Algebraic Geometry and Topological Properties of Map Germs

Map germs are the local models of mappings between algebraic varieties or manifolds, studied in arbitrarily small neighbourhoods of a point. Algebraic geometry provides powerful tools for understanding the polynomial or analytic structure of such germs, while topology furnishes invariants and classification schemes that capture their qualitative behaviour. Singularities of map germs arise when the local rank of the differential drops, giving rise to intricate phenomena such as cusps, folds and higher‐order degeneracies. Central concepts in this field include the Milnor number, which measures the complexity of singularities, and image Milnor numbers, which quantify the topology of a germ’s image. Recent advances have leveraged stratification theory, blow-up techniques and computable modules to classify germs up to equivalence relations such as \mathscr{A}-equivalence. These classifications underpin applications in areas as diverse as differential topology, robotics path planning and real‐world modelling of optical caustics. Beyond pure classification, ongoing work explores how deformations and unfoldings of germs inform questions of equisingularity, monodromy and cohomological connectivity, revealing deep links between local algebraic structure and global topological invariants.

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Algebraic Geometry and Topological Properties of Map Germs publication trend

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Technical terms

Map germ: An equivalence class of a map defined near a point, capturing its local behaviour under arbitrarily small perturbations.

Isolated instability (isolated singularity): A singular point of a map germ at which no other singularities occur arbitrarily close by.

Milnor number: An integer invariant measuring the complexity of an isolated hypersurface singularity by counting vanishing cycles.

Image Milnor number: A generalisation of the Milnor number that quantifies the topology of the image of a map germ under perturbation.

ICIS (isolated complete intersection singularity): A space defined by the vanishing of several polynomials in which the singularity at a point is isolated.

Topological invariant: A property of a space or map germ that remains unchanged under continuous deformations.

References

1. Recognition and Implementation of Contact Simple Map Germs from (C2,0)→(C2,0). Mathematics (2023).
2. Cohomological connectivity of perturbations of map‐germs. Mathematische Nachrichten (2024).
3. A weak version of the Mond conjecture. Collectanea Mathematica (2023).