The Mathematics PhD at UIUC nurtures pure and applied scholars tackling problems from Langlands reciprocity to topological data analysis. Students prove new theorems on algebraic K-theory, simulate stochastic PDEs for climate models on Blue Waters 2, and translate category theory into safer type systems with CS collaborators. Weekly seminars, visiting Fields medalists, and pedagogical training craft versatile thinkers who publish in *Annals* and guide next-gen learners.
Explicit reciprocity laws for Hilbert modular forms
Persistent-homology invariants quantifying brain-connectome development
Rigorous error bounds for neural-network solvers of Navier–Stokes
KAM-theory extension for quasi-periodic solutions in climate oscillations
Probabilistic proof of combinatorial identities via random walks on groups
Topos-theoretic framework for constructive quantum mechanics
Large-deviation principles for stochastic gradient descent
Cluster-algebra structures in scattering amplitudes
Mathematical modeling of pandemic dynamics on time-varying networks
Quantum-safe cryptosystem design using supersingular isogeny graphs
Grant proposal for open-source software teaching algebraic topology
White paper on ethical use of predictive math models in public policy
Public lecture series on mathematics of voting fairness
Machine-learning conjecture generator for integer partitions
Interactive visualization of hyperbolic tessellations in VR
Categorical semantics for reversible computing languages
Prove, model, and illuminate with Math @ UIUC.
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