Mathematics Review:
Complex Analysis - Tibra Ali
- Lecture 1 - Review of the basics of complex numbers. Geometrical interpretation in terms of teh Argand-Wessel plane. DeMoivre's theorem and applications. Branch points and branch cuts.
- Lecture 2a & Lecture 2b - Cauchy-Riemann equations. Holomorphic functions and harmonic functions. Contour integration.
- Lecture 3 - Cauchy's theorem and integral formula. Taylor's theorem. Singularities. Laurent's series. Residues.
- Lecture 4 - Applications of the integral formula to evaluate integrals. Trigonometric integrals, semi-circular contours, mousehole contours, keyhole contours.
Linear Algebra - Anna Kostouki
- Lecture 1 - Linear Vector Spaces, Linear Operators, Scalar Products, Dual Spaces, Adjoint Operators, Eigenvalues & Eigenvectors, Hermitian & Unitary Operators.
- Lecture 2a & Lecture 2b - Abstract Algebras, Structure Constants, Homomorphisms, Clifford & Grassmann Algebras.
- Lecture 3 - Group Theory: Finite groups and the permutation group. SU(2) and SO(3).
- Lecture 4 - The Lorentz and Poincaré groups (a short review of Special Relativity).
Differential Equations - Sarah Croke
- Lecture 1 - First order differential equations; examples: Einstein theory of radiation, optical attenuations; methods of solution.
- Lecture 2 - Second order differential equations, homogeneous and inhomogeneous; reduction of order; variation of parameters; the Wronskian.
- Lecture 3 - Series solutions; Euler's equation; Extended power series method, form of solutions in different cases; Bessel's equation.
- Lecture 4 - Bessel Functions; Separation of variables; Spherical Harmonics; WKB approximation.
Evaluation of Integrals and Calculous of Variations - Denis Dalidovich
- Lecture 1 - Gaussian Integrals in one and many dimensions. Averages with the Gaussian weight.
- Lecture 2 - Wick's Theorem. Imaginary Gaussian Integral. Gaussian Integral with Grassman variables.
- Lecture 3- Functionals and functional derivatives. Euler-Lagrange equations; examples from classical mechanics.
- Lecture 4 - Noether's theorem. Functionals describing continuous systems; Lagrangian density. Extrema of functionals subject to contraints; Lagrange multipliers.
Special Functions and Distributions - Dan Wohns
- Lecture 1 - Dirac delta; Test functions; Distributions and their derivatives.
- Lecture 2 - Orthogonal polynomials; Recurrence relations; Weights.
- Lecture 3 - Generalized Rodrigues' formulae; Classification of orthogonal polynomials; Sturm-Liouville theory.
- Lecture 4 - Gamma; Zeta; Hypergeometric functions
Integral Transforms and Green's Functions - David Kubiznak
Lie Groups and Lie Algebra's - Freddy Cachazo
- Lecture 1 - Introduction to Lie Groups and Lie Algebras in Physics. Lie groups, representations, structure constants.
- Lecture 2 - The Poincaré group & algebra. Representations on Hilbert space. Massless and massive irreps. The Little Group.
- Lecture 3 - Classifications of Lie Algebras. Helicity and Spin. Highest weight representations of SU(2).
- Lecture 4 - The Adjoint representation. Classification of (simple) Lie Algebras. Roots diagrams and Dynkin diagrams.
- Lecture 5 - The group associated with the standard model of particle physics. Weights. Highest weight representations. Fundamental and anti-fundamental representations of su(3). Tensor products of representations. Clebsch-Gordan Decomposition. Young's Tableaux.
Mathematica - Pedro Vieira
RESEARCHER PRESENTANTIONS:
Presentation 1 - Paul Fendley
Presentation 2 - Joseph Minahan
Presentation 3 - Matthias Staudacher
Presentation 4 - Yakir Aharonov
Presentation 5 - Vladimir Kazakov
Presentation 6 - Itay Yavin
Presentation 7 - Guifre Vidal
Presentation 8 - Lucien Hardy
Presentation 9 - Laurent Freidel
Presentation 10 - Robert Spekkens
Presentation 11 - Lee Smolin
Presentation 12 - Cliff Burgess
Presentation 13 - Avery Broderick
Presentation 14 - Pavel Kovtun
Presentation 15 - Subir Sachdev