Mathematical Physics Courses

List of graduate courses offered at the Department of Mathematics, by members of our research group.

Mathematical Quantum Mechanics

Level: Master (Fall).

Goal: Introduce several mathematical topics in quantum mechanics, with emphasis on non-perturbative aspects and the role of different types of symmetries.

Syllabus:

Basic Principles: Operators and Hilbert space, quantization, observables and probability; Wave function and Schroedinger equation; Bosonic, grassmannian and super/conformal d-dimensional oscillators; Witten index; Elliptic potentials; Bound-states vs scattering; Feynman integral and propagators.

Symmetries: Wigner's theorem, self-adjoint operators, spectral theory; Angular momentum; Symmetries, representations and quantum statistics; Addition of angular momenta; Basics of quantum groups.

Resurgence and Summability: Perturbation theory; Instantons and tunneling; Double and multiple wells; Complex/exact WKB approximation; Borel summability, resurgence and transseries.

Coherence vs Decoherence: EPR and quantum entanglement; Coherent states; Decoherence; Quantum-to- classical transition; Quantum information, quantum chaos and its black hole bounds.

Quantization on Manifolds: Dirac constraints; Schroedinger equation, oscillators and scattering; Bogoliubov transformations; Geometric and deformation quantization; Non-commutative geometry.

Introduction to Supersymmetry

Level: Master (Fall).

Goal: A gentle introduction to supersymmetry, focusing on supersymmetric quantum mechanics and exploring some of its various applications, particularly in classical index theorems.

Syllabus:

Super vector space, Lie superalgebra. Grassmann calculus.

Fermionic harmonic oscillator. Representation of the partition function and of the supertrace of a fermionic system using path integrals.

Supersymmetric quantum mechanics (SUSY QM). Properties of supersymmetry. The Witten index and its representation using a path integral. The supersymmetric harmonic oscillator.

Elliptic differential operators on compact manifolds. The De Rham complex.

SUSY QM on a compact, orientable, Riemannian manifold without boundary. The supercharge operator Q and De Rham cohomology. The corresponding Witten index and the Euler characteristic of the complex defined by the operator Q. The Chern–Gauss–Bonnet theorem.

SUSY QM on a compact spin manifold of even dimension, without boundary. The Dirac operator and the corresponding Witten index. The Atiyah–Singer index theorem.

The Atiyah–Patodi–Singer index theorem. The eta invariant and pseudo-modular forms.

Algebraic and Geometric Methods in Engineering and Physics

Master (Fall).

Goal: To introduce topics of algebra, geometry and topology and to give examples of their applications in physics and engineering.

Syllabus:

Topics of algebra and applications: Rings, fields and modules. Groups, actions and representations of finite groups. Applications: Information security. Vibrations (of buildings, etc) of symmetric structures. Structural optimization using representations of finite groups.

Topics of geometry and topology and applications: Elements of topology. Topological spaces and metric spaces. Fundamental group and coverings. Simplicial complexes and homology. Manifolds and tensor fields. Riemannian manifolds. Forms and integration. De Rham cohomology. Flows of vector fields. Lie derivatives and symmetry group of a tensor field. Morse theory. Aplications: Data science and persitent homology. Applications of Morse theory to big data. Cosmological models.

Lie algebras, Lie groups and applications: Lie groups and Lie algebras. Compact Lie groups and their Lie algebras. Root systems. Elements of the theory of representations. Lie group actions on manifolds. Applications: Statics and dynamics of robots and grassmanians and flag manifolds. Particle physics and theories of unification.

Path Integral and Applications: From Neural Networks to Quantum Fields

Level: Doctoral (Spring).

Goal: The goal of the course is to dive into the mathematics of Feynman’s path integrals and explore some its recent applications to areas such as Physics, Artificial Intelligence, Engineering and Economics.

Syllabus:

Probability theory preliminaries.

Application 1: Wide neural networks at initialization as random functions and the geometry associated with training.

Stochastic processes.

Application 2: Feynman path integral approach to quantum mechanics.

Stochastic differential equations.

Application 3: remarks on PDE’s and boundary value problems.

Application 4: generative diffusion models.

Application 5: Quantum scalar field theory.

Other possible applications: Interacting particle systems and random matrix theory.

Conformal Field Theory
Level: Doctoral (Fall).
Goal: Introduce the basics of conformal field theory and construct the corresponding formalism for minimal models (including the relation to matrix models) and WZW models.
Syllabus:
Conformal Symmetry: Phase transitions and scale invariance; Conformal group, Virasoro algebra, OPE's and Ward identities; Superconformal symmetry, RNS algebras and spin fields.
Operator Formalism: Free fields, conformal families and bootstrap; Bosons and fermions on the torus, modular invariance, CFT and Riemann surfaces; Boundary conditions and Verlinde formula.
Minimal Models: Verma modules, representations and Kac determinant; Minimal models, unitarity and fusion rules; Coulomb gas formalism; Modular invariance.
Matrix Models: The 1/N expansion, semi-classical approximation, resolvent and spectral curve; Orthogonal polynomials and correlation functions; Critical points and DSL; Non-perturbative phenomena.
Current Algebras and WZW Models: Simple Lie algebras and affine Lie algebras; WZW models, Sugawara construction and Knizhnik-Zamolodchikov equation; Ishibashi states.

Mathematical Relativity
Level: Doctoral (Spring).
Goal: Introduce several mathematical topics in general relativity.
Syllabus:
Examples: the Schwarzschild solution and cosmological models; matching and Oppenheimer-Snyder collapse; Penrose diagrams.
Causality: chronological and causal past and future, domain of dependence; chronological, stably causal and globally hyperbolic spacetimes.
Singularity Theorems: Jacobi equation and conjugate points; energy conditions; existence of maximizing geodesics; Hawking and Penrose singularity theorems.
The Cauchy Problem: wave equation; Cauchy problems with constraints; Gauss-Codazzi relations and 3+1 decomposition of the Einstein equation; Choquet-Bruhat theorem; constraint equations.
Positive Mass Theorem: Komar mass; Einstein-Hilbert action; Lagrangian and Hamiltonian formulation of the Einstein equations; ADM mass; positive mass theorem; Penrose inequality.
Black Holes: Kerr solution; Killing horizons; surface gravity; Smarr formula; area theorem; black hole thermodynamics.

Geometry and Gauge Theory
Level: Doctoral (Spring).
Goal: Introduce the basic notions of gauge theory and its interaction with both geometry and topology, with a particular emphasis towards the non-perturbative structure of the theory.
Syllabus:
Core topics:
Gauge Theory: Fibre-bundle geometry; Yang-Mills theory and Yang-Mills-Higgs theory; Chern-Simons theory; Self-dual Yang-Mills equations, BPS equations.
BRST and BV: First and second class constraints, quantization of constrained systems; BRST symmetry, ghosts and Koszul-Tate differential; Feynman integral, antifield formalism and BV.
Optional topics:
Anomalies: Fermions, classical and quantum symmetries; Fujikawa method; Index theorem; Global anomalies.
Monopoles: Solitons, semi-classical methods; Topological conservation laws; The 't Hooft-Polyakov monopole; Moduli spaces, scattering, Nahm transform and the spectral curve.
Instantons: Tunnel effect, gauge theory and θ-vacua; Moduli spaces, ADHM construction; Divergent series and Borel summability.
Qunatization of Gauge Theories: Perturbative quantization of 4D Yang-Mills theory and of Chern-Simons theory; Quantization of Dijkgraaf-Witten theory; TQFT.

String Theory
Level: Doctoral (Spring).
Goal: Introduce the basic notions of string theory.
Syllabus:
Core topics:
Bosonic Strings: Polyakov action, covariant quantization, open strings and closed strings; S-matrix, tree-level and one-loop amplitudes; Riemann surfaces and CFT.
D-Branes and Dualities: Toroidal compactification, closed strings and T-duality; Orbifolds; D-branes, T-duality and Wilson lines; Gauge theory and Born-Infeld electrodynamics.
Superstrings: Superstrings of type I and II, Ramond and Neveu-Schwarz sectors, modular invariance and GSO projection; Superstring interactions; Calabi-Yau compactifications.
Optional topics:
More on D-Branes and Dualities: T-duality; D-brane interactions: kinematics, dynamics and bound states; S-duality, U-duality, M-theory and other dualities; Black holes and AdS/CFT.
Topological String Theory: Chern-Simons theory; Kähler and Calabi-Yau geometry; Topological σ-models, A and B models; Mirror symmetry; Large N dualities and matrix models.