Peter F. Stiller

Peter F. Stiller
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Peter F. Stiller
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Mathematics - Algebraic Geometry (2)
 
Statistics - Machine Learning (2)
 
Computer Science - Learning (2)
 
Mathematics - Metric Geometry (1)

Publications Authored By Peter F. Stiller

This work draws its inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Memoli to partition-based clustering, and the Isbell/Dress school's study of injective envelopes. Carlsson and Memoli introduce the idea of viewing clustering methods as functors from a category of metric spaces to a category of clusters, with functoriality subsuming many desirable properties. Read More

We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. Read More

In his pioneering work on injective metric spaces Isbell attempted a characterization of cellular complexes admitting the structure of an injective metric space, following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure. Considerable advances in the understanding, classification and applications of injective envelopes have been made by Dress, Huber, Sturmfels and collaborators (producing, among other results, many specific examples of injective polyhedra), and most recently by Lang, yet a combination theory explaining how to glue injective polyhedra together to produce large families of injective spaces is still unavailable. In this paper we apply the duality theory of cubings -- simply connected non-positively curved cubical complexes -- to provide a more principled and accessible proof of a result of Mai and Tang on the injective metrizability of collapsible simplicial complexes. Read More

Given a torsion section of a semistable elliptic surface, we prove equidistribution results for the components of singular fibers which are hit by the section, and for the root of unity (identifying the zero component with ${\Bbb C}$) which is hit by the section in case the section hits the zero component. Read More

We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an elliptic curve defined over a function field in one variable. An interesting conjecture concerning Galois actions on the relative de~Rham cohomology of these surfaces is discussed. Read More