# 2005 Midwest Geometry Conference

## Ohio State University, Columbus

## April 29 - May 1, 2005

# ABSTRACTS

## Nicolae Anghel

#### Department of Mathematics, University of North Texas, Denton, TX 76203

*The spectrum of spherical Dirac-type operators*

###
A general way of investigating the spectrum of operators induced on Euclidean
spheres by graded Euclidean Dirac-type operators will be presented. The key
ingredients in accomplishing this will be a separation of variables and a
description of the spaces of homogeneous harmonic functions on punctured
Euclidean spaces. Standard results regarding the spectrum of the classical
spherical Dirac operator or the spherical Laplace-Beltrami operator will then
be easily recovered.

## Ivan Avramidi

#### Department of Mathematics, New Mexico Tech, Socorro, NM 87801

*Spectral asymptotics in matrix geometry*

### We describe a “matrix” generalization of Riemannian geometry when
instead of a Riemannian metric there is a matrix valued self-adjoint
symmetric two-tensor that plays a role of a “non-commutative” metric.
We construct invariant first-order and second-order self-adjoint
elliptic partial differential operators, that can be called
“non-commutative” Dirac operator and non-commutative Laplace
operator.We construct the corresponding heat kernel for the non-commutative
Laplace type operator and compute its first two spectral invariants. A
linear combination of these two spectral invariants gives a functional
that can be considered as a non-commutative generalization of the
Einstein-Hilbert action. The extremals of this functional naturally
define the “non-commutative” generalization of Einstein spaces.

## Jean Cortissoz

#### Department of Mathematics, University of Toledo, Toledo, OH 43606

*Some remarks on the Ricci flow in manifolds with boundary*

### In this talk I will discuss a Boundary Value Problem (BVP) for the Ricci
flow in three manifolds with boundary. I will sketch proofs of short time
existence and subconvergence of the solution of this BVP on certain family of
rotationally symmetric metrics on the ball. Also, I will digress a little on a
possible approach to face more general cases.

## Gordon Craig

#### Department of Mathematics, Bishop’s University, Lennoxville,
Québec J1H 1A8, Canada

*Dehn filling and asymptotically hyperbolic Einstein manifolds*

### I will discuss a construction which allows us to find infinitely many
nonhomotopic asymptotically hyperbolic Einstein manifolds with the same
conformal structure at infinity.

## Daniel C. Galehouse

#### Department of Physics, University of Akron, Akron, OH 44230

*Spinor coordinate space and Dirac theory*

### Coordinates from a real, conformally flat, eight-dimensional Riemannian
space are combined pairwise to give a four element complex valued vector that
transforms locally as a spinor. If the scalar curvature is set to zero, a
linear transformation to five dimensions, using anti-commuting matrices,
generates the Dirac equation. The single particle wave function is given by
the spinor gradient of the three-halves power of the conformal factor.
Gravitational and electromagnetic interactions are carried over from
five-dimensional theory. Additional physical effects are evident in the
geometry, and can be shown to involve the transmutation of an electron into a
neutrino by the weak interaction. Further developments may allow for the
description of more complicated phenomenology in particle physics.

## Igor Mineyev

#### Department of Mathematics, University of Illinois at Urbana-Champaign,
Urbana, IL 61801

*Metric conformal structures in hyperbolic groups*

### For any (uniformly locally finite) hyperbolic complex
*X* I will present a construction of a visual metric
$\check{d}$ on the ideal boundary of *X* that
makes the Isom(*X*)-action on the boundary bi-Lipschitz,
Möbius, and conformal. All this in particular applies to groups that are
hyperbolic in the sense of Gromov.
The definition of $\check{d}$ is based on a “nice”
metric $\hat{d}$ on a hyperbolic group that was constructed in a
joint paper with G. Yu. On a very informal level, the averaging process
involved in the construction of these metrics can be thought to resemble the
Ricci flow on manifolds, though we work with cell complexes and in the absence
of any Riemannian structure. The visual metric is of interest in particular
because of its relation to the Cannon's conjecture that is a group theoretic
analog of Thurston's hyperbolization conjecture for 3-manifolds.

## Jason Parsley

#### Department of Mathematics, University of Georgia, Athens, GA 30605

*Biot-Savart operator, helicity, and linking on
S*^{3}

### We extend the Biot-Savart law from physics to an operator BS
acting on all vector fields on *S*^{3}, a geometric
setting for electrodynamics in positive curvature. We show that Maxwell’s
equations hold and that BS acts as a right inverse to curl. We
then discuss its application to energy-minimization problems in geometry and
physics that depend on curl eigenvalues. Also, BS allows us to
construct linking integrals on *S*^{3}. As one further
application, we can express the helicity of a vector field, a measure of how
much it coils around itself, as
*H*(*V*) = <*V*, BS(*V*)>. We
find upper bounds for helicity on the three-sphere; our bounds are not sharp
but within an order of magnitude.

## Victor Patrangenaru

#### Department of Mathematics and Statistics, Texas Tech University, Lubbock,
TX 79409

*Metric classification of geometries of positive Ricci curvature
in* 3*D*

### Using Cartan's approach, we give an explicit formula of the metric tensor
of a simply connected 3*D* homogeneous space of positive
scalar curvature in terms of the principal Ricci curvatures at one point. The
principal Ricci curvatures at a single point fully determine such an isometry
class of a 3*D* Riemannian homogeneous space, and each
isometry class can be represented by a Lie group with a left invariant metric.
The problem whether a 3*D* compact connected simply
connected manifold has a Lie group structure, raised in my paper "On the
3*D* Riemannian homogeneous spaces of positive sectional
curvature" [Algebra Geom. Appl. Semin. Proc. 2(2002), 5-13], is an open
alternative to the Ricci flow approach to Poincaré’s conjecture.

## Alexander Ramm

#### Mathematics Department, Kansas State University, Manhattan, KS 66506

*On symmetry problems*

### Suppose that *S*, a smooth surface homeomorphic to a sphere,
is the boundary of a domain *D*, and the overdetermined problem
(*L*+*k*^2) *u* = 0 in
*D*, *u* = 1 on
*S* and *u*_*N* = 0 on
*S* has a solution. Does it follow that
*S* is a sphere? This is a problem equivalent to the
Pompeiu problem. Other problems of this type are considered.

## Xiaochun Rong

#### Department of Mathematics, Rutgers University, Piscataway, NJ 08854

*Fundamental groups of positively curved manifolds with symmetry*

### We will report a recent progress in the study of positively curved
manifolds which admit isometric actions by abelian groups. This is joint work
with Yusheng Wang.

## John Ryan

#### Department of Mathematics, University of Arkansas, Fayetteville, AR 72701

*Ahlfors matrices, fundamental domains in n
real variables and hyperbolic Dirac operators*

### In this talk we will use a construction of Ahlfors using Clifford algebras
to construct Möbius transformations in *n* real
variables. These in turn will be used to introduce analogues of fundamental
domains for analogues of subgroups of the modular group, SL(2,**Z**).
These are examples of conformally flat manifolds. A Dirac operator with
respect to the hyperbolic metric on these spaces will be set up and the
function theory, including Hardy spaces, for these operators will be examined.

## Ralf Spatzier

#### Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

*About the classification of Anosov actions*

### The classification of Anosov systems is a deep and central problem in
dynamics, going back to the 70's for the case of single Anosov
diffeomorphisms. While this conjecture is outstanding, much progress has been
made for the case of commuting Anosov diffeomorphisms and Anosov
*R*^{n}-actions. Essentially, one conjectures that such
actions are smoothly conjugate to algebraic actions. We prove this for the
special case of Cartan actions under additional technical assumptions.

## Joseph Towe

#### Department of Physics, The Antelope Valley College, Lancaster, CA 93536

*A vacuum in which super-unification preserves quark triplets*

### A specific vacuum (of the hererotic superstring) is postulated in which
quark-lepton transitions occur only as loop corrections in the asymptotic
limit of QCD. ‘Super-unified’ interactions therefore preserve quark
triplets so that proton decay is not indicated. The vacuum from which
interacting fields emerge is characterized by a specific 3-dimensional
representation of the SU(3) sub-group of SO(6). The proposed model
accounts for exactly three fermionic generations, and becomes a theory by
predicting a new quark - a left-handed (non-strange) version of the strange
quark.

## Lina Wu

#### Department of Mathematics, University of Oklahoma, Norman, OK 73019

*Regularity for stable-stationary p-harmonic maps into an
ellipsoid*

### In this paper we investigate the regularity of stable-stationary
*p*-harmonic maps from an open set in an Euclidean space
into an ellipsoid. By assuming that the target manifolds do not carry any
homogeneous properties, we obtain some compactness results and regularity
theorems. As an application we prove that the stationary
*p*-harmonic maps into an ellipsoid are regular, except possibly
for a closed singular set of
(*n* - *p*)-dimensional Hausdorff measure zero.