Georgia Institute of Technology College of Sciences

The Ptolemy coordinates are efficient coordinates for computingboundary-unipotent representations of a 3-manifold group in SL(2,C). Wedefine a slightly modified version which allows you to computerepresentations that are not necessarily boundary-unipotent. This givesrise to a new algorithm for computing the A-polynomial.

Suppose that F is a field with p elements, and let G be the finite-index congruence subgroup of SL(n, F[t]) obtained as the kernel of the homomorphism that reduces entries in SL(n, F[t]) modulo the ideal (t). Then H^(n-1)(G;F) is infinitely generated. I'll explain the ideas behind the proof of the above result, which is a special case of a result that applies to any noncocompact arithmetic group defined over function fields.

A non-trivial group G is called left-orderable if there exists a strict total ordering < on its elements such that g

The question of what conditions guarantee that a symplectic$S^1$ action is Hamiltonian has been studied for many years. Sue Tolmanand Jonathon Weitsman proved that if the action is semifree and has anon-empty set of isolated fixed points then the action is Hamiltonian.Furthermore, Cho, Hwang, and Suh proved in the 6-dimensional case that ifwe have $b_2^+=1$ at a reduced space at a regular level $\lambda$ of thecircle valued moment map, then the action is Hamiltonian. In this paper, wewill use this to prove that certain 6-dimensional symplectic actions whichare not semifree and have a non-empty set of isolated fixed points areHamiltonian. In this case, the reduced spaces are 4-dimensional symplecticorbifolds, and we will resolve the orbifold singularities and useJ-holomorphic curve techniques on the resolutions.