**Graduate curriculum**

A standard graduate level curriculum in theoretical and mathematical physics typically covers a number of **core subjects**, while offering a list of **electives**. All subjects should be covered at the intermediate and advanced levels.

**Note.** Graduate level courses are usually much more demanding than those offered in the undergraduate curriculum.

**I. Core subjects**

**1. Quantum field theory in Minkowski spacetime**

This is usually divided in two courses:

**a.** **Quantum field theory I** ("Introductory QFT")

The Dirac equation, canonical quantization, basic scattering theory, basic perturbation theory, basic renormalization theory, introduction to path integral techniques and QED (as the main example of the formalism), elements of Euclidean field theory.

**b.** **Quantum field theory II** ("Advanced QFT")

Non-Abelian gauge theory, renormalization, instantons and monopoles, the large N expansion, non-perturbative effects and the Standard Model.

**2. General relativity and introduction to cosmology**

Foundations of GR, the Einstein equation, basic exact solutions (black holes, AdS, de Sitter, FLRW etc.) and an introduction to cosmology.

**3. Advanced statistical mechanics**

Classical and quantum statistical mechanics, well-known models (Ising, percolation, BCS etc.), mean field theory, introduction to the Wilsonian renormalization group.

**4.** **Particle physics and astrophysics**

Elementary particle physics, topics in scattering theory, introduction to astrophysics and astroparticle physics, topics in cosmology.

**II. Physics electives**

These subjects are usually assigned a number of points according to their difficulty and time burden. A student must choose a few such courses each year so as to fulfill a minimal number of points requirement. Electives typically state a set of pre-requisites (courses that should be taken before the elective, since they provide necessary background knowledge).

**1. Quantum field theory on curved spacetime**

Basic formalism of background quantization on curved spacetime, backreaction, Bekenstein-Hawking entropy, applications to cosmology (cosmological scattering theory etc).

**2. Manybody theory and Euclidean QFT**

Basic formalism, theory of linear response, applications to gases, critical phenomena and condensed matter physics.

Basic theory of classical and quantum integrable systems, integrable hierarchies.

**4. Advanced general relativity**

Mathematical foundations of general relativity, numerical gravity etc.

Introduction to conformal field theory, including relevant subjects in the representation theory of infinite-dimensional Lie algebras and vertex algebras; applications to surface physics and string theory.

Wilsonian theory of critical phenomena; applications to condensed matter physics, astrophysics etc.

**7. Rigorous quantum mechanics and rigorous QFT**

Operator theory techniques, semigroups of operators, abstract index theory, compact perturbation theory, rigorous scattering theory, axiomatic, constructive and algebraic QFT, applications.

**8. Perturbative and lattice QCD**

Perturbative QCD, introduction to quark-gluon plasma. Euclidean formalism, exotic phases, applications to cosmology and astrophysics. Basic lattice formalisms, various ways to include fermions, Monte Carlo sampling, relevant numerical methods, overview of current state of the art.

**9.** **Constrained systems, BV-BRST methods and topological field theories**

**10. String theory and supergavity**

This is generally divided in two courses:

**a. String theory I ("Introduction to String Theory")**

Quantization of the free bosonic strings, superstring and heterotic strings; elements of (super)conformal field theory; elements of string perturbation theory; D-branes, nonlinear sigma models, elements of supergravity theory and M-theory, the AdS-CFT correspondence.

**b. String theory II ("Advanced String Theory")**

(Super)string perturbation theory and string field theory, supergravity and string theory compactifications (classical Calabi-Yau compactifications, flux compactifications, "abstract' compactifications based on a CFT), elements of Teichmuller theory, algebraic geometry and moduli space theory.

**III. Mathematics electives**

These are usually offered by the Mathematics department of the same university. The subjects are usually assigned a number of points according to their difficulty and time burden. A student must choose a few such courses each year so as to fulfill a minimal number of points requirement. Mathematics electives typically state a set of mathematics pre-requisites.

**1. Lie groups, Lie algebras and their representations**

**3. Riemannian and pseudo-Riemannian geometry**

**5. Complex and Kahler geometry**

**7. Functional analysis and operator theory**

**9. Measure and probability theory**

**10. Computational and numerical mathematics**