The University of Chicago’s Ph.D. in Mathematics program is world-renowned for its depth in pure mathematics and expansion into applied areas. These dissertation ideas help students explore critical mathematical phenomena across logic, geometry, algebra, and computational frameworks.
Homological Techniques in Derived Algebraic Geometry
Representation Theory of Lie Algebras and Quantum Groups
Moduli Spaces in Algebraic Geometry and String Theory Connections
Spectral Graph Theory and Its Applications in Network Analysis
Topological Data Analysis for High-Dimensional Data Structures
Probabilistic Number Theory and the Distribution of Prime Gaps
Category Theory in Homotopy Type Theory Frameworks
Existence and Uniqueness of Solutions to Nonlinear Elliptic PDEs
Ergodic Theory and Its Application in Dynamical Systems
Applications of Model Theory in Algebraic Geometry
Analytic Number Theory and Zeta Function Hypotheses
Wavelet Methods for Solving Multiscale PDEs
Algebraic Topology in Computational Neuroscience Models
Random Matrix Theory in Statistical Physics and Finance
Geometric Group Theory and Hyperbolic Manifold Structures
Knot Theory and Quantum Invariants in Low-Dimensional Topology
Optimal Transport Problems and Wasserstein Metric Geometry
Fixed Point Theorems in Nonlinear Functional Analysis
Symplectic Geometry and Floer Homology Applications
Combinatorial Approaches to Ramsey Theory
Sheaf Theory in the Study of Differential Equations
Fourier Analysis Techniques in Signal and Image Processing
Algebraic Curves and Riemann Surfaces in Complex Geometry
Stochastic Differential Equations in Mathematical Finance
Mathematical Logic and the Foundations of Computability Theory
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