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.

Renormalization Group

Level: Master (Fall).

Goal: Introduce the basic notions of quantum field theory, the Feynman integral and the renormalization group, with applications to gauge theory.

Syllabus:

Finite Dimensional Integrals: Partition function and correlation functions; Feynman diagrams; Effective action; Berezin integral, supersymmetry and localization.

Feynman Integral: Classical and quantum action functional; Definition of Feynman integral; Green functions and propagators; Correlation functions and operator formalism; Wick's theorem.

Scalar Field Theory: Perturbative expansion; Cross-sections, Feynman diagrams and Feynman rules; Divergences and regularization; Renormalization and beta-functions.

Renormalization Group: Real and momentum space; Fixed points, anomalous dimensions and critical exponents; Renormalization group, effective action, effective potential; Wilson-Polchinski RG equation; Zamolodchikov's c-theorem; Composite operators and OPE's.

Gauge Theory: QED and QCD; Ward-Takahashi identities; Euler-Heisenberg Lagrangian; Renormalized perturbation theory; Beta-functions, asymptotic freedom; Renormalizable and non-renormalizable theories.

Algebraic and Geometric Methods in Engineering and Physics

Level: 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.

Feynman Integral and Applications

Goal: Imaginary time Feynman integral as a measure on a functional space. Stochastic processes, stochastic differential equations and Feynman-Kac formula. Rigorous applications to quantum field theory, statistical physics, finance and engineering. Some examples of applications of the Feynman integral, with a definition not based (exclusively) on measure theory, to quantum field theory and mathematics.

Syllabus:

Elements of measure theory and probability theory.

Stochastic processes and stochastic differential equations: Kac-Feynman formula.

Measures on spaces of tempered distributions.

Applications of the path integral: with a definition based on measure theory to Quantum Field Theory, Statistical Physics, Engineering and Finance.

Examples of applications of the path integral: with a definition not based (exclusively) on measure theory.

Conformal Field Theory

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

Goal: Introduce several mathematical topics in general relativity, with emphasis on singularity theorems and black hole solutions in several dimensions.

Syllabus:

Examples: de Sitter, Anti-de Sitter, FLRW, Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman solutions with/without cosmological constant; Carter-Penrose diagrams; d-dimensional generalizations.

Singularity Theorems: Causal structure, properties of global hyperbolicity and complete geodesics; Definition, description and the character of singularities; Hawking and Penrose theorems.

The Cauchy Problem: Einstein equations, initial data and second order hyperbolic equations; Existence and uniqueness in empty space and with matter; Positive mass theorem, Penrose inequality, cosmic censorship.

The 4 Laws of Black Holes: The wave equation in curved spacetime; Classical and quantum fields in curved spacetime; Area theorem and Hawking effect; Thermodynamical laws; Wald's formula.

Horizon Topology in Several Dimensions: Unicity theorems in d=4; Solutions in dimension d=5 and dimension d≥6; Gregory-Laflamme instability; Black rings, black saturn and blackfolds.

Geometry and Gauge Theory

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:

Gauge Theory: Fibre-bundle geometry; Yang-Mills theory and Yang-Mills-Higgs theory; Chern-Simons theory; Self-dual Yang-Mills equations, BPS equations and gauge theory.

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.

Anomalies: Fermions, classical symmetries, quantums symmetries and the axial current; Fujikawa method; Index theorem; Non-perturbative considerations, anomalies and BRST; Global anomalies.

Monopoles: Solitons, semi-classical methods and collective coordinates; 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; Topology and boundary conditions; Moduli spaces, ADHM construction and holomorphic vector bundles; Divergent series and Borel summability.

String Theory

Goal: Introduce the basic notions of string theory, together with an introduction to selected advanced research topics.

Syllabus:

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.

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; OSV conjecture.