Phase 1: Mathematical Foundations (2-3 months)

Prerequisites

Linear Algebra

• Vector spaces, linear transformations
• Eigenvalues and eigenvectors
• Inner products and orthogonality
• Tensor products

Multivariable Calculus

• Partial derivatives and gradients
• Multiple integrals
• Line and surface integrals
• Jacobians and change of variables
• Implicit and inverse function theorems

Point-Set Topology

• Open and closed sets
• Continuity and homeomorphisms
• Compactness and connectedness
• Quotient spaces

Abstract Algebra (Basic)

• Groups, rings, and fields
• Homomorphisms and isomorphisms

Phase 2: Introduction to Manifolds (3-4 months)

Smooth Manifolds

• Topological manifolds
• Smooth structures and atlases
• Smooth maps and diffeomorphisms
• Partitions of unity
• Tangent spaces and tangent bundles
• Cotangent spaces and differential forms
• Vector fields and flows
• Submanifolds and embeddings

Calculus on Manifolds

• Differential forms
• Exterior derivative
• Wedge product
• Pull-backs and push-forwards
• Integration on manifolds
• Stokes' theorem

Phase 3: Riemannian Geometry (3-4 months)

Riemannian Metrics

• Definition and examples
• Length and distance
• Riemannian volume forms
• Isometries

Connections and Covariant Derivatives

• Affine connections
• Levi-Civita connection
• Parallel transport
• Geodesics and exponential map
• Normal coordinates

Curvature Theory

• Riemann curvature tensor
• Sectional curvature
• Ricci curvature and scalar curvature
• Gaussian curvature (surfaces)
• Weyl tensor
• Curvature operator

Classical Theorems

• Gauss-Bonnet theorem
• Myers' theorem
• Bonnet-Myers theorem
• Cartan-Hadamard theorem

Phase 4: Advanced Topics (4-6 months)

Lie Groups and Lie Algebras

• Matrix Lie groups
• Lie algebra structure
• Exponential map
• Adjoint representation
• Homogeneous spaces

Fiber Bundles and Connections

• Principal bundles
• Associated bundles
• Connections on bundles
• Gauge theory basics
• Characteristic classes (Chern, Euler)

Comparison Geometry

• Jacobi fields
• Conjugate points
• Cut locus
• Rauch comparison theorems

Geometric Analysis

• Laplace-Beltrami operator
• Hodge theory
• Harmonic forms
• Spectral geometry
• Heat kernel

Phase 5: Specialized Topics (Ongoing)

Complex Geometry

• Complex manifolds
• Kähler manifolds
• Hermitian metrics
• Dolbeault cohomology

Symplectic Geometry

• Symplectic manifolds
• Hamiltonian mechanics
• Moment maps
• Symplectic reduction

Differential Topology

• Morse theory
• Cobordism
• Surgery theory
• Index theorems

Geometric Flows

• Ricci flow
• Mean curvature flow
• Yamabe flow

Modern Applications

• Optimal transport
• Geometric deep learning
• Shape analysis
• General relativity

Geometric Deep Learning (2017-Present)

• Graph neural networks on manifolds
• Equivariant neural networks using symmetry groups
• Riemannian optimization for machine learning
• Geometric representation learning
• Applications: protein folding, molecular dynamics, 3D computer vision

Optimal Transport Theory (2010s-Present)

• Wasserstein geometry
• Computational optimal transport algorithms (Sinkhorn)
• Applications in machine learning and data science
• Connections to gradient flows

Ricci Flow and Geometric Analysis

• Post-Poincaré conjecture developments
• Ricci flow on singular spaces
• Ancient solutions and classification
• Numerical Ricci flow algorithms

Quantum Geometry

• Non-commutative differential geometry
• Quantum groups and deformation theory
• Connections to quantum field theory and string theory

Geometric Statistics and Shape Analysis

• Statistical methods on Riemannian manifolds
• Shape spaces and diffeomorphism groups
• Medical imaging applications
• Computational anatomy

Synthetic Geometry

• Metric spaces with synthetic curvature bounds
• RCD spaces (Riemannian Curvature Dimension)
• Applications to optimal transport and analysis

Topological Data Analysis

• Persistent homology
• Mapper algorithm
• Geometric and topological inference
• Applications in data science

Discrete Differential Geometry

• Discrete exterior calculus
• Discrete curvature notions
• Polyhedral surfaces
• Applications in computer graphics and physics simulations

Gauge Theory and Mathematical Physics

• Mirror symmetry
• Calibrated geometries (G2, Spin(7))
• Supersymmetric field theories
• Geometric quantization

Information Geometry

• Fisher-Rao metric on probability distributions
• Applications in statistics and machine learning
• Divergence functions and dual connections

Textbooks

• Beginner: "Differential Geometry of Curves and Surfaces" by Do Carmo
• Intermediate: "Introduction to Smooth Manifolds" by Lee
• Advanced: "Riemannian Geometry" by Do Carmo or Petersen
• Reference: "Foundations of Differential Geometry" by Kobayashi & Nomizu

Online Resources

• MIT OCW: Differential Geometry courses
• YouTube: XylyXylyX channel, MathTheBeautiful
• nLab: Comprehensive wiki for advanced topics
• ArXiv: Research papers (differential geometry section)

Programming Practice

• Start with Python + NumPy/SciPy
• Progress to specialized libraries (GeomStats, PyTorch Geometric)
• Learn symbolic computation (SymPy, SageMath)

Estimated Timeline

18-24 months for solid foundation, lifetime for mastery. Adjust pace based on your background and goals. Focus on understanding concepts deeply before moving forward, and always implement computational projects to solidify theoretical knowledge.