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I will discuss a novel framework of the very early universe which addresses the traditional horizon and flatness problems of big bang cosmology and predicts a scale invariant spectrum of perturbations. Unlike inflation, this scenario requires no exponential superluminal expansion of space-time. Instead, the early universe is described by a conformal field theory minimally coupled to gravity. The conformal fields develop a time-dependent expectation value which breaks the flat space so(4,2) conformal symmetry down to so(4,1), the symmetries of de Sitter, giving perturbations a scale invariant spectrum. The solution is an attractor, at least in the case of a single time-dependent field. Meanwhile, the metric background remains approximately flat but slowly contracts, which makes the universe increasingly flat, homogeneous and isotropic. The essential features of the scenario depend only on the symmetry breaking pattern and not on the details of the underlying lagrangian.

I analyze the various roles of infinity in current thinking about cosmology. Topics include initial conditions, attractor behavior, inflation and the precision and meaning of quantum measurements. I review the de Sitter equilibrium cosmology as an example of a finite cosmology, and present some new predictions that permit observable tests.

In many respects, de Sitter space behaves like a system at finite temperature in finite volume. I will extend this to include the lack of first-order phase transitions. This rules out exponential decay in the de Sitter landscape, which changes the global structure in a significant way.

Inflationary cosmology not only provided a simple solution to various cosmological problems, but also made predictions later confirmed by observations. Despite of its success, a straightforward extrapolation of the theory to higher energy scales led to new problems and seems to require new physics. In this talk I review the new problems, discuss their possible resolutions and speculate on possible predictions of the new physics.

I will introduce a simple 6d model of flux compactification that shows a remarkable rich landscape of vacua with different number of large and compact dimensions. I will then describe the instantons interpolating between these different vacua as well as some the implications of a transdimensional multiverse of this form.

In its best understood version, the Steinhardt-Turok cyclic universe contains two crucial ingredients: an unstable field trajectory during the ekpyrotic phase, and the subsequent brane collision corresponding to the crunch/bang transition. These two features act as strong selection principles and determine the broad physical properties of the universe emerging from the bang. As such, they significantly alleviate (and perhaps resolve) the measure problem that is inherent to all cosmological models that produce universes with a range of physical properties.

Galaxy clusters are the biggest gravitationally bound structures in the Universe. Simple features of these objects can help us reconstruct the initial conditions at the Big-Bang and test the fundamental laws of physics.

If the universe is a quantum mechanical system it has a quantum state. This state supplies a probabilistic measure for alternative histories of the universe. During eternal inflation these histories typically develop large inhomogeneities that lead to a mosaic structure on superhorizon scales consisting of homogeneous patches separated by inflating regions. As observers we do not see this structure directly. Rather our observations are confined to a small, nearly homogeneous region within our past light cone. This talk will describe how the probabilities for these observations can be calculated from the probabilities supplied by the quantum state without introducing a further ad hoc measure.

In this talk, I will describe how the collision of Minkowski or crunching bubbles can re-start inflation in a portion of the bubble interior. Consistent with various singularity theorems, such collisions can only seed a lasting inflationary phase with energy density lower than that of the parent vacuum.