Course

This course will introduce you to some of the geometrical structures underlying theoretical physics. Previous knowledge of differential geometry is not required. Topics covered in the course include: Introduction to manifolds, differential forms, symplectic manifolds, symplectic version of Noether’s theorem, integration on manifolds, fiber bundles, principal bundles and applications to gauge theory.

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

In this mini-course we will describe some recent integrability developments in N=4 SYM. Pedro will start with some overview of three point functions in this theory. Paul will introduce the powerful Quantum Spectral Curve formalism describing the full planar spectrum of N=4 SYM starting with some elementary spin chain introduction. In this formalism, each operator in the theory is governed by a (set of) Q-function(s). In his last lecture Paul will walk us through an explicit example from beginning to end of a QSC solution. Pedro will then describe some explorations on three point correlation functions in this theory. The goal would be to have a machine where three Q-functions are given as input and a three-point function is spit out as output. We will describe where we are in this quest. 

No Zoom link or hybrid participation available. Registration is not required.

A quantum field theory is deemed topological if it exhibits the remarkable property of being independent of any background metric. In contrast to most other types of quantum field theories, topological quantum field theories possess a well-defined mathematical framework, tracing its roots back to the pioneering work of Atiyah in 1988. The mathematical tools employed to define and study topological quantum field theories encompass concepts from category theory, homotopy theory, topology, and algebra.
In this course, we will delve into the mathematical foundations of this field, explore examples and classification results, especially in lower dimensions. Subsequently, we will explore more advanced aspects, such as invertible theories, defects, the cobordism hypothesis, or state sum models in dimensions 3 and 4 (including Turaev-Viro and Douglas-Reutter models), depending on the interests of the audience.
Today, the mathematics of topological quantum field theories has found numerous applications in physics. Recent applications include the study of anomalies, non-invertible symmetries, the classification of topological phases of matter, and lattice models. The course aims to provide the necessary background for understanding these applications.

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