How can one thing remain itself while its representation keeps changing?
Geometric Transport studies representation, local frames, transport, topology, gluing, equivalence, and coherence in a setting where those words have precise mathematical content.
The progression
The research moves from simple phase invariance to local-to-global geometry and finally to equivalence between whole representation systems and ordered proof-path coherence.
Quantum rays, multiple dynamics descriptions, local gauge choices, Berry transport, and holonomy.
Tiny gauge-invariant loops assemble into first Chern topology; the integer changes only through gap closure.
Degenerate subspaces produce matrix-valued transport where order matters, extending into second-Chern topology.
Several local descriptions can define one global object; even the atlas itself may change without changing the underlying bundle.
Two atlases can be explicitly related through common refinement rather than canonicalized into one privileged representation.
Different witness paths are compared directly; the non-Abelian case shows why ordered composition must remain explicit.
Coordinates can change while the object does not.
The research repeatedly scrambles local representation choices and asks whether the physically meaningful result survives.
Global structure need not have one global coordinate system.
Patch compatibility and transition data can carry information that no single chart contains.
Same object need not mean same representation.
Explicit equivalence witnesses preserve both sides rather than erasing difference through canonicalization.
How you got there can be part of the result.
Direct and composed witnesses can be tested for agreement, and composition order is never silently discarded.