NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA

March - May 2015

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Go to April 10 talks      Go to April 23 talk      Go to May 8 talk

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FRIDAY, MARCH 27:   Ralph Kopperman will discuss "Decimals and Aspigories". At CCNY; tea 12:30 p.m. at NAC 8/130, talk 1:00 to 2:00 p.m. at NAC 5/150. For more information, contact Ralph Kopperman, rdkcc@ccny.cuny.edu.

     ABSTRACT:   We examine the construction of the reals from finite spaces of decimals.

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WORKSHOP ON CATEGORICAL ASPECTS OF PARTIAL METRIC SPACES

Friday, April 10, 11 a.m.-.2p.m., at CCNY, NAC 5/150

SCHEDULE:

     11 a.m. - 12 noon, Michael Bukatin, Nokia Corporation, will discuss "Topics in bicontinuity, partial inconsistency, and vector semantics".

     12 noon - 12:30 p.m., tea.

     12:30 p.m. - 1:30 p.m., Steve Matthews, University of Warwick, will speak on "Intensional Partial Metric Spaces".

     1:30 p.m. - 2 p.m., tea.

ABSTRACTS:

     Michael Bukatin, Nokia, "Topics in bicontinuity, partial inconsistency, and vector semantics"

      The Scott topology allows modeling the meaning of programs with continuous functions. Admitting the elements expressing partial contradictions, such as negative probabilities and segments of negative length, allows us to bridge the gap between Scott domains and vector spaces, raising hopes that standard methods of applied mathematics will become applicable to the tasks of program learning, program transformation, and program synthesis.

      In this talk I hope to cover three topics within this context.

     1) Bicontinuity and the domain of arrows. A partially ordered set D is a bicontinuous domain, if both D and D^* (D with the reverse order) are continuous domains. The domain of arrows, D^* \times D, is a remarkably interesting structure. Our original inspiration for the domain of arrows came from the desire to consider limits and continuity in the intensional partial metric spaces (functors into a category of partial metrics) proposed by Steve Matthews. Important constructions on the domain of arrows resemble distributors (profunctors) and hopefully can be lifted back to the context of intensional spaces.

     2) Negative probability and quantum algorithms. The phase space formulation of quantum mechanics uses the Wigner quasiprobability distribution rather than complex amplitude. There is a body of research reformulating quantum algorithms in terms of quasiprobabilities. At the same time there are surprising parallels between Grover's quantum algorithm and a scheme of computations in the domain of arrows involving a mixture of monotonic steps and order-reversing involutions.

     3) Linear combinations of computations. I am aware of two large classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. Working with those types of computations seems to give the most direct shot at the practical use of vector semantics in the tasks of program learning, program transformation, and program synthesis.

     Steve Matthews, University of Warwick, "Intensional Partial Metric Spaces"

     Partial metric spaces [4,5] generalise metric spaces [1] by allowing self-distance to be a non-negative number. Originally motivated by the goal to reconcile metric space topology with the logic of computable functions and Dana Scott's innovative theory of topological domains, they are now too rigid a form of mathematics to be of use in modelling contemporary applications software (aka Apps) which is increasingly concurrent, pragmatic, interactive, rapidly changing, and inconsistent in nature.

      This presentation aims to further develop partial metric spaces in order to catch up with the modern computer science of Apps. Our illustrative working example is that of the Lucid programming language [2,3] and a temporal generalisation using Wadge's hiaton, an innovative notion of pause in a computation.

      References:

      [1] M. Fr\'{e}chet, "Sur quelques points du calcul fonctionnel", Rend.Circ. Mat. Palermo 22 (1906), pp. 1-74.

      [2] E.A. Ashcroft \& W.W. Wadge, "Lucid, a nonprocedural language with iteration", Comm. ACM 20, (1977), pp. 519-526.

      [3] W.W. Wadge \& E.A. Ashcroft, "Lucid, the Dataflow Programming Language", APIC Studies in Data Processing 22, Academic Press (1985).

      [4] S.G. Matthews, "Partial metric topology", in General Topology \& its Applications, Proc. 8th Summer Conf. Queen's College (1992), Annals of the New York Academy of Science, eds. Andima, S. et al, Vol 728, (1994), pp. 183-197.

      [5] M. Bukatin, R. Kopperman, S. Matthews, \& H. Pajoohesh, "Partial Metric Spaces", American Mathematical Monthly, Vol. 116, No. 8 (2009), pp. 708-718.

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THURSDAY, APRIL 23:   Aaron R. Todd, Baruch College, CUNY, will discuss "The Separable Quotient Problem, Barrelled Spaces and Cc(X)". At CCNY; tea 3:30 p.m. NAC 8/130, talk 4 p.m. NAC 6/110. For more information, contact Ralph Kopperman, rdkcc@ccny.cuny.edu.

     ABSTRACT:   A long standing problem is whether every infinite dimensional Banach space has an infinite dimensional separable quotient. We discuss joint work with J. Kakol and S. A. Saxon on this problem for barrelled locally convex spaces and for the spaces, Cc(X), of realvalued functions continuous on a T{7/2}-space X, where Cc(X) has the compact-open topology. Of course, this topic provides for useful interactions/questions relating topological properties for X and analytic properties for Cc(X).

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FRIDAY, MAY 8:   Daniel R. Patten, Syracuse University, will speak on "Differential calculus on discrete structures". At CCNY; tea 12:30 p.m. NAC 8/130, talk 1 p.m. NAC 5/150. For more information, contact Ralph Kopperman, rdkcc@ccny.cuny.edu.

     ABSTRACT:   We obtain a conservative extension of classical elementary differential calculus on real and complex vector spaces to arbitrary convergence spaces and, in particular, discrete structures. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to obtain a differential calculus on groups. As an application, we obtain a Boolean differential calculus, in which the differentials satisfy both linearity and the Leibniz identity.

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For more information, contact:

CCNY: R. Kopperman (845-915-0914), S. Popvassilev (646-257-7197)
College of Staten Island (718-982-3626): P. R. Misra
Baruch College (646-312-4136): A. Todd
LIU C. W. Post Center (516-299-2447): S. Andima
Medgar Evers College, (718-270-6416): H. Pajoohesh
Queensborough Community College, (718-281-5291): F. Jordan