Topology
Bases, Denseness, & Closure Operators
Quotient Map, Mobius Band, & Klein Bottle
Countability, Hausdorff, & Open Covers
Homotopy Theory & Fundamental Groups
Product, Subspace, & Zariski Topologies
Covering, Monodromy, & Conjugacy Class
Abstract Algebra
Cosets, Normal Subgroups, Lagrange’s Theorem
Isomorphism Theorems, Generators, Composition Series
Group Rings, Prime Ideals, Maximal Ideals
Ring Isomorphisms, Principal Ideal Domains, Irreducibles
Fractals, Dynamics, & Chaos Theory
Newton’s Method & Box-Counting Dimension
Markov Chains, Stochastic Matrices, & Distributions
Vector Field Dynamics & Periodic Orbits
Graph Theory, Conjugacies, & Chaotic Maps
Real Analysis
σ-Algebras
Measure Spaces
Premeasures & Extensions
Lebesgue Measure
Complex Analysis
Mobius Transformations, Riemann Sphere, Rational Functions
Holomorphic Functions, Power Series, Radius of Convergence
Riemann Surfaces, Analytic Continuation, Conformal Maps
Meromorphic Functions, Residue Calculus, Harmonic Functions
Functional Analysis
Index Theory, Fredholm Operators, & K-Theory
Toeplitz Matrices, Symbol, & Winding Number
Hilbert Spaces, Unitary Mappings, & Fourier Series
Toeplitz C*-Algebra, Atkinson's Theorem
K-Theory & C*-Algebras
Uniqueness of Norm, Hilbert spaces
Projections & *-Homomorphisms
K0 Group of Compact Operators
Riemannian Geometry
Riemannian Metrics, Connections
Geodesics, Convex Neighborhoods
Sectional Curvature, Jacobi Fields
Conjugate Points, Hopf-Rinow
Symplectic Geometry & Lie Theory
I learned topics in symplectic geometry and Lie theory to prove certain results of a 2-dimensional topological quantum field theory.
