The New York Category Theory Seminar and the New York Haskell Meetup is sponsoring a

The list for announcements and scheduling regarding this group is also at the hott-nyc google group:

https://groups.google.com/forum/#!forum/hott-nyc

In this talk, I will first give a slightly more detailed description of the Stone-Priestley duality theorem. I will then broadly describe two recent mathematical contributions, which we obtained by building on this duality: (1) an analysis of sheaf representations of lattice-ordered algebraic structures (joint work with Gehrke, CNRS); (2) a model-theoretic perspective on profinite semigroups (joint work with Steinberg, CCNY).

→"On Connectivity Spaces", Cahiers de topologie et géométrie différentielle catégoriques, LI, 4 (2010) 282-315. arXiv : 1001.2378.

→"Espaces connectifs : représentations, feuilletages, ordres, difféologies", Cahiers de topologie et géométrie différentielle catégoriques, LIV (2013). arXiV : 1610.07366.

→"Structures connectives de l'intrication quantique", juillet 2014, arXiv : 1407.5920.

→"Définition du topos d'un espace connectif", octobre 2016, arXiv : 1610.04422.

→"Dynamiques en interaction : une introduction à la théorie des dynamiques sous-fonctorielles ouvertes", août 2016, arXiv : 1608.07938.

What is the relation to the usual Shannon theory of information? The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform requantifying transformation from the corresponding formulas of logical information theory. The transformation replaces the counting of distinctions with the counting of the number of binary partitions (bits) it takes, on average, to make the same distinctions by uniquely encoding the distinct elements--which is why the Shannon theory perfectly dovetails into coding and communications theory. Moreover, logical entropy is a measure (in the sense of measure theory) while Shannon entropy is not--even though Shannon carefully defined the compound notions of his entropy so that they would satisfy the usual Venn diagram formulas (e.g., inclusion-exclusion) as if it were a measure on a set. Since logical entropy automatically satisfies those formulas as a (probability) measure, and the uniform requantifying transform preserves Venn diagrams, that explains how the Shannon notions can have those relationships. In short, the logical theory of information-as-distinctions is grounded on logic and it displaces the Shannon theory from being the foundational theory of information to being a higher-order specialized theory that requantifies logical information for the purposes of coding and communications theory, where it has been so successful.

We show how to mathematically model the two versions starting just with a subset S of U and then focus on the less familiar #2 interpretation where S is viewed not as a set of distinct elements but as a more abstract entity 'S-ness' that is definite on what is common between the elements of S and indefinite on how they differ. The point is that the #2 case dovetails precisely with the quantum mechanics notion of a superposition of distinct eigenstates of some observable which is a state definite only on what is common to the superposed eigenstates and is objectively or onticly indefinite between them. The process in general of classifying by some attribute to make the #2 indefinite state more definite then emerges as the process of measurement in QM.

Joint with Primoz Skraba and Joao Pita Costa, we have developed a topos-based foundation for persistent homology, where the underlying Heyting algebra P captures the *shape* of the persistent homology theory, and persistent homology emerges as the internal theory of (semi)simplicial homology within the set theory determined by the topos of sheaves over P.

In this talk, I will give an introduction to persistent homology, and demonstrate how the theory fits with a topos-based foundation.

Previous Semesters:

##### Fall 2015 - Fall 2016 Topos Theory Reading Group

##### Fall 2013 - Spring 2015 Homotopy Type Theory Reading Group

##### Spring 2012

##### Fall 2011

##### Spring 2011

##### Fall 2010

##### Spring 2010

##### Fall 2009

##### Spring 2009

Other Category Theory Seminars: