CAUTION: The Sender of this email is not from within Dalhousie. THE TALKS WILL ALL BE ON ZOOM THIS SEMESTER. THE ZOOM LOGIN INFORMATION WILL BE EMAILED A FEW DAYS BEFORE THE TALK. ITWILL ALSO BE POSTED ON THE SEMINAR WEB PAGE ON THE DAY OF THE TALK. http://www.sci.brooklyn.cuny.edu/~noson/CTseminar.html PLEASE SPREAD THE WORD. Wednesdays 7:00 - 8:30 PM. Some of the talks are videoed and available <https://www.youtube.com/channel/UCNOfhimbNwZwJO2ltv1AZOw/videos> here. Contact <mailto:noson@sci.brooklyn.cuny.edu> N. Yanofsky to schedule a speaker or to add a name to the seminar mailing list. _____ Fall 2020 _____ * Speaker: Rick Jardine, University of Western Ontario. * Date and Time: Wednesday September 16, 2020, 7:00 - 8:30 PM., on Zoom. * Title: Posets, metric spaces, and topological data analysis. * Abstract: Traditional TDA is the analysis of homotopy invariants of systems of spaces V(X) that arise from finite metric spaces X, via distance measures. These spaces can be expressed in terms of posets, which are barycentric subdivisions of the usual Vietoris-Rips complexes V(X). The proofs of stability theorems in TDA are sharpened considerably by direct use of poset techniques. Expanding the domain of definition to extended pseudo metric spaces enables the construction of a realization functor on diagrams of spaces, which has a right adjoint Y |--> S(Y), called the singular functor. The realization of the Vietoris-Rips system V(X) for an ep-metric space X is the space itself. The counit of the adjunction defines a map \eta: V(X) --> S(X), which is a sectionwise weak equivalence - the proof uses simplicial approximation techniques. This is the context for the Healy-McInnes UMAP construction, which will be discussed if time permits. UMAP is non-traditional: clusters for UMAP are defined by paths through sequences of neighbour pairs, which can be a highly efficient process in practice. _____ * Speaker: David Ellerman, University of Ljubljana. * Date and Time: Wednesday September 30, 2020, 7:00 - 8:30 PM., on Zoom. * Title: The Logical Theory of Canonical Maps: The Elements & Distinctions Analysis of the Morphisms, Duality, Canonicity, and Universal Constructions in Sets. * Paper: <http://www.sci.brooklyn.cuny.edu/~noson/Ellerman2020.pdf> Available here. * Abstract: Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Sets, the category of sets and functions. The analysis extends directly to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and refinement) in the lattices for the dual logics define morphisms. The thesis is that the maps that are canonical in Sets are the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps. _____ * Speaker: Jonathon Funk, Queensborough CUNY. * Date and Time: Wednesday October 14, 2020, 6:00 - 7:30PM (NOTICE DIFFERENT TIME) on Zoom. * Title: Pseudogroup Torsors. * Abstract: We use sheaf theory to analyze the topos of etale actions on the germ groupoid of a pseudogroup in the sense that we present a site for this topos, which we call the classifying topos of the pseudogroup. Our analysis carries us further into how pseudogroup morphisms and geometric morphisms are related. Ultimately, we shall see that the classifying topos classifies what we call a pseudogroup torsor. In hindsight, we see that pseudogroups form a bicategory of `flat' bimodules. Joint work with Pieter Hofstra. _____ * Speaker: Andrei V. Rodin, Saint Petersburg State University. * Date and Time: Wednesday October 21, 2020, 7:00 - 8:30 PM., on Zoom. * Title: ****. * Abstract: *** _____ * Speaker: Larry Moss, Indiana University. * Date and Time: Wednesday October 28, 2020, 7:00 - 8:30 PM., on Zoom. * Title: Coalgebra in Continuous Mathematics. * Abstract: A slogan from coalgebra in the 1990's holds that 'discrete mathematics : algebra :: continuous mathematics : coalgebra' The idea is that objects in continuous math, like real numbers, are often understood via their approximations, and coalgebra gives tools for understanding and working with those objects. Some examples of this are Pavlovic and Escardo's relation of ordinary differential equations with coinduction, and also Freyd's formulation of the unit interval as a final coalgebra. My talk will be an organized survey of several results in this area, including (1) a new proof of Freyd's Theorem, with extensions to fractal sets; (2) other presentations of sets of reals as corecursive algebras and final coalgebras; (3) a coinductive proof of the correctness of policy iteration from Markov decision processes; and (4) final coalgebra presentations of universal Harsanyi type spaces from economics. This talk reports on joint work with several groups in the past 5-10 years, and also some ongoing work. _____ * Speaker: Luis Scoccola, Michigan State University. * Date and Time: Wednesday November 4, 2020, 7:00 - 8:30 PM., on Zoom. * Title: Locally persistent categories and approximate homotopy theory. * Abstract: In applied homotopy theory and topological data analysis, procedures use homotopy invariants of spaces to study and classify discrete data, such as finite metric spaces. To show that such a procedure is robust to noise, one endows the collection of possible inputs and the collection of outputs with metrics, and shows that the procedure is continuous with respect to these metrics, so one is interested in doing some kind of approximate homotopy theory. I will show that a certain type of enriched categories, which I call locally persistent categories, provide a natural framework for the study of approximate categorical structures, and in particular, for the study of metrics relevant to applied homotopy theory and metric geometry. _____ * Speaker: Noah Chrein, University of Maryland. * Date and Time: Wednesday November 11, 2020, 7:00 - 8:30 PM., on Zoom. * Title: Yoneda ontologies. * Abstract: TBA _____ * Speaker: Enrico Ghiorzi, Appalachian State University. * Date and Time: Wednesday November 18, 2020, 7:00 - 8:30 PM., on Zoom. * Title: Internal enriched categories. * Abstract: Internal categories feature a notion of completeness which is remarkably well behaved. For example, the internal adjoint functor theorem requires no solution set condition. Indeed, internal categories are intrinsically small, and thus immune from the size issues commonly afflicting standard category theory. Unfortuntely, they are not quite as expressive as we would like: for example, there is no internal Yoneda lemma. To increase the expressivity of internal category theory, we define a notion of internal enrichment over an internal monoidal category and develop its theory of completeness. The resulting theory unites the good properties of internal categories with the expressivity of enriched category theory, thus providing a powerful framework to work with. _____ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]