We consider the conformal decomposition of Einstein's
constraint equations introduced by Lichnerowicz and York, on a
compact manifold with boundary. We show that there exists a solution
to the coupled Hamiltonian and momentum constraint equations when the
derivative of the mean extrinsic curvature is small enough, and
assuming that the Ricci scalar of the background metric is bounded,
though it can change sign on the manifold. The solutions are in
general not uniquely determined by the source functions and boundary
data. The proof technique is based on finding barriers for the
Hamiltonian constraint equation which are independent of the
solutions of the momentum constraint equation, and then using
standard fixed-point methods for increasing operators in Banach
spaces. This work generalizes a previous work by Isenberg and
Moncrief on closed manifolds.

A major advance in combinatorics occured in 1972,
when E. Szemeredi provided a complicated and ingeneous combinatorial
proof of the Erd\"os-Turan conjecture: every subset

Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

\noindent Suppose we view the three-dimensional sphere as: $S^3 = \{(z,w) \subset \mathbb{C}^2|\ |z|^2 + |w|^2 = 1\}. $ If we are given a complex curve $V_f = \{(z,w)|0 = f(z,w) \in \mathbb{C} [z,w]\},$ we can then examine the intersection $K = V_f \cap S^3.$
In the transverse case, this intersection $K$ will be a link i.e. an embedded
one-manifold in the three-sphere. This talk will be interested in the question:
Question: Which links can arise from complex curves in the above manner?
I will discuss the history of this problem, focusing first on the case where
$f(z,w)$ has an isolated singularity at the origin where the question is completely
answered. I’ll then discuss how a powerful set of knot invariants
defined by Ozsvath and Szabo and independently by Rasmussen using the
theory of pseudo-holomorphic curves can provide information on the above
question. More precisely, Ozsvath and Szabo and Rasmussen defined a numerical
invariant of knots, denoted $\tau(K)$, which we show provides an obstruction
to knots arising in the above manner. More surprisingly, suppose
we focus on knots whose exteriors, $S^3 - K$, admit the structure of a fiber
bundle over the circle, the so-called $fibered$ knots. In this case we show
that $\tau(K)$ detects exactly when a fibered knot arises as the intersection of
the three-sphere with a complex curve satisfying a certain genus constraint.
Our proof relies on connections between Ozsvath-Szabo theory and certain
geometric structures on three-manifolds called contact structures.