Non-perturbative methods have become a mainstay in the research toolkit of theoretical and mathematical physics. Among them, methods that draw on gravity and conformal field theory play a central role.

This course, intended for students familiar with differential geometry, the basic notions of gravity, and quantum field theory, surveys the most classical and widely used results in this line of research.

Starting from conformal field theory, we will move toward its connections with negatively curved spaces — anti-de Sitter (AdS) spaces — and their links to random matrices, scattering theory, and various physical phenomena, including quark physics, quantum information, and the renormalization group.

Syllabus

• Conformal field theory and quantum field theory—physical and mathematical aspects.
• Introduction to string theory and anti-de Sitter space: how are they related?
• Computing correlation functions via the holographic correspondence (the GKPW and BDHM dictionary). Operator reconstruction (the HKLL prescription).
• The holographic renormalization group and its aspects.
• Scattering theory and “celestial holography.”
• Applications of the AdS/CFT correspondence—quarks and superconductors.
• Random matrices, low dimensions, integrable systems, and deformations.
• AdS/CFT and quantum information: entanglement entropy, error-correcting codes, complexity.

Lecturer
Ageev Dmitry Sergeevich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2025-303).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center