1. Initial Data Set formulation The initial data set is given by (gij, pij) on a three dimensional manifold where gij is asymptotically euclidean (so that asymptotic quantities can be defined) and pij falls off quadratically. We also study the case when g^ is asymptotically hyperbolic and the three manifold is asymptotically null. Several important questions remain unsolved for the initial data set and its Cauchy development.

A bstract We conjecture that every three dimensional canonical singularity defines a five dimensional N N= 1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a) a chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b) Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

We study Hawking radiation from three-dimensional black holes. For this purpose the emission of charged scalar and charged fermionic particles is investigated from charged BTZ black holes, with and without rotation. We use the quantum tunneling approach incorporating WKB approximation and spacetime symmetries. Another class of black holes which is asymptotic to a Sol three-manifold has also been investigated. This procedure gives us the tunneling probability of outgoing particles, and we compute the temperature of the radiation for these black holes. We also consider the quantum tunneling of particles from black hole asymptotic to Sol geometry.

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We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+1 dimensions, based on a generic finite group G and a 4-cocycle ω over G. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z) generators as the modular S and T matrices of the ground states, which yield a set of topological quantum numbers classified by ω and quantities derived from ω. Our model fulfills a Hamiltonian extension of the 3+1-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group G. This work is presented to be accessible for a wide range of physicists and mathematicians.