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The standard Hamiltonian formulation of (first order) gravity breaks manifest covariance both in its retention of the Lorentz group as a local gauge group and in its discrepant treatment of spacelike and timelike diffeomorphisms. Here we promote more covariant alternatives for canonical quantum gravity that address each of these problems, and discuss the implications for both the classical and the quantum theory of gravity. By retaining the full local Lorentz group, one gains significant insight into the geometric and algebraic properties of the Hamiltonian dynamics. As an example, we discuss the possibility of computing the internal spin angular momentum of asymptotically flat spacetimes, which may lead to insight into the nature of spin in quantum gravity. By treating the spacelike and timelike diffeomorphisms on equal footing, using techniques from geometric quantization we find a new representation of the quantum constraints where the total Hamiltonian is kinematical in the same vein as the Gauss and diffeomorphism constraints. Finally, we discuss the possibility of a manifestly 4-dimensional symplectic form on the Lagrangian phase space.

This course is aimed at advanced undergraduate and beginning graduate students, and is inspired by a book by the same title, written by Padmanabhan. Each session consists of solving one or two pre-determined problems, which is done by a randomly picked student. While the problems introduce various subjects in Astrophysics and Cosmology, they do not serve as replacement for standard courses in these subjects, and are rather aimed at educating students with hands-on analytic/numerical skills to attack new problems.

There are a number of arguments in the philosophical, physical and cosmological literatures for the thesis that time is not fundamental to the description of nature. According to this view, time should be only an approximate notion which emerges from a more fundamental, timeless description only in certain limiting approximations. My first task is to review these arguments and explain why they fail. I will then examine the opposite view, which is that time and change are fundamental and, indeed, are perhaps the only aspects of reality that are not emergent from a more fundamental, microscopic description. The argument involves several aspects of contemporary physics and cosmology including 1) the problem of the landscape of string theory, 2) cosmological inflation and the problem of initial conditions, 3) the interpretation of the “wavefunction of the universe,” and the problem of what is an observable in classical and quantum general relativity. It also involves issues in the foundations of mathematics and the issue of the proper understanding of the role of mathematics in physics. The view that time is real and not emergent is, I will argue, supported by considerations arising from all these issues It leads finally to a need for a notion of law in cosmology which replaces the freedom to choose initial conditions with a notion of laws evolving in time. The arguments presented here have been developed in collaboration with Roberto Mangabeira Unger .

This course is aimed at advanced undergraduate and beginning graduate students, and is inspired by a book by the same title, written by Padmanabhan. Each session consists of solving one or two pre-determined problems, which is done by a randomly picked student. While the problems introduce various subjects in Astrophysics and Cosmology, they do not serve as replacement for standard courses in these subjects, and are rather aimed at educating students with hands-on analytic/numerical skills to attack new problems.

I discuss how we can give a satisfactory account of theory confirmation for theories with random data, such as Copenhagen quantum theory, despite the lack of a completely satisfactory definition of probabilistic theories of nature. I also explain why neither this nor any other proposed account of scientific confirmation works for many-worlds theories

I will identify six \'problems of time\' that arise in connection with quantum gravity and review the extent to which some of them can be regarded as solved, highlighting the very different aspects that they assume depending on one\'s starting point: Hamiltonian vs. path-integral, discrete vs continuous.

There is now a great deal of evidence confirming the existence of a very hot and dense early stage of the universe. Much of this data comes from a detailed study of the cosmic microwave background (CMB) - radiation from the early universe that was most recently measured by NASA\'s WMAP satellite. But the information presents new puzzles for scientists. One of the most blatant examples is an apparent paradox related to the second law of thermodynamics. Although some have argued that the hypothesis of inflationary cosmology solves some of the puzzles, profound issues remain. In this talk, Professor Penrose will describe a very different proposal, one that suggests a succession of universes prior to our own. He will also present a recent analysis of the CMB data that has a profound bearing on these issues.

Exactly half a century after Minkowski’s justly famous lecture, Dirac’s efforts to quantize gravity led him “to doubt how fundamental the four-dimensional requirement in physics is”. Dirac does not appear to have explored this doubt further, but I shall argue that it needs to be considered seriously. The fact is that Einstein and Minkowski fused space and time into a four-dimensional continuum but never directly posed the two most fundamental questions in dynamics: What is time? What is motion? It was an historical accident that Einstein attempted to implement Mach’s principle after he had created special relativity; otherwise he would have been forced to address these questions, which have never been properly considered. I shall show how they can be answered and suggest that: 1) time and space are utterly different; 2) the dynamical law of the universe may define absolute simultaneity in a manner that is still consistent with local validity of Minkowski’s marvellous notion of spacetime.