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We will discuss mathematical aspects of classical and quantum field theory, including topics such as: symplectic manifolds and the phase space, symplectic reduction, geometric quantization, Chern-Simons theory, and others.

The course centers on an in-depth study of the symmetry structure of General Relativity and how this is intimately related to its dynamics and to the challenges posed to its quantization. To achieve this understanding, we will introduce a host of concepts and techniques, broadly (and loosely) known under the name of “Covariant Phase Space Method”. This provides a different perspective on GR’s physics, a perspective in which phase space, rather than spacetime, is front and center. We will apply these ideas and techniques to discuss the so-called Problem of Time, Wald's approach to black hole entropy as a Noether charge, and the relationship between Dirac's Hypersurface Deformation Algebra and GR's symmetries and dynamics. We will also discuss the problem of detecting single gravitons as well as crucial analogies and differences between a quantum electromagnetic and gravitational field. Lecture notes specific for the course will be provided.

Machine learning has become a very valuable toolbox for scientists including physicists. In this course, we will learn the basics of machine learning with an emphasis on applications for many-body physics. At the end of this course, you will be equipped with the necessary and preliminary tools for starting your own machine learning projects.

Classical probability theory makes the (mostly, tacit) assumption that any two random experiments can be performed jointly.  This assumption seems to fail in quantum theory.  A rapidly growing literature seeks to understand QM by placing it in a much broader mathematical landscape of ``generalized probabilistic theories", or GPTs,  in which incompatible experiments are permitted.   Among other things, this effort has led to  (i) a better appreciation that many "characteristically quantum" phenomena (e.g., entanglement)  are in fact generic to non-classical probabilistic theories, (ii) a suite of reconstructions of (mostly, finite-dimensional) QM from small packages of assumptions of a probabilistic or operational nature, and (iii) a clearer view of the options available for generalizing QM.  This course will offer a survey of this literature,  starting from scratch and concluding with a discussion of recent developments. 

Mathematical prerequisites: finite-dimensional linear algebra, ideally including tensor products and duality, plus some exposure to category theory (though I will briefly review this material as needed).  

Scheduling note: There will be 5 lectures from March 12-26, then a gap of two weeks before the final 2 lectures held April 16 & 18.

Format: In-person only; lectures will be recorded for PIRSA but not live on Zoom.

This course introduces quantum field theory from scratch and then develops the theory of the quantum fluctuations of fields and particles. We will focus, in particular, on how quantum fields are affected by curvature and by spacetime horizons. This will lead us to the Unruh effect, Hawking radiation and to inflationary cosmology. Inflationary cosmology, which we will study in detail, is part of the current standard model of cosmology which holds that all structure in the universe - such as the distribution of galaxies - originated in tiny quantum fluctuations of a scalar field and of space-time itself. For intuition, consider that quantum field fluctuations of significant amplitude normally occur only at very small length scales. Close to the big bang, during a brief initial period of nearly exponentially fast expansion (inflation), such small-wavelength but large-amplitude quantum fluctuations were stretched out to cosmological wavelengths. In this way, quantum fluctuations are thought to have seeded the observed inhomogeneities in the cosmic microwave background - which in turn seeded the condensation of hydrogen into galaxies and stars, all closely matching the increasingly accurate astronomical observations over recent years. The prerequisites for this course are a solid understanding of quantum theory and some basic knowledge of general relativity, such as FRW spacetimes.

https://uwaterloo.ca/physics-of-information-lab/teaching/quantum-field-theory-cosmology-amath872phys785-w2024

https://pitp.zoom.us/j/96567241418?pwd=U3I1V1g4YXdaZ3psT1FrZUdlYm1zdz09

 

Review of elementary general relativity. Timelike and null geodesic congruences. Hypersurfaces and junction conditions. Lagrangian and Hamiltonian formulations of general relativity. Mass and angular momentum of a gravitating body. The laws of black-hole mechanics.

Zoom: https://pitp.zoom.us/j/97183751661?pwd=T0szNnRjdUM2dENYNTdmRmJCZVF1QT09

This course will cover phenomenological studies and experimental searches for new physics beyond the Standard Model, including: naturalness, extra dimension, supersymmetry, grand unification, dark matter candidates (WIMPs and axions) and their detection.

This survey course introduces some advanced topics in quantum field theory and string theory. Topics may include anomalies, conformal field theory, and bosonic string theory and are subject to change depending on the topics covered in the TBD elective course.

This Cosmology course will provide a theoretical overview of the standard cosmological model.
Topics will include: FRW universe, Thermal History, Inflation, Cosmological Perturbation Theory, Structure Formation and Quantum Initial Conditions.

Quantum phases have become a staple of modern physics, thanks to their appearance in fields as diverse as condensed matter physics, quantum field theory, quantum information processing, and topology. The description of quantum phases of matter requires novel mathematical tools that lie beyond the old symmetry breaking perspective on phases. Techniques from topological field theory, homotopy theory, and (higher) category theory show great potential for advancing our understanding of the characterization and classification of quantum phases. The goal of this workshop is to bring together experts from across mathematics and physics to discuss recent breakthroughs in these mathematical tools and their application to physical problems. 

 

Scientific Organizers

Lukas Mueller
Alex Turzillo
Davide Gaiotto
 

Sponsored in part by the Simons Collaboration on Global Categorical Symmetries