## Departmental PhD Thesis Exam – James Uren

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 $(S1)^{3g-3+n'}$ 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.