Multifold sums and products over R, and combinatorial problems on sumsets

We prove a new bound on a version of the sum-product problem studied by Chang. By introducing several combinatorial tools, this expands upon a method of Croot and Hart which used the Tarry-Escott problem to build distinct sums from polynomials with specific vanishing properties. We also study other aspects of the sum-product problem such as a method to prove a dual to a result of Elekes and Ruzsa and a conjecture of J. Solymosi on combinatorial geometry. Lastly, we study two combinatorial problems on sumsets over the reals. The first involves finding Freiman isomorphisms of real-valued sets that also preserve the order of the original set. The second applies results from the former in proving a new Balog-Szemeredi type theorem for real-valued sets.Ph.D