2005 Midwest Geometry Conference

Ohio State University, Columbus

April 29 - May 1, 2005


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  S3

We extend the Biot-Savart law from physics to an operator  BS  acting on all vector fields on  S3, 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  S3. 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  3D

Using Cartan's approach, we give an explicit formula of the metric tensor of a simply connected  3D  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  3D  Riemannian homogeneous space, and each isometry class can be represented by a Lie group with a left invariant metric. The problem whether a  3D  compact connected simply connected manifold has a Lie group structure, raised in my paper "On the  3D  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  Rn-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.

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