We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for an arbitrary compact semisimple gauge group in the so-called regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang-Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity. * Supported in part by NSERC grant A8059.

We prove that there exists a class of non-stationary solutions to the Einstein-Euler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of the Einstein-Euler equations which contains a singular parameter ǫ = vT /c where vT is a characteristic velocity scale associated with the fluid and c is the speed of light. The symmetric hyperbolic formulation allows us to derive ǫ independent energy estimates on weighted Sobolev spaces. These estimates are the main tool used to analyze the behavior of solutions in the limit ǫ ց 0.

We discuss certain recent mathematical advances, mainly due to Perelman, in the theory of Ricci flows and their relevance for renormalization group (RG) flows. We consider nonlinear sigma models with closed target manifolds supporting a Riemannian metric, dilaton, and 2-form B-field. By generalizing recent mathematical results to incorporate the B-field and by decoupling the dilaton, we are able to describe the 1-loop β-functions of the metric and B-field as the components of the gradient of a potential functional on the space of coupling constants. We emphasize a special choice of diffeomorphism gauge generated by the lowest eigenfunction of a certain Schrödinger operator whose potential and kinetic terms evolve along the flow. With this choice, the potential functional is the corresponding lowest eigenvalue, and gives the order α ′ correction to the Weyl anomaly at fixed points of (g(t), B(t)). The lowest eigenvalue is monotonic along the flow, and since it reproduces the Weyl anomaly at fixed points, it accords with the c-theorem for flows that remain always in the first-order regime. We compute the Hessian of the lowest eigenvalue functional and use it to discuss the linear stability of points where the 1-loop β-functions vanish, such as flat tori and K3 manifolds.

We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.Constants of this type will always be non-negative, non-decreasing, continuous functions of their arguments.Given four vector bundles V , W , Y and Z that sit over Σ, and mapsFor situations, where we want to bound f (t, w, v) by g(t, v) up to an undetermined constant of proportionality, we defineif there exists a R ∈ (0, R) and a map f ∈ C 0 [T 0 , 0), C ∞ (B R (W ) × B R (V ), L(Y, Z))such that