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We construct classical tilting objects in derived categories of equivariant sheaves on quasi-projective varieties, which give equivalences with derived categories of modules over algebras. Our applications include a conceptual explanation of the importance of the McKay quiver associated to a representation of a finite group G and the development of a McKay correspondence for the cotangent bundle of the projective line. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-09-04 14:42:25.099

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Six experiments tested the assumption that, in visual perception, observers have reliable and direct access to the equivalence of shapes in projective geometry (I call this "the invariance hypothesis in the theory of shape constancy"). This assumption has been made in the study of vision since Helmholtz's time. Two experiments tested recognition of the projective equivalence of planar shapes. In another four experiments, subjects estimated the apparent shape of a solid object from different perspectives. Departure from projective equivalence was assessed in each study by measuring the cross ratio for the plane. This measure of projective invariance is new to perceptual research. Projective equivalence was not found to be perceived uniformly in any of the studies. A significant effect of change in perspective was found in each study. These results were construed as supporting the classical theory of depth cues against the invariance hypothesis.

Orientation – the process by which animals determine their position in an environment – can be accomplished by using the visually distinct properties of objects or surfaces, known as features (i.e., colour or pattern) or the relationship among objects and surfaces, known as geometry (i.e., wall length or angular information). Although features have been shown to facilitate the encoding of geometry, little is known as to whether restricting one’s viewpoint to include fewer features will still facilitate the encoding of geometry. During this experiment, men and women were trained to search near either an acute or an obtuse corner of a virtual parallelogram-shaped room that contained either three or four discrete and distinctive features. Participants were subsequently tested for their encoding of wall length and angles when the cues were presented in isolation, together, or in conflict. Results showed that the number of features present during training did not influence the encoding of geometry. However, the discrete and distinctive properties of the features overshadowed the encoding of angles by women as well as by participants who were trained with the obtuse corner. Although some groups of participants did not encode angular information when this was the only available geometric cue, all groups weighed angles more heavily than wall length when the cues provided conflicting information. This result suggests that one type of geometric cue (i.e., wall length) can facilitate the encoding of another (i.e., angles).

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Any Piecewise-Linear (PL) surface can be formed from a regular polygon (including the interior) with an even number of edges, where the edges are identified in pairs to form a two-dimensional manifold. The resulting surfaces can be distinguished by algebraic means. An analysis of the construction algorithm can also be used to determine the resulting surface. Knowledge of the polygon used can also yield information about the surfaces formed.In this thesis, an algorithm is developed that will analyze all possible edge pairings for an arbitrary regular polygon. The combination of this data, along with known techniques from geometric topology, will categorize the constructions of these PL surfaces. A procedure using matrices is developed that will determine the Euler number and establish which algebraic words are equivalent.This topic extends to two-dimensional manifolds a classical method of analysis for three-dimensional manifolds. It therefore provides a more geometrical approach than has traditionally been used for two dimensional surfaces. / Department of Mathematical Sciences