G-structures and singularities of map-germs

Plática dada por Shyuichi Izumiya (Hokkaido University) en el evento Singularities in generic geometry and its applications: Valencia V, para celebrar el 60 aniversario de Federico Sánchez Bringas en agosto del 2017 en el Auditorio Carlos Graef del Amoxcalli en la Facultad de Ciencias de la UNAM

Abstract:
We consider equivalence relations among smooth section-germs of vector bundles with respect to structure . One of these equivalence relations is the G-equivalence among smooth map germs introduced by Tougeron [5]. Alltough Gervais [1, 2, 3] investigated some properties of G-equivalence, there are no proper examples in their contexts. We give several interesting applications including quantum chemistry and spintronics [4], which are expected to have an application to the theory of topological insulators and so on. We emphasize that we can open a door for new important applications of singularity theory. Moreover we define other equivalence relations with respect to structure of vector bundles. Since a vector space can be identified with the tangent space of itself, we can consider the structure of the vector bundle as a G-structure of the fiber (the vector space).
Then we discover that there are a lot of previous applications or new applications of the theory of singularities as examples of these equivalence relations. Some of these equivalence relations are geometric subgroups of A or K in the sense of Damon. However, some of them are not.
References
[1] J. J. Gervais, Suffuciency of jets. Pacific Journal of Mathematics, 72, 419–422, (1977)
[2] J. J. Gervais, G-stability of mappings and stability of bifurcation diagrams. J. London Math. Soc. 25, 551–563, (1982)
[3] J. J. Gervais, Deformations G-verselles et G-stables. Can. J. Math. XXXVI, 9–21, (1984)

[4] H. Teramoto, K. Kondo, S. Izumiya, M. Toda and T. Komatsuzaki, Classification of Hamiltonians in neighborhoods of band crossings in terms of the theory of singularities. To appear in J. of Math. Phys.
[5] J. C. Tougeron, Ideaux fonctions differentiables. Springer-Verlag, Berlin, Heiderberg, New York 1972

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Agradecemos apoyo de
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