## Science Honors Program

I have been an instructor in the Science Honors Program at Columbia since Fall 2016. This page records the content and notes for my latest classes, on the basics of representation theory and its role in modern (mathematical) physics.

### Spring 2020: Representation Theory and Physics

I will update the notes (PDF) weekly, after every class. These notes are accumulated from previous versions of the course and may contain more material than what is covered in class.

- Week 1: Symmetries, what is representation theory, groups (e.g. dihedral, symmetric)
- Week 2: Homomorphisms and isomorphisms, products of groups, classification of finite abelian groups
- Week 3: (Normal) subgroups, quotients of groups, Lagrange's theorem, vector spaces, bases
- Week 4: Trace, determinant, group representations and examples, direct sums and irreps
- Week 5: Quantum states, wavefunctions, state spaces and decomposition into irreps
- Week 6: Quantum measurement, Stern-Gerlach experiment
- Week 7: Tensor product and entanglement (EPR paradox), special relativity, time evolution
- Week 8:
**class canceled** - Week 9: Manifolds, (matrix) Lie groups, examples (GL, SL, O, SO)
- Week 10: Tangent spaces, Lie algebras and commutators, representation theory of su(2), semisimple Lie algebras
- Week 11: SO(3,1) representations, spin, gauge theories, standard model
- Week 12: Standard model, grand unified theories, supersymmetry

### Fall 2019: Representation Theory and Physics

I will update the notes (PDF) weekly, after every class.

- Week 1: Continuous vs discrete symmetries, what is representation theory, groups (e.g. symmetry groups)
- Week 2: Dihedral group, symmetric group, homomorphisms and isomorphisms
- Week 3: (Normal) subgroups, products and quotients of groups, Lagrange's theorem, classification of finite abelian groups
- Week 4: Vector spaces, bases, trace, determinant, group representations
- Week 5: Examples of representations, direct sums and irreps, intertwiners, Schur's lemma
- Week 6: Quantum states, wavefunctions, Hilbert space (decomposition into irreps), Stern-Gerlach experiment
- Week 7: Quantum measurement, tensor product and entanglement (EPR paradox), time evolution
- Week 8: Manifolds, (matrix) Lie groups, tangent spaces, Lie algebras and commutators
- Week 9: Representation theory of su(2) and sl(2,C) (spin, raising/lowering operators), semisimple Lie algebras
- Week 10: SO(3,1) representations, spin, gauge theories, standard model
- Week 11 (notes): Qubits, quantum computation, symmetric and exterior powers, Schur-Weyl duality
- Week 12: Perturbation theory, renormalization (counterterms), gravity, grand unified theories

### Before Fall 2019

From Fall 2016 to Spring 2019, I taught the Geometry and Topology course.