Recent techniques from Bridgeland Stability have given new, interesting results about stable vector bundles on surfaces. However, in higher dimensions the theory is substantially harder, so there has been significantly less progress in this case. In this thesis, we develop theory for a related notion-tilt stability-then apply this theory to stable vector bundles.In the first part of this thesis, we recall and further develop the theory of tilt stability. This development culminates in a wall-crossing result for tilt stability.In the second part of this thesis, we apply our wall-crossing result to study stable vector bundles. Our first application is a criterion for when the restriction of a slope stable bundle to an integral subvariety is still slope stable. Our second application is to the theory of Lazarsfeld-Mukai bundles. Specifically, we show the Lazarsfeld-Mukai bundle associated toa Gieseker stable bundle is slope stable, and that slope stability and slope semistability are equivalent for Lazarsfeld-Mukai bundles associated to ample line bundles.
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