Northwestern’s Mathematics PhD ranges from algebraic geometry and number theory to topological data analysis. Doctoral researchers prove new theorems on Langlands functoriality, model knot energies, and collaborate with machine-learning theorists. A communication fellowship trains students to translate abstraction for broader audiences.
Prove modularity of certain Calabi–Yau threefold Galois representations
Analyze persistence-homology stability in noisy point-cloud data
Develop new bounds on expander graph spectra via representation theory
Study dynamics of rational maps on moduli spaces of Riemann surfaces
Model stochastic PDEs arising in turbulent fluid interfaces
Extend rough-path theory to manifold-valued signals
Investigate combinatorial game theory applications in network security
Derive novel inequalities in harmonic analysis for multilinear operators
Apply p-adic Hodge theory to Selmer groups of elliptic curves
Explore knot homology invariants under satellite operations
Construct asymptotic-safe discretizations for integrable PDEs
Examine probabilistic zeta functions of infinite groups
Optimize low-rank tensor decompositions for quantum many-body systems
Design outreach modules teaching infinity via set-theory visualizations
Write an expository survey on machine learning in pure-math proof discovery
Explore deep structures and new horizons with Northwestern’s Mathematics PhD.
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