[en] The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.

[en] Compasses are called rusty, if one can draw only the unit circle with them. We prove that: From two points A and B, with only rusty compasses, one can draw the points of k-section of AB, and all the vertices of a regular n-gon which has a side AB, where k is any integer greater than 1, and n=3,4,5,6,8,12,17,257,..etc. Generally, let A be (0,0) and B be (λ,0), then one can draw all the points (λx,λy) where x and y are any elements in some regular 2^m-extension of the rational field, for m=1,2,3,... (author). 10 refs, 11 figs

[en] The space-time connections giving rise to the same curvature tensor are constructed and the corresponding geometries compared. The notion of gauge and non-gauge copies in the context of tetrad formulation is elucidated and illustrated by an explicit calculation. Some comments are made on the copies in Einstein-Cartan and Weyl-Cartan geometries. (Author)

[en] Published in summary form only

[en] In this paper we develop a picture of Quantum Mechanics based on the description of physical observables in terms of expectation value functions, generalizing thus the so called Ehrenfest theorems for quantum dynamics. Our basic technical ingredient is the set of tools which has been developed in the last years for the geometrical formulation of Quantum Mechanics. In the new picture, we analyze the problem of the dynamical equations, the uncertainty relations and interference and illustrate the construction with the simple case of a two-level system.

[en] A comprehensive introduction to the field of supersymmetry and supergravitation is provided. A chapter on tetrad formulation and Einstein-Cartan theory of gravitation has been included to make the exposition self-contained. (Author)

[en] Using the Lamb formalism, we show that some completely integrable homogeneous and inhomogeneous nonlinear Schroedinger (NLS) type equations such as derivative NLS, extended NLS, higher-order NLS, inhomogeneous NLS, circularly and radially symmetric NLS, and generalized inhomogeneous radially symmetric NLS equations can be related to certain types of moving helical space curves. copyright 1997 The American Physical Society

[en] Problems of investigating the Universe space-time geometry are described on a popular level. Immediate space-time geometries, corresponding to three cosmologic models are considered. Space-time geometry of a closed model is the spherical Riemann geonetry, of an open model - is the Lobachevskij geometry; and of a plane model - is the Euclidean geometry. The Universe real geometry in the contemporary epoch of development is based on the data testifying to the fact that the Universe is infinitely expanding

[en] Plane-symmetric Lorentzian manifolds are classified according to their homothetic vector fields and metrics or classes of metrics. It is found that these manifolds can admit 4-, 5-, 7- and 11-dimensional homothetic Lie algebras. Explicit self-similar metrics are obtained corresponding to the homothetic Lie algebras of dimensions greater than or equal to five. For 4-dimensional homothetic Lie algebras, there appear classes of self-similar metrics, with metric coefficients related by non-linear differential constraints. Explicit examples are provided to ensure the existence of proper homothetic vector fields in the cases where the metrics are given implicitly. Later, using the classification, self-similar metrics are found which represent perfect-fluid solutions of the Einstein field equations.

[en] The emergence of the new non-Euclidean geometry of Bolyai, Gauβ , and Lobachevskii (BGL) and its impact on modern sciences is the subject of a series of biennial conferences. Below, I briefly review the history. (author)