Curvature and the Shape of the Universe

We may have an intuitive idea of what it means for surfaces to be curved, but what does it mean for higher dimensional spaces to be curved? In this talk, we will try to quantify curvature on a surface and try to extend this notion to three dimensional spaces. We will show that with an understanding of curvature, we can make sense of a universe which is finite but without boundary

Riemannian Geometry (Abstract)

Riemannian Geometry studies the geometry of curved spaces. It originated with the ideas of the Bernhard Riemann in the 19th century extending Gaussian geometry, or the study of geometry of curves and surfaces contained in 3 dimensional Euclidean space

Curvature (Abstract)

We may have an intuitive idea of what it means for surfaces to be curved, but what does it mean for higher dimensional spaces to be curved

Floquet calculations of rates of frequency conversion and multiphoton Ionisation in intense laser field

Nonperturbative rates of harmonic generation or frequency mixing and rates of multiphoton ionisation were obtained for atomic hydrogen and for a one-electron model of the negative hydrogen ion, using the Floquet-Sturmian method. The following cases were investigated: elliptically polarised, monochromatic incident field, two-colour incident field with incommensurable frequencies, coherent super position of an incident field and its third harmonic, and superposition of a laser field and a static electric field. For elliptical polarisation, the ellipticity angle of the harmonics differ from the ellipticity of the incident laser and we established that there is an offset angle between the major axis of the incident laser's polarisation ellipse and that of the harmonics generated. The variation is greatest in the plateau region. Resonance-enhanced ionisation was studied when a high frequency field was applied in addition to a fundamental field. In contrast to these changes in the ionisation rate only small variations have been obtained in the harmonic generation rate. Results for frequency-mixing exhibit a difference in the strength of the harmonics, although the behaviour is the same in terms of the length of the plateau and the cut-off order. Also we found a marked change between sum-frequency and difference frequency processes, which depends on the harmonic order. In addition to a strong production of even harmonics, it was found that the variation of the rate of ionisation and harmonic generation in the presence of a static field oscillates as a function of the strength of the latter. In the case of hydrogen, we observed dc-stark shift induced resonance enhancements. The results obtained for H(^-) are in very good agreement with those obtained for two-colour (w, 3w) mixing. We concluded that the application of a static field in general leads to a reduction in the conversion efficiency

Curvature and the Shape of the Universe

We may have an intuitive idea of what it means for surfaces to be curved, but what does it mean for higher dimensional spaces to be curved? In this talk, we will try to quantify curvature on a surface and try to extend this notion to three dimensional spaces. We will show that with an understanding of curvature, we can make sense of a universe which is finite but without boundary