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This is an advanced graduate course which develops the math and physics of general relativity from scratch up to the highest level. The going will sometimes be steep but I try to be always careful. The purpose is to prepare for studies in quantum gravity, relativistic quantum information, black hole physics and cosmology. Quick summary of the contents: - Coordinate-free Differential Geometry, Weyl versus Ricci curvature versus Torsion, Vielbein Formalism, Spin-connections, Form-valued Tensors, Spectral Geometry, some Cohomology. - Derivations of General Relativity including as a Gauge Theory, Diffeomorphism Invariance vs. Symmetries, Bianchi Identities vs. Local and Global Conservation Laws. - Penrose Diagrams for Black Holes and Cosmology, Types of Horizons, Energy Conditions and Singularity theorems, Properties and Classification of Exact Solutions. - Cosmology and Models of Cosmic Inflation

Matter is quantum. Growing experimental results on materials, natural and synthetic (ion traps, cold atoms etc.,) and concomitant theoretical developments make `quantum matter' an exciting field. There is also a growing interplay of quantum matter physics and quantum information/computation. With a focus on concepts I plan to discuss key phenomenology, quantum models and theory.

The main focus of this course is the exploration of the symmetry structure of General Relativity which is an essential step before any attempt at a (direct) quantization of GR. We will start by developing powerful tools for the analysis of local symmetries in physical theories (the covariant phase space method) and then apply it to increasingly complex theories: the parametrized particle, Yang--Mills theory, and finally General Relativity. We will discover in which ways these theories have similar symmetry structures and in which ways GR is special. We will conclude by reviewing classical results on the uniqueness of GR given its symmetry structure and discuss why it is so hard to quantize it. In tutorials and homeworks, through the reading of articles and collegial discussions in the classroom---as well as good old exercises---you will explore questions such as "Should general relativity be quantized at all? Is a single graviton detactable (even in principle)?", "What is the meaning of the wave functions of the universe?", "Can we do physics without time?".

We will cover the basics of the gauge/gravity duality, including some of the following aspects: holographic fluids, applications to condensed matter systems, entanglement entropy, and recent advances in understanding the black hole information paradox.

Title: An introduction to twistors Course Description: Twistor theory, introduced by Penrose many years ago, is a way to reformulate massless fields on four-dimensional space-time in terms of an auxiliary 6-dimensional complex manifold, called twistor space. This course will introduce twistor space and the Penrose correspondence (relating fields on twistor space and space-time), at both classical and quantum levels. We will discuss the twistor realization of self-dual Yang-Mills theory and of self-dual gravity. If time permits we will discuss the connection between twistors and celestial holography.

In this mini course, I shall introduce the basic concepts in 2D topological orders by studying simple models of topological orders and then introduce topological quantum computing based on Fibonacci anyons. Here is the (not perfectly ordered) syllabus.                   

• Overview of topological phases of matter
• Z2 toric code model: the simplest model of 2D topological orders
• Quick generalization to the quantum double model
• Anyons, topological entanglement entropy, S and T matrices
• Fusion and braiding of anyons: quantum dimensions, pentagon and hexagon identities
• Fibonacci anyons
• Topological quantum computing

Can the effectiveness of a medical treatment be determined without the expense of a randomized controlled trial? Can the impact of a new policy be disentangled from other factors that happen to vary at the same time? Questions such as these are the purview of the field of causal inference, a general-purpose science of cause and effect, applicable in domains ranging from epidemiology to economics. Researchers in this field seek in particular to find techniques for extracting causal conclusions from statistical data. Meanwhile, one of the most significant results in the foundations of quantum theory—Bell’s theorem—can also be understood as an attempt to disentangle correlation and causation. Recently, it has been recognized that Bell’s result is an early foray into the field of causal inference and that the insights derived from almost 60 years of research on his theorem can supplement and improve upon state-of-the-art causal inference techniques. In the other direction, the conceptual framework developed by causal inference researchers provides a fruitful new perspective on what could possibly count as a satisfactory causal explanation of the quantum correlations observed in Bell experiments. Efforts to elaborate upon these connections have led to an exciting flow of techniques and insights across the disciplinary divide. This course will explore what is happening at the intersection of these two fields. zoom link: https://pitp.zoom.us/j/94143784665?pwd=VFJpajVIMEtvYmRabFYzYnNRSVAvZz09

The course is an introduction to quantum field theory in curved spacetime. Upon building up the general formalism, the latter is applied to several topics in the modern theory of gravity and cosmology where the quantum properties of fundamental fields play an essential role.

Topics to be covered:
1) Radiation of particles by moving mirrors 
2) Hawking radiation of black holes  
3) Production of primordial density perturbations and gravity waves during inflation
4) Statistical properties of the primordial spectra  

Required prior knowledge:
Foundations of quantum mechanics and general relativity

We will review the notion of information in the most possible general sense. Then we will revisit our definitions of entropy in quantum physics from an informational point of view and how it relates to information theory and thermodynamics. We will discuss entanglement in quantum mechanics from the point of view of information theory, and how to quantify it and distinguish it from classical correlations. We will derive Bell inequalities and discuss their importance, and how quantum information protocols can use entanglement as a resource. We will introduce other notions of quantum correlations besides entanglement and what distinguishes them from classical correlations. We will also analyze measurement theory in quantum mechanics, the notion of generalized measurements and their importance in the processing and transmission of information. We will introduce the notions of quantum circuits and see some of the most famous algorithms in quantum information processing, as well as in quantum cryptography. We will end with a little introduction to the notions of relativistic quantum information and a discussion about quantum ethics.