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This is an introduction to background independent quantum theories of gravity, with a focus on loop quantum gravity and related approaches. Basic texts: -Quantum Gravity, by Carlo Rovelli, Cambridge University Press 2005 -Quantum gravityy with a positive cosmological constant, Lee Smolin, hep-th/0209079 -Invitation to loop quantum gravity, Lee Smolin, hep-th/0408048 -Gauge fields, knots and gravity, JC Baez, JP Muniain Prerequisites: -undergraduate quantum mechanics -basics of classical gauge field theories -basic general relativity -hamiltonian and lagrangian mechanics -basics of lie algebras

This is an introduction to background independent quantum theories of gravity, with a focus on loop quantum gravity and related approaches. Basic texts: -Quantum Gravity, by Carlo Rovelli, Cambridge University Press 2005 -Quantum gravityy with a positive cosmological constant, Lee Smolin, hep-th/0209079 -Invitation to loop quantum gravity, Lee Smolin, hep-th/0408048 -Gauge fields, knots and gravity, JC Baez, JP Muniain Prerequisites: -undergraduate quantum mechanics -basics of classical gauge field theories -basic general relativity -hamiltonian and lagrangian mechanics -basics of lie algebras

Asymptotic statements like the almost-equi-partition law, the theorm of Shannon Mc -Millan-Breiman, the theorem of Sanov have all natural quantum analogs. They all talk about the thermodynamik limit of quantum spin systems. I will try to summarize these results and sketch the main ideas of proof.

I look at the information-processing involved in a quantum computation, in terms of the difference between the Boolean logic underlying a classical computation and the non-Boolean logic represented by the projective geometry of Hilbert space, in which the subspace structure of Hilbert space replaces the set-theoretic structure of classical logic. I show that the original Deutsch XOR algorithm, Simon's algorithm, and Shor's algorithm all involve a similar geometric formulation. In terms of this picture, I consider the question of where the speedup relative to classical algorithms comes from.