Principal bundles in algebraic geometry

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When confronted with a complex problem one technique that has proven useful is to exploit its symmetries. In many naturally occurring problems the collection of symmetries is often quite large, infinite in nature. To understand the collection of symmetries in this situation, it is useful to see if the collection symmetries, has extra structure. One such structure is that the collection be defined by some polynomial equations. This is known as an algebraic group in mathematics. This proposal is about study the geometry of algebraic groups and the spaces that they act upon. The geometry can often be very complicated and we need the appropriate machinery to be able to say anything about these spaces. One such machine is homological algebra. This proposal seeks to use such homological techniques to further the study of algebraic groups and their associated spaces. The most immediate consequences of this work will be felt in mathematics. Secondary effects will be felt in mathematical physics which has seen a merger with algebraic geometry in recent decades. Part of the proposal is about rational algebraic varieties. These objects have seen applications in the computer age, for example to computer aided design where they can be used to compute intersections quickly.