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Everyone welcome. Refreshments will be served in the Math Lounge before the exam. DEPARTMENTAL PHD THESIS EXAM ---------------------------- Friday, June 24, 2011, 2:10 p.m., in BA 6183, 40 St. George Street PhD Candidate: James Uren PhD Advisors: Lisa Jeffrey and Paul Selick Thesis Title: Toric Varieties Associated with Moduli Spaces Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a hamiltonian action of a compact torus $(S^{1})^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$ A copy of the thesis can be obtained by contacting jjuren@math.toronto.edu.

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