The PhD in Mathematics at MIT trains scholars in abstract reasoning and problem-solving across theoretical and applied domains. These project ideas support deep inquiry into number theory, topology, logic, algebra, analysis, and modern mathematical modeling.
Advancements in Prime Gap Theorems via Sieve Theory
Topological Invariants in 4-Manifolds Using Seiberg–Witten Theory
Categorical Models for Homotopy Type Theory
Algebraic Structures in Elliptic Cohomology
Fourier Analysis Applications in PDE Solvability
Modular Forms and Their Applications in Cryptographic Systems
Ergodic Theory and Measure-Preserving Transformations
Explicit Constructions in the Langlands Program
Symplectic Geometry in Classical and Quantum Mechanics
Hardness of Approximation Results in Combinatorics
Lattice-Based Cryptographic Schemes from Geometry of Numbers
Graph Limits and Sparse Random Structures
Algebraic Combinatorics in Schubert Calculus
Topos Theory and Sheaf-Theoretic Logic
Nonlinear Optimization via Variational Inequalities
Quantum Groups and Their Categorification
Spectral Theory for Large Sparse Operators
Stochastic PDEs and Applications in Financial Modeling
Geometric Representation Theory in Lie Algebras
Analytic Number Theory in Additive Problems
Braid Groups and Applications in Low-Dimensional Topology
Mathematical Models for Epidemic Dynamics on Networks
Reinforcement Learning Modeled with Stochastic Processes
Combinatorial Proofs in Ramsey Theory and Extremal Graph Theory
Applications of p-Adic Analysis in Arithmetic Geometry
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