When do local structures admit a global realization?
I study how local geometric and combinatorial constraints govern global existence, rigidity, and stability under degeneration.
Geometric analysis
Papers answer questions of Komendarczyk (2008) and Friedman–Park (2016, 2024), prove the magnetic frustration conjecture (Chakradhar et al., 2025), and refute fat-bundle reduction conjectures (Ziller, 2001; Florit–Ziller, 2011). Local volume can vanish while spectral memory survives; gauge mirrors reveal topology invisible to paired curvature.
Rigidity theory
Resolving a conjecture open for nearly two decades: a partition criterion for rigid bodies joined at pins, built from stress degeneracy and collinearity flags.
Mentioned by Bill Jackson at Lancaster’s September 2026 rigidity workshop.
Matroid theory
Resolving a conjecture open for nearly four decades: infinitely many excluded minors rule out a finite forbidden-minor characterization of weak orientability, with a companion Lean formalization.
Dillon Mayhew (Leeds) checked the final Lean theorem statement. Seminar invitation from Jesús De Loera · August 14, 2026.
NeurIPS 2026 Spotlight (Top 1.3% of submissions; Top 5% of accepted papers)
Riemannian Hodge Message Passing (RHMP) learns positive-definite metrics on discrete field spaces, making geometry adaptable while keeping topological identities exact, Hodge energies nonnegative, and propagation equivariant under orthogonal changes of hidden feature basis.
International Conference on Machine Learning (ICML) 2026 · Regular Paper
Introducing a hybrid Eulerian–Lagrangian architecture, Hodge Spectral Duality (HSD), that exploits Hodge orthogonality to decouple unlearnable topological features from learnable geometric dynamics. Combining discrete differential forms with an auxiliary ambient space, the method markedly improves accuracy, efficiency, and fidelity to physical invariants when solving field equations on geometric meshes.
Journal of Physics: Condensed Matter (JPCM)
Resolving the phenomenological coexistence of charge density wave-driven anomalous Hall effect (AHE) and zero-field superconducting diode effect (SDE) in kagome metals. This work establishes a unified theoretical framework connecting microscopic orbital current patterns to macroscopic transport signatures.
Journal of Physics A: Mathematical and Theoretical
Turning the founding question of the fractional Chern insulator (FCI) field—in what precise sense is a Chern band "close to" the lowest Landau level?—from a decade of heuristic band-geometry correlations into rigorous, quantitative control: operator-norm bounds on the projected density algebra that cleanly separate geometric effects (Berry curvature, Fubini–Study metric) from interband-mixing dispersion, with quasi-adiabatic continuation then tying single-particle algebraic deviation to many-body gap stability. A new finite-volume indicator \(\mathcal{E}_{\Lambda,L}\) correlates with the many-body gap in Harper–Hofstadter and kagome models far more sharply than the conventional Berry-curvature proxy \(\sigma_B\), giving a quantitative single-particle criterion for zero-field topological many-body phases.
Journal of Physics: Condensed Matter (JPCM)
Proving that in time-reversal-symmetry-breaking superconductors, the unique symmetry-allowed channel through which a pseudoscalar background field activates the Higgs mode is odd-frequency mixing—static coupling is strictly forbidden. This identifies a previously unrecognized selection rule for TRSB collective-mode coupling.
IEEE Control Systems Letters (L-CSS) · IEEE Conference on Decision and Control (CDC) 2026
Providing a posteriori second-order sufficient conditions for Bolza-type optimal control problems solved via collocation methods, enabling rigorous local optimality verification of numerically computed trajectories.