Doctoral Thesis Defense – Jake Smith

April 15, 2024 at 1:00pm, Lippitt Hall Room 205

Title: “Fitting Ideals Without a Presentation”

Abstract: For any finite holomorphic map-germ, a classic paper of Mond and Pellikaan established that the presentation matrix of the source as a module over the target can be chosen square and symmetric, and provide an algorithm for producing such a presentation.Moreover, they showed that the Fitting ideals of such a presentation provide an appropriate analytic structure on the multiple point spaces in the target, that is, one which is calculable, behaves well under deformation and which is reduced at ordinary k-tuple points. The multiple point spaces in the target, endowed with the above analytic structure, provide key insights to the topology and geometry of the image of finite map-germs. For the Fitting ideals defining the single (image), double and triple point spaces there are known methods for computing the Fitting ideals without an explicit matrix presentation (due to work of Mond-Pellikaan and Piene), suggesting that the same may exist for higher multiple point spaces. In this thesis, we generalize a result of Mond and Pellikaan to higher multiple point spaces. We also present, as a result of our investigation, a related inductive description of the Fitting ideals of square matrices with generic entries, suggesting the possibility of further generalizing our main result.