OCTOBER 16: Frederic Mynard, New Jersey City University, will discuss "Pointfree convergence".
At CCNY; tea 3:15pm NAC 8/130, talk 4pm SH 205. For more information, contact Ralph Kopperman,
ABSTRACT: This is a preliminary report on joint work with Jean Goubault-Larrecq. To investigate pointfree convergence, a pointfree analog of the theory of convergence spaces, we introduce the category of convergence lattices and its opposite category Foc of focales. Let Conv denote the category of convergence spaces and continuous maps. Each convergence space determines a focal via a functor L:Conv -> Foc, and each focal determines a convergence space on its set of points via a functor pt:Foc -> Conv.
We show that L -| pt is an adjunction between Conv and Foc, and we characterize the image of pt in Conv, and investigate the fixed points of ptL, among which are in particular T0 sober topological spaces. We further examine the relationship between this duality and the traditional Stone duality between the categories of topological spaces and of locales.
NOVEMBER 13: Ralph Kopperman, CCNY, will discuss "Fixed points and fixed subspaces". At CCNY;
tea 3:15 p.m., NAC 8/130, talk 4 p.m., SH 381. For more information, contact Ralph Kopperman,
ABSTRACT: We look at topological methods to solve the equation f(A)=A, for f, a function and A, a set or point.
DECEMBER 11: Nate Ackerman, Harvard University, will discuss "Completeness in G-Ultrametric
Spaces". At CCNY; tea 3:15pm NAC 8/130, talk 4pm SH 381. For more information, contact Ralph Kopperman,
ABSTRACT: For a complete lattice, a G-ultrametric spaces is a generalization of an ultrametric space where the distances take values in G. For such spaces there are three natural notions of completeness.
First, in a G-ultrametric space we can generalize the notion of a Cauchy sequence, except the sequence no longer is indexed by the natural numbers, but rather by a cofinal subset of G. If every Cauchy sequence converges then we say a space is Cauchy complete.
Second, there is the notion of spherical completeness, i.e. that any nested sequence of closed ball has non-empty intersection. This property is fundamental for proving generalizations of the Banach fixed point theorem.
Finally, a third notion is that of injectivity. An object in a category is injective if every partial map can be extended into a total map.
In this talk we discuss the relationships between these three notions.
For more information, contact:
CCNY: R. Kopperman (845-915-0914), S. Popvassilev (212-650-5346)
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