NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA
August - December 2007
CHECK FOR CAMPUS AND LOCATION
AUGUST 30 : Steve Matthews, U. of Warwick, UK, "Efficiency oriented
programming languages and semantics". Joint meeting with CUNY Graduate Center Computer
Science Colloquium, 4:15 - 5:30pm, CUNY Graduate Center, Room 9204/9205,
followed by a wine and cheese reception in Room 4102 (Science Center).
ABSTRACT:
Following the example of Michel Schellekens, we consider how two familiar
specialisms of theoretical computer science may find more common ground.
Any given algorithm can be studied for properties such as its complexity, but,
when coded up as a program, all that the typical programming language facilitates
is retention of sufficient information to enable the computer to (automatically)
execute each step in the correct order. For example, how could a coding of Bubble
Sort be automatically distinguished from a coding of Quick Sort to determine which
program is using the algorithm having the best average time complexity? Such questions
are undoubtedly difficult to address, and will remain so as long as there remains an
enormous gap between the complexity of algorithms and the denotational semantics of
programming languages. The smaller the gap, the more we can contemplate programs
continually analysing themselves at run time for their own algorithmic behaviour, and
making improvements accordingly. In today's talk we will combine research from
Mike Smyth on discrete spatial models, from Ulrich Hohle on many valued topology, with our
own work on partial metric spaces to present a complementary approach to that of Schellekens
to establishing connections between the discrete mathematics of present day algorithms
research (i.e. graph theory) and the continuous mathematics traditionally used in
denotational semantics (i.e. topology and domain theory).
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SEPTEMBER 6: Laszlo Zsilinszky, U. of North Carolina at Pembroke,
"The Baire property of hyperspace topologies". At Queens College, Tea 3:30 in Kiely 508,
talk 4:30 in Kiely 422. For local information and parking, contact G. Itzkowitz,
718-997-5849, gitzkowitz@prodigy.net.
ABSTRACT:
The hyperspace of a topological space X is the space of nonempty closed subsets of X.
Properties of the Vietoris hyperpspace topology have been thoroughly studied throughout
the last century, and characterizations of various completeness properties in terms of the
base space X are well-established. However, until recently, Baireness of the Vietoris
topology turned out to be an elusive property.
The Wijsman hypertopology is another well-studied topology due to its
applicability to various branches of mathematics. One remarkable property of the Wijsman
topology is that equivalent metrics on X may generate different Wijsman topologies;
moreover, taking the supremum of all the Wijsman topologies corresponding to equivalent
metrics on X gives the Vietoris topology.
The talk will discuss the current status of research on Baireness and related
properties of these hyperspaces in light of new developments. Open problems will be given.
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OCTOBER 4: Ralph Kopperman, City College of CUNY, "Normal non-Hausdorff spaces",
at C. W. Post Center of LIU. Tea 3:15, Pell Hall 240; talk at 4:00 Pell Hall, room to be
arranged. For more information, call the C. W. Post Math Department at 516-299-2448 or
write to SAndima@liu.edu.
ABSTRACT: Beginning with the Sierpinski
space - the smallest non-trivial space - examples of such spaces abound. Far from
simply being curiosities, they are needed to approximate compact Hausdroff spaces with
finite T0 spaces.
We give examples and a partial characterization of these
spaces. We also discuss this type of approximation and their role in it.
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OCTOBER 18: Homeira Pajoohesh, Medgar Evers College, CUNY,
"Generalized metric topologies on lattice ordered groups", at CCNY. Tea 3pm, NAC 8/133,
talk 4pm NAC 4/205. For more information, parking: R. Kopperman,
rdkcc@ccny.cuny.edu .
ABSTRACT: "Intrinsic" metrics into
lattice ordered groups have long been considered. Topologies on such groups have
been considered as well. But topologies often arise from metrics into the reals, and
getting topologies from these metrics into lattice ordered groups has not been discussed.
A key issue must be overcome: properties of the strictly positive reals key to
defining the metric topology fail for the strictly positive elements of a typical
lattice ordered group. We solve this problem, and show how some important topologies
indeed arise from metrics.
We define partial metrics and discuss their uses in computer
science, in obtaining intrinsic topologies on lattice ordered groups, and elsewhere.
A central use for them is to split certain metric topologies on ordered sets, into upper
and lower subtopologies. Among these is the Euclidean topology on the lattice ordered
group of real numbers.
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NOVEMBER 1: Venu Menon, Univ. of Connecticut. "Continuity in partially
ordered sets". At Baruch Coll. Vertical Campus, Lexington Ave. and 24 St. Tea 3:15,
talk 4:00, Dean's Conference Room, 8-250. For information contact A. Todd,
artbb@cunyvm.cuny.edu .
ABSTRACT: Continuous lattices and
their generalizations, continuous domains, have been studied for more than three decades.
Recall that given elements of a poset, p is way below q, if whenever q is less than or equal
to each upper bound of a directed set D, then p is less than or equal to an element of D.
Continuous lattices are complete lattices where each element is the supremum of elements
way below it. A poset is a continuous domain, if it has sups of directed sets and satisfies
the following two conditions:
(i) each element is the sup of elements way below it, and
(ii) for each element, the set of elements way below it is directed.
A continuous poset is any poset in which the conditions (i)
and (ii) are satisfied. In a complete lattice, in fact in any sup-semilattice,
condition (ii) above is automatically satisfied. The purpose of this talk is to look at
posets which need not be dcpos or lattices but which satisfy the condition that each
element is the sup of elements way below it. Since we don't require the set of elements
way below an element to be directed, we will require a condition slightly stronger than
condition (i) above. Several of the pleasing algebraic and topological properties of
continuous domains extend to this setting.
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NOVEMBER 29: Geta Techanie, SUNY College at Old Westbury.
"Images of Two to One Maps", at C. W. Post Center of LIU. Tea 3:15, Pell Hall 240;
talk at 4:00 Pell Hall, 202. For more information, call the C. W. Post Math
Department at 516-299-2448 or write to SAndima@liu.edu.
ABSTRACT: A function f:X->Y is two-to-one
if for each point y in Y, there are exactly two points of X that map to y. For
some spaces X, there does not exist a two-to-one continuous function f:X->Y for any choice
of Y. Another situation is when there are two-to-one continuous functions f:X->Y
defined on a space X, but given any such function the image space Y is determined up to a
homeomorphism.
We discuss two-to-one continuous images of N*, the
remainder beta(N)-N of the Stone-Cech compactification of the discrete space of natural
numbers. In particular, we show that every two-to-one continuous image of N* is
homeomorphic to N* when the continuum hypothesis is assumed.
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DECEMBER 6: Scott Williams, SUNY at Buffalo, "New Results in box Products".
At Medgar Evers College, CUNY. Earlier talk for students at 2, Tea 3, talk at 4
all Carroll 408 (on Carroll near Nostrand). For more information, write to Homeira Pajoohesh,
hpajoohesh@mec.cuny.edu.
ABSTRACT: Given the box topology,
is the product of countably many copies of the reals normal? This problem is over 80 years old,
and yet unsolved by some of the strongest topologists and logicians of the 20th century.
What can be said if we loosen the hypothesis and allow some set theory? There is a
class H of separable metric spaces X such that:
1. If some finite power of X is not in H, then the box product of countably
many copies of X is not normal.
2. If each finite power of X is in H, then the box product of countably many
copies of X is normal.
Yes, we need set theory for (2).
Earlier talk: "My Favorite Functions".
ABSTRACT: We quickly brush over
from continuous functions to those differentiable; however, there is a universe between,
especially when the domains are allowed to vary. I shall discuss some famous, and not so
famous, but unusual continuous functions.
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DECEMBER 20: Francis Jordan, Georgia Southern U., "Dualities between
covering properties of a space and local properties of its space of real-valued functions".
Tea 3pm, NAC 8/133, talk 4pm NAC 4/125. Information and parking: R. Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT:
We will discuss the duality between a topological space X and the space C(X) of continuous
real valued functions on X with the topology of point-wise convergence. In particular,
statements about coverings of X by open sets become statements about the neighborhood
filters of points in C(X). We identify a large class of properties for which these duality
results can be proven.
This approach allows us to unify a number of past results and
prove some new theorems which answer open questions. We will also consider subsets of the real
line that satisfy some of the global properties we will discuss.
For more information, contact:
CCNY (212-650-5346): R. Kopperman
College of Staten Island
(718-982-3626): P. R. Misra
Baruch College (212-387-1463): A. Todd
LIU C.
W. Post Center (516-299-2447): S. Andima
Queens College, Math. (718-380-1832):
G. Itzkowitz
Queens College, Comp. Sci. (718-997-3478): T. Y. Kong
Medgar Evers
(718-270-6125): H. Pajoohesh