Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Ricci flows on surfaces related to the Einstein weyl and Abelian vortex equations

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

On the chaotic instability of a nonsliding liquid-filled top with a small spheroidal base via Melnikov-Holmes-Marsden integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.

Optimal control, geometry, and quantum computing

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.

Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 19-dimensional data sets.

Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.

Constructing localized non-linear projection manifolds in a principled way:hierarchical generative topographic mapping

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis (Bishop98a) in several directions: 1. We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. 2. We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. 3. Using tools from differential geometry we derive expressions for local directionalcurvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model.We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set andapply our system to two more complex 12- and 19-dimensional data sets.

On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

∗ Partially supported by INTAS grant 97-1644

On the Stabilization of the Wave Equation by the Boundary

* Partially supported by CNPq (Brazil)

On the 3-Wave Equations with Constant Boundary Conditions

2010 Mathematics Subject Classification: 37K40, 35Q15, 35Q51, 37K15.

Denoising Manifolds for Dimension

2000 Mathematics Subject Classification: 68T01, 62H30, 32C09.

Contact CR-Submanifolds of Kenmotsu Manifolds

2000 Mathematics Subject Classification: 53C15, 53C42.

Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.

Tamed and Compatible Symplectic Forms on Four-Dimensional Almost Complex Lie Algebras

We are able to give a complete description of four-dimensional Lie algebras g which satisfy the tame-compatible question of Donaldson for all almost complex structures J on g are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are tamed but not compatible with symplectic forms.? Note that Donaldson asked his question for compact four-manifolds. In that context, the problem is still open, but it is believed that any tamed almost complex structure is in fact compatible with a symplectic form. In this presentation, I will define the basic objects involved and will give some insights on the proof. The key for the proof is translating the problem into a Linear Algebra setting. This is a joint work with Dr. Draghici.