Topics in algebraic geometry

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This research is centered around principal bundles on algebraic varieties. In studying bundles, tangential subjects of independent interest, such as derived categories, K-theory and Gromov-Witten invariants are touched upon. These subjects have found applications in mathematical physics and we intend to both contribute to and be guided by this literature. This research makes systematic use of algebraic stacks. These are geometric objects with added symmetries. Their virtue lies in that many problems have their natural solutions as algebraic stacks rather than spaces. Stacks show up in the work as parameter spaces and as curves with added structure. The next tool that we make use of is the derived category and its various enhancements. The derived category is a kind of abelianizing object. It takes a complicated object and replaces it with something more accessible which at the same time encapsulates all its important invariants and properties. Using these tools we aim to study three fundamental questions. The first question is of basic interest to number theorists. General theory says that certain curves and maps can be defined via equations that have coefficients in a finite extension of the rational numbers. However not much is known about which particular extension. In other words, we have an existential theorem that we wish to make constructive. The second question is about counting are calculation of a particular number. We wish to count maximal subbundles of a fixed parabolic bundle. The approach that we intend to use will make use of ideas originating in mathematical physics. We intend to generalise some important combinatorial formulas in the process of the calculation. The last question that will be addressed is about calculating parameters. We have an important universal object, the moduli stack of principal bundles that appears in many parts of mathematics and physics. A basic question that can be asked about it is, how many parameters does in generally depend on. Given the objects ubiquity, this is an important question that should be addressed.