The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Each week the Department of Mathematics hosts a wide variety of seminars and events. For a comprehensive list of events, please consult our departmental calendar.
Failure of some (important) properties of a holomorphic mapping manifests as degeneracies in the family of fibres of the mapping. Among these properties are openness and flatness. The first goal in my thesis is to develop criteria such that first, they effectively (i.e., computationally) detect such degeneracies in the family of fibres, and second, they are applicable to the case of mappings with singular targets. Particularly for flatness, no such algorithms that work in the general setting of singular targets were known before. We prove that a mapping (with a locally irreducible target) is flat (resp. open) if and only if no (resp. isolated) irreducible component is mapped to the origin in the pullback of the mapping by the blowing-up. The second goal is to characterize different modes of such degeneracies. We take an index that measures the level of non-openness of mappings, and obtain some results on its behaviour.
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