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 mini-course will introduce twisted holography, which is holography for BPS subsectors of gauge theory and gravity. We will start by introducing the B-model topological string from the space-time perspective, before discussing branes, backreaction, and the holographic duality.

Zoom: https://pitp.zoom.us/j/98839130613?pwd=SExFK0ZVYzJ3NmJhU1RFa21PWU1qQT09

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

This course has two main goals: (1) to introduce some key models from condensed matter physics;  and (2) to introduce some numerical approaches to studying these (and other) models.  As a  precursor to these objectives, we will carefully understand many-body states and operators from  the perspective of condensed matter theory.  (However, I will cover only spin models.  We will not  discuss or use second quantization.)

Once this background is established, we will study the method of exact diagonalization and write  simple python programs to find ground states, correlation functions, energy gaps, and other  properties of the transverse-field Ising model.  We will also discuss the computational limitations  of exact diagonalization.  Finally, I will introduce the concept of matrix product states, and we will  see that these can be used to study ground state properties for much larger systems than can be  studied with exact diagonalization.

Each 90-minute session will include substantial programming exercises in addition to lecture.  Prior programming experience is not expected or required, but I would like everyone to have  python (version 3) installed on their computer prior to the first class, including Jupyter notebooks;  see “Resources” below.