The New York City Category Theory Seminar: The Fall Line-Up of Talks.

The New York City Category Theory Seminar: The Fall Line-Up of Talks.

[Noson Yanofsky]

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.

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Fall 2020

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*  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.

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*  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.

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*  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.

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*  Speaker:     Andrei V. Rodin, Saint Petersburg State University.

*  Date and Time:     Wednesday October 21, 2020, 7:00 - 8:30 PM., on Zoom.

*  Title:    ****.

*  Abstract: ***

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*  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.

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*  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.

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*  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

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*  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.

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