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Sonja Henze will defend her doctoral thesis on Thursday, October 31st

  • The defense will take place at the “Miguel de Guzmán” room of the Faculty of Mathematics of Complutense University of Madrid, at 12:30 pm

Sonja Henze obtained a Bachelor’s degree in Mathematics from RWTH Aachen University (Germany) in 2011 and a Master’s degree in Mathematics from the same university in 2013. During her master’s she did a one year stay at the University of Nottingham (UK) thanks to the Erasmus program.

In 2013 she obtained PhD grant from La Caixa Foundation and started working on her doctoral thesis at ICMAT (Madrid, Spain). During her PhD she did a research stay at IMPA (Brazil) and visited the Basque Center for Applied Mathematics – BCAM several times, where she is a member of the ERC project led by Ikerbasque Research Professor Javier Fernández de Bobadilla, New methods and interactions in Singularity Theory and beyond.

Her PhD thesis has been supervised by Javier Fernandez de Bobadilla, leader of BCAM’s research line in Singularity Theory and Algebraic Geometry, and Maria Pe Pereira from the Complutense University of Madrid.

On behalf of all BCAM members, we would like to wish Sonja the best of luck in her upcoming thesis defense.

PhD thesis Title: Moderately Discontinuous Algebraic Topology for Metric Subanalytic Germs

The talk is about new subanalytic bi-Lipschitz invariants of subanalytic germs equipped with a metric. If the germs are real or complex analytic equipped with the inner or outer metric, the invariants are analytic invariants. One of the invariants is a homology theory, the other one a homotopy theory. Both of them are functors into a category that has families of groups and group homomorphisms as objects and commutative diagrams as morphisms. The parameter for the family of groups is given by b ∈ (0, ∞]. We call the group corresponding to  b ∈ (0, ∞] the b-MD homology and b-MD homotopy, respectively.

The b-MD homology shares several properties with the singular homology of topological spaces for any b ∈ (0, ∞].: it is invariant by suitable metric homotopies; it allows a relative and absolute Mayer-Vietoris long exact sequences for a suitable cover of the metric subanalytic germ; and as a consequence, we have a certain theorem of excision and a Ĉech spectral sequence; and we have a long exact relative homology sequence. Those tools make concrete computations possible. We illustrate that by means of the example of complex plane algebraic curve germs equipped with the outer metric. For irreducible complex plane algebraic curve germs equipped with the outer metric our homology theory is a complete invariant.

The b-MD homotopy shares several properties with the ordinary homotopy of punctured topological spaces for b ∈ (0, ∞]. In particular, it admits a Hurewicz homomorphism from the b-MD homotopy to the b-MD homology; in degree one, the Hurewicz homomorphism is an isomorphism when abelianizing the domain.