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