Fall 2020, Jacobus Verbaarschot
Lecture 1
Functionals, Functional Derivatives, Euler-Lagrange equations, Constraints
Lecture 2
Catenary
Lecture 3
Action principle, Noether's theorem, Linear Vector spaces, linear maps
Lecture 4
Dual space, Orthonormal basis, Scalar product
Lecture 5
Direct Sum, Qoutient space, Co-kernel
Lecture 6
Projection operator, Linear equation, Fredholm alternative
Lecture 7
Determinant, Adjugate matrix, Inverse, Cailey's theorem
Lecture 8
Derivative, Diagonalization, Jordan Canonical form
Lecture 9
Quadratic Form, Symplectic Form
Lecture 10
Function spaces, Banach space, Hilbert space, Cauchy sequence, Complete space,
Cauchy-Schwarz inequality
Lecture 11
Parseval's theorem, Orhtogonal polynmials
Lecture 12
Three step recurrence relations, Legendre, Hermite and
Tchebychev polynomials
Lecture 13
Distributions
Lecture 14
Axioms of roup theory
Lecture 15
Subgroups, Cosets, Normal subgroup, Quotient group, Examples
Lecture 16
Conjugacy classes, Permutation group
Lecture 17
Questions
Lecture 18
Group action on sets, tansitive, faithful, free
Lecture 19
Representations of groups, real, pseudo-real
Lecture 20
Direct sum, Direc product
Lecture 21
Schur's Lemma
Lecture 22
Unitarity of representations, Orthogonality
Lecture 23
Characters, orthogonality relations, character table of S3
Lecture 24
Completeness of characters, character table of S4
Lecture 25
Group Algebra
Lecture 26
Group theory in Quantum Mechanics
Lecture 27
Vibrational Spectrum of H_2 O
Lecture 28
Midterm
Lecture 29
Vibrational Spectrum of H_2 O
Lecture 30
Vibrational Spectrum of H_2 O
Lecture 31
Lie Groups
Lecture 32
Symplectic Group
Lecture 33
Time reversal invariance, SU(2)
Lecture 34
Invariant vector fields
Lecture 35
Maurer-Cartan form
Lecture 36
Euler angles, group integration, Haar measure
Lecture 37
Relation between O(3) and SU(2)
Lecture 38
SO(N) spinor representation of SU(2), Spin(N)
Lecture 39
Adjoint representation, Peter Weyl theorem, character of SU(2)
Lecture 40
Lie Algebras, Killing form
Lecture 41
Killing Metric, Casimir Operator, SU(2)
Lecture 42
SU(3), General Semi-Simple Lie Algebras