Mathematics / research
Modern mathematical structures
Approach research mathematics through precise introductory models of symmetry, homological structure, transport, information geometry, and inverse problems.
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Module 01 research
Modern mathematical structures
Approach research mathematics through precise introductory models of symmetry, homological structure, transport, information geometry, and inverse problems.
- 01
Representation theory and symmetry
Turn abstract composition into linear maps, then distinguish invariant pieces, faithful actions, and irreducibility.
30 min - 02
Lie algebra representations
Represent infinitesimal symmetries through bracket-preserving maps and verify the smallest noncommutative example.
32 min - 03
Categories, functors, and natural transformations
Organize mathematical constructions by identity, composition, and compatibility across every object.
31 min - 04
Chain complexes and cohomology
Compute cycles modulo boundaries and see why a differential squaring to zero does not make homology vanish.
32 min - 05
Symplectic geometry and Hamiltonian flow
Generate dynamics from a closed nondegenerate two-form and distinguish area-preserving flow from damping.
33 min - 06
Optimal transport and Wasserstein distance
Compare probability measures through admissible couplings and compute a transport distance without confusing cost with a metric.
34 min - 07
Information geometry and Fisher metrics
Measure local statistical distinguishability while tracking regularity, identifiability, and coordinate changes.
34 min - 08
Inverse problems and regularization
Identify unstable inverse directions and quantify how regularization exchanges noise amplification for bias.
34 min - 09
Random matrices and spectral laws
Normalize a Wigner ensemble correctly and distinguish an asymptotic empirical law from individual eigenvalues or a finite plot.
35 min - 10
Research conjectures and proof strategies
Turn a plausible pattern into a precise statement, search for counterexamples, and distinguish evidence from a complete argument.
33 min