Columbia’s Ph.D. in Mathematics is designed for advanced research across pure and applied mathematical disciplines. These dissertation topics span number theory, differential geometry, algebraic structures, and stochastic processes.
Arithmetic Geometry: Diophantine Equations over Function Fields
Homotopy Type Theory and Foundations of Mathematics
Ergodic Theory and Applications in Symbolic Dynamics
Spectral Graph Theory and Quantum Walks on Networks
Modular Forms and Their Role in Modern Number Theory
Geometric Group Theory and Hyperbolic 3-Manifolds
Machine Learning and Optimization in Banach Spaces
Random Matrix Theory and Universality in Eigenvalue Distributions
Algebraic Topology Approaches to Data Shape Analysis
P-adic Analysis and Applications in Cryptography
Category Theory and Monoidal Structures in Computer Science
Lie Groups and Their Representations in Particle Symmetry
Partial Differential Equations in Fluid Dynamics Simulation
Stochastic Calculus in Mathematical Finance Models
Nonlinear Wave Equations in General Relativity
The Langlands Program and Duality in Number Theory
Sheaf Cohomology in Complex Algebraic Varieties
Mathematical Logic: Forcing and Large Cardinals
Topos Theory and Its Role in Modern Logic
Variational Methods in Elasticity and Continuum Mechanics
Tropical Geometry and Its Combinatorial Applications
Symplectic Geometry in Classical and Quantum Mechanics
Percolation Theory and Statistical Phase Transitions
Algebraic K-Theory and Homological Invariants
Geometric Flows and the Ricci Flow Conjecture
Arithmetic Dynamics in Rational Maps over Finite Fields
Hodge Theory and Period Mappings in Algebraic Geometry
Dynamical Systems on Fractal Structures
Wavelet Theory and Its Applications in Image Compression
Topological Data Analysis in High-Dimensional Statistics
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