## Science Honors Program: Geometry and Topology

I have been an instructor in the Science Honors Program at Columbia since Fall 2016. From Fall 2016 to Spring 2019, I taught a course on geometry and topology. This page records the content and notes for each semester's class.

### Spring 2019: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic and invariance, planar models, new surfaces (Klein bottle, projective plane)
- Week 2: no class
- Week 3: Cutting and pasting, connected sums, orientability, word representations, classification theorem for surfaces
- Week 4: Definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces), Euler characteristic of manifolds
- Week 5: First homotopy groups (computation for the cylinder/circle), crash course on group theory (isomorphisms, commutativity, classification of finitely generated abelian groups)
- Week 6: Classification of manifolds, Poincaré conjecture, topological degree, Brouwer's fixed point theorem, Nash's equilibrium theorem
- Week 7: Borsuk-Ulam theorem, (discrete) ham sandwich theorem, vector fields, hairy ball theorem, Poincaré-Hopf theorem
- Week 8: Isometry, geodesics, curvature (osculating circles), Gaussian curvature, Gauss's Theorema Egregrium, local Gauss-Bonnet theorem
- Week 9: Global Gauss-Bonnet theorem, moduli spaces, uniformization theorem, Teichmuller space, string theory
- Week 10: Plane curves, enumerative geometry, projective space (homogeneous coordinates, homogenization), Bezout's theorem
- Week 11 (no notes): classical mechanics (Lagrangian and Hamiltonian), Stern-Gerlach experiment, quantum mechanics, Heisenberg uncertainty principle, quantum field theory

### Fall 2018: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic
- Week 2: Invariance of Euler characteristic, cutting and pasting, connected sums and their Euler characteristics
- Week 3: Orientability, word representations, classification theorem for surfaces
- Week 4: Knots, unknotting number, crossing number, Jones polynomial, chirality, lower bound on crossing number
- Week 5: Definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces), Euler characteristic of manifolds
- Week 6: First homotopy groups (computation for the cylinder/circle), crash course on group theory (isomorphisms, commutativity, classification of finitely generated abelian groups)
- Week 7: Classification of manifolds, Poincaré conjecture, topological degree, Brouwer's fixed point theorem, Nash's equilibrium theorem
- Week 8: Proof of Nash's equilibrium theorem, Borsuk-Ulam theorem, (discrete) ham sandwich theorem
- Week 9: Poincaré-Hopf theorem, isometry, geodesics, curvature (osculating circles)
- Week 10: Gaussian curvature, Gauss's Theorema Egregrium, local and global Gauss-Bonnet theorems
- Week 11: Plane curves, projective space (homogeneous coordinates, homogenization), enumerative geometry, Bezout's theorem
- Week 12 (no notes): classical mechanics (Lagrangian and Hamiltonian), Stern-Gerlach experiment, quantum mechanics, Heisenberg uncertainty principle, quantum field theory, path integrals, Feynman diagrams

### Spring 2018: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic and invariance, planar models, new surfaces (Klein bottle, projective plane)
- Week 2: Cutting and pasting, connected sums and their Euler characteristics, orientability, word representations
- Week 3: Classification theorem for surfaces, definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces)
- Week 4: Euler characteristic of manifolds, first homotopy groups (computation for the cylinder/circle), crash course on group theory (isomorphisms, commutativity, classification of finitely generated abelian groups)
- Week 5: Classification of manifolds, Poincaré conjecture, topological degree, Brouwer's fixed point theorem, Nash's equilibrium theorem
- Week 6: Borsuk-Ulam theorem, (discrete) ham sandwich theorem, vector fields, hairy ball theorem, Poincaré-Hopf theorem
- Week 7: Isometry, geodesics, curvature (osculating circles), Gaussian curvature, Gauss's Theorema Egregrium, local and global Gauss-Bonnet theorems
- Week 8 (no notes): Metrics, curvature, uniformization theorem
- Week 9: Knots, unknotting number, crossing number, Jones polynomial
- Week 10: Plane curves, projective space (homogeneous coordinates, homogenization), enumerative geometry, Bezout's theorem
- Week 11: Apollonius problem, Schubert's principle, Grassmannians, Young diagrams, Pieri rule, Schubert calculus
- Week 12 (no notes): classical mechanics (Lagrangian and Hamiltonian), Stern-Gerlach experiment, quantum mechanics, Heisenberg uncertainty principle, quantum field theory, path integrals, Feynman diagrams

### Fall 2017: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic
- Week 2: Invariance of Euler characteristic, planar models, new surfaces (Klein bottle, projective plane), cutting and pasting, connected sums and their Euler characteristics
- Week 3: Orientability, word representations, classification theorem for surfaces
- Week 4: Knots, unknotting number, crossing number, Jones polynomial
- Week 5: Definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces)
- Week 6: Euler characteristic of manifolds, first homotopy groups (computation for the cylinder/circle)
- Week 7: Crash course on group theory (isomorphisms, commutativity, classification of finitely generated abelian groups), classification of manifolds, Poincaré conjecture
- Week 8: Topological degree, Brouwer's fixed point theorem, Nash's equilibrium theorem
- Week 9: Borsuk-Ulam theorem, (discrete) ham sandwich theorem, vector fields, hairy ball theorem, Poincaré-Hopf theorem
- Week 10: Isometry, geodesics, curvature (osculating circles), Gaussian curvature, Gauss's Theorema Egregrium
- Week 11: Local and global Gauss-Bonnet theorems, angles and areas of spherical triangles, non-Euclidean geometry
- Week 12 (no notes): Algorithmic complexity, P vs NP, principal component analysis, randomized algorithms and Yao's principle, elliptic curve cryptography

### Spring 2017: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic (and invariance)
- Week 2: Planar models, new surfaces (Klein bottle, projective plane), cutting and pasting, connected sums and their Euler characteristics
- Week 3: Orientability, word representations, classification theorem for surfaces
- Week 4: Definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces)
- Week 5: Euler characteristic of manifolds, first homotopy groups (computation for the cylinder/circle)
- Week 6: Crash course on group theory (isomorphisms, commutativity, classification of finitely generated abelian groups), classification of manifolds, Poincaré conjecture
- Week 7: Topological degree, Brouwer's fixed point theorem, Nash's equilibrium theorem
- Week 8: Borsuk-Ulam theorem, (discrete) ham sandwich theorem, vector fields, hairy ball theorem, Poincaré-Hopf theorem
- Week 9: Isometry, geodesics, curvature (osculating circles), Gaussian curvature, Gauss's Theorema Egregrium
- Week 10: Local and global Gauss-Bonnet theorems, angles and areas of spherical triangles, non-Euclidean geometry
- Week 11: Knots, unknotting number, crossing number, Jones polynomial, chirality, lower bound on crossing number
- Week 12 (no notes): classical mechanics (Lagrangian and Hamiltonian), Stern-Gerlach experiment, quantum mechanics, Heisenberg uncertainty principle, quantum field theory, path integrals and amplitudes, Feynman diagrams, renormalization

### Fall 2016: Geometry and Topology

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

- Week 1: Introduction, surfaces, homeomorphism, Euler characteristic
- Week 2: Invariance of Euler characteristic, planar models, new surfaces (Klein bottle, projective plane), cutting and pasting
- Week 3: Connected sums and their Euler characteristics, n-holed tori, orientability, word representations
- Week 4: Operations on word representations, classification theorem for surfaces, manifolds
- Week 5: Definition and examples of manifolds, products and quotients (higher-dimensional spheres, tori, and projective spaces)
- Week 6: Euler characteristic and orientability of manifolds, first homotopy groups (computation for the circle)
- Week 7: Higher homotopy groups, complex projective space, holonomy and spherical triangles
- Week 8: Classification of manifolds, Poincaré conjecture, topological degree, Brouwer's fixed point theorem
- Week 9: Borsuk-Ulam theorem, ham sandwich theorem, vector fields, Poincaré-Hopf theorem, hairy ball theorem
- Week 10: Isometry, geodesics, curvature (osculating circles), Gaussian curvature, Gauss's Theorema Egregrium
- Week 11: Local and global Gauss-Bonnet theorems, angles and areas of spherical triangles, non-Euclidean geometry
- Week 12 (no notes): classical mechanics (Lagrangian and Hamiltonian), Stern-Gerlach experiment, quantum mechanics, Heisenberg uncertainty principle, quantum field theory, path integrals and amplitudes, Feynman diagrams, renormalization