# NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA

## CHECK FOR CAMPUS AND LOCATION

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MARCH 7:   J. Bak, CCNY, "On the Newman definition of simple-connectedness in the complex plane." At CCNY. Tea 3:15 p.m., NAC 8/133; talk 4:00 p.m., NAC 6/113. Information,parking: R. Kopperman, rdkcc@ccny.cuny.edu .

ABSTRACT:   Strashimir Popvassilev's recent proof of the equivalence of Donald J. Newman's definition to the more common ones for simple-connectedness justifies our use of that definition. That is very gratifying, since Newman's definition leads to such a quick proof of the General Closed Curve Theorem, which in turn leads to a very elegant proof of the Riemann Mapping Theorem for open simply-connected domains. We give this proof, and as time allows, we may discuss the related question of conformal mappings between closed domains.

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MARCH 14:  Strashimir G. Popvassilev, CCNY, "On the `Connected within \epsilon to \infty' definition of simple connectedness," At CCNY. Tea 3:15 p.m., NAC 8/133; talk 4:00 p.m., NAC 6/113. Information, parking: R. Kopperman, rdkcc@ccny.cuny.edu .

ABSTRACT:   At the end-of-semester party at CCNY in December 2012 Professor Joseph Bak commented that the following definition, used in his book "Complex Analysis", coauthored with Donald J. Newman, is very convenient to work with:

A connected open subset D of the (complex) plane C is simply connected iff its complement CD is "connected within epsilon to infinity." That is, if for any \epsilon>0 and z in CD, there is a continuous curve g(t), 0<=t<\infty such that:

(a) dist(g(t),C\D)<\epsilon for all t\geq 0,

(b) g(0)=z,

(c) lim g(t)=\infty as t->\infty.

Professor Newman felt that this definition is equivalent to the standard definitions. However, with his passing, there remained the unfinished business of proving (or disproving) the equivalence. We present a proof of the equivalence. We also state a conjecture relating conformal maps to Toeplitz' question whether every Jordan curve has an inscribed square.

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MARCH 21:  Frederic Mynard, Georgia Southern University, "Regularity in convergence-approach spaces". At Baruch College, CUNY, 24th Street and Lexington Avenue. Tea 3:15 p.m., VC 6-215, talk 4:00 p.m. VC 6-215. For local information: A. Todd, aaron.todd@baruch.cuny.edu.

ABSTRACT:   After discussing generalizations of topological spaces in the direction of convergence spaces and in the direction of approach spaces, we will focus on the common generalization: convergence approach spaces.

In this context, we study two variants of regularity and characterize them in two ways: in terms of a property of contractive extensions of contractive partial maps into the space, and in terms of contractivity of "near limits" in the canonical function space structure.

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RELATED EVENTS:

Wednesday, April 10 Medgar Evers Mini Conference.

Thursday, April 11 Joint Computer Science-Math Colloquium.

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APRIL 11:  S. G. Matthews, Department of Computer Science, University of Warwick, Coventry, UK, "Temporal Partial Metric Spaces". At Baruch College, CUNY, 24th Street and Lexington Avenue. Tea 3:15 p.m. VC 6-215, talk 4:00 p.m. VC 6-215. For local information: A. Todd, aaron.todd@baruch.cuny.edu.

ABSTRACT:   Partial metric spaces generalise metric spaces by allowing self-distance to be a positive 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 static a form of mathematics to be of use in modelling contemporary application software (aka apps), which is increasingly pragmatic, interactive, and inconsistent, in nature. This paper addresses the reality that if partial metric spaces are to survive in future research, then they must firstly progress by means of a computable temporal generalisation. How can this be achieved?

(1) The induced partial ordering is temporally ordered in the sense of the Lucid algebra,

(2) time is quantifiable in the sense of the time complexity of algorithms, and

(3) time can be paused in the sense of Wadge's hiaton.

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APRIL 18:  J.D. Mireles James, Rutgers University, "Computer Assisted Analysis of Invariant Manifolds and Connecting Orbits for Nonlinear Dynamical Systems. At Baruch College, CUNY, 24th Street and Lexington Avenue. Tea 3:15 p.m. VC 6-215, talk 4:00 p.m. VC 6-215. More information: A. Todd, aaron.todd@baruch.cuny.edu .

ABSTRACT:   I will discuss some constructive methods for approximating invariant sets associated with differential equations and discrete time dynamical systems. I am especially interested in "a-posteriori" methods for estimating the resulting approximation errors and also computational implementation of the estimates which allow one to pass from "classical" numerics to mathematically rigorous numerical results. I will discuss a number of example applications such as computation of invariant manifolds, computer assisted proof of existence of connecting orbits, and computer assisted proof of the existence of chaotic dynamics.

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MAY 9 (Joint with CUNY Graduate Center CS Colloquium): Krzysztof Ciesielski (West Virginia University, and University of Pennsylvania), "Delineating objects in images via minimization of \ellp energies: Fuzzy Connectedness, Graph Cut, and Random Walk algorithms." Talk from 4:15 p.m. to 5:30 p.m., 9204/04. Tea follows at 5:30 p.m.

ABSTRACT:   In the talk we briefly discuss the general problem of image segmentation: How to find a precise location of objects of interest in a given digital image?

Then we turn attention to the algorithms that find the objects via energy minimization, with particular attention to the energies defined with \ellp-norm based energies, 1\leq p\leq \infty. We emphasize the fact that several widely used segmentation algorithms return objects that minimize such energies. This includes: Graph Cut, Random Walker, Relative Fuzzy Connectedness, and Iterative Relative Fuzzy Connectedness algorithms.

Finally, we discuss strengths and weaknesses of these algorithms from the theoretical point of view. Some experimental data may also be presented.

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MAY 16:   Ralph Kopperman, CCNY, "Decimal Approximation". At Queensboro College, CUNY, 1:30-2:30 p.m. Room S-314. For more information, contact F. Jordan, fejord@hotmail.com .

ABSTRACT:   The usual decimal approximation of the reals represents them as a categorical limit of finite neighborhood spaces and special continuous maps.

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Friday, MAY 31:   Jerzy Kakol (Faculty of Mathematics and Computer Science, Poznan, Poland), "Angelicity and compactness in spaces C(X)". At Baruch College, CUNY, 24th Street and Lexington Avenue. Light Tea 9:45 a.m. VC 6-215, talk 10:30 a.m. VC 6-215. For local information: A. Todd, aaron.todd@baruch.cuny.edu.

ABSTRACT:   The aim of the talk is to summarize several (older and very recent) re-sults, concepts and ideas concerning the behaviour of compact sets in many of the locally convex spaces, including spaces C p(X) and C c(X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with the pointwise and compact-open topology, respectively. In the theory of locally convex spaces E, working with compact sets K in E, two essential questions may arise, namely, about (i) metrizability of sets K, and (ii) (weakly) angelicity of spaces E. Positive answers concerning part (i) (apart famous Smulians theorem) are covered by results of Pfister (about (DF)-spaces), Valdivia (about dual metric spaces), Cascales and Orihuela (about (LF)-spaces) and a general result due to Cascales-Orihuela referring to the class G(which contains, among others, all (DF) and (LM)-spaces). A simple approach with re-proving the above mentioned cases (results) was noticed by Kakol and Saxon who do not require the machinery of (quasi- and semi-) Suslin spaces, upper semicontinuous compact-valued maps, generalized inductive limits, etc. We characterize analytic sets in class C p(X) which leads to generalizations of the former results mentioned above. Several results extend earlier works of Talagrand, Preiss, Amir-Lindenstrauss, Corson, etc. Question (ii) refers to a very useful concept of angelic spaces (introduced by Fremlin) for which several variants of compactness coincide. We present a number of results, including a remarkable and applicable Orihuela theorem for spaces C p(X) over web-compact spaces X, providing angelicity. Methods of descriptive topology will be used. Applications of this concept both in topology and functional analysis are provided.