refines the Jacobson-Morozov result. In particular, we extend the results of Hitchin on E6 to this generality.
Speaker: Jens Christensen
Title: Characterization of function spaces using representations
Abstract:
This is a presentation of a representation theoretic approach to characterization of classical function spaces. The approach is
linked to square integrable representations. The characterized spaces include modulation spaces (related to the Short Time Fourier Transform) and Besov spaces (widely used in wavelet theory and PDE's). This is ongoing work joint with Gestur Olafsson.
Speaker: Rafe Mazzeo
Title: CMC Foliations and Area Minimizing Hypersurfaces in AH manifolds
Abstract:
I will first describe a general result about the existence of sequences of constant mean curvature hypersurfaces, or foliations by such hypersurfaces, converging to infinity in general conformally compact manifolds. The key to this construction is the reduction of this problem in extrinsic geometry to a prescribed scalar curvature problem in the conformal infinity on the boundary. This is joint work with Frank Pacard.
I will then describe new work with Spyros Alexakis concerning the moduli space of complete proper area minimizing hypersurfaces in Poincare-Einstein spaces and extremal properties of the renormalized area functional.
Speaker: Changzheng Qu
Title: Invariant Flows in Klein Geometry
Abstract:
In this talk, we study motions of curves and surfaces in Klein geometry. First, we show that many integrable equations are associated with invariant curve or surface motions in Klein geometry. Second, we show that Ermakov systems arise from curve flows in affine geometry or conformal geometry. The symmetric property to solutions of some Ermakov systems is also studied. Finally, motions of curves in geometries $S^1 \times R$ and geometric inequalities related to isometry groups of the geometries are studied.
Speaker: Justyn
Title: Nikko
Abstract:
68eb8466ef76f125ea4f224c6f60eb86
http://njdokj.info/55d05e529a8f1e8a6bb81a24a58a9891/68eb8466ef76f125ea4f224c6f60eb86
http://njdokj.info/55d05e529a8f1e8a6bb81a24a58a9891/68eb8466ef76f125ea4f224c6f60eb86
[url]http://njdokj.info/55d05e529a8f1e8a6bb81a24a58a9891/68eb8466ef76f125ea4f224c6f60eb86[url]
