NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA
February - May 2007
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FEBRUARY 22 : Gabor Herman, CUNY Graduate Center,
"Boundaries in Digital Spaces". Joint meeting with CUNY Graduate
Center Computer Science Colloquium, 4:15-5:30 at CUNY Graduate Center,
Room 9204/9205, followed by a wine and cheese reception in Room 4102 (Science Center).
ABSTRACT: Intuitively a boundary
in an N-dimensional digital space is a connected component of the (N-1)-dimensional
surface of a connected object. In this talk I explain these concepts and show that the
boundaries so specified have properties that are intuitively desirable. I describe
some efficient algorithms for tracking such boundaries and show that they can be used
for computer graphic displays of internal structures of the human body based on the
output of a medical imaging device.
MARCH 1 : Bojana Pejic, University of Pittsburgh, "Uniqueness of Polish
group topologies," at C. W. Post Center of LIU. Tea 3:15, Pell Hall
240; talk 4:00, Pell Hall 204. For parking and more information,
contact Susan Andima, sandima@liu.edu.
ABSTRACT: Polish groups are separable
completely metrizable topological groups. They are ubiquitous in mathematics, arising
naturally as the symmetry groups of other mathematical objects. A fundamental question
in the theory of Polish groups is that of the uniqueness of a Polish group
topology: when is the topology of a Polish group uniquely determined by
its algebraic structure?
The key to unlocking this problem is to determine which sets in a Polish
group are `definable' both algebraically and topologically (Borel). It
turns out that the answer to this question is more complex, and
interesting, than it first appears. For example the set of squares {\it is
}Borel in the group of all autohomeomorphisms of the unit interval, but is
{\it not} Borel in the group of all autohomemorphisms of the circle. Even
so we now know that certain Lie groups and all finitely generated
profinite groups have a unique Polish group topology.
MARCH 8 :Don Johnson, Ramsey, NJ, "Adjoining an identity to a reduced
archimedean f-ring." At CCNY; tea 3:15 NAC8/134, talk 4:00 NAC6/114.
For parking contact R. Kopperman, 212-650-5125,
rdkcc@ccny.cuny.edu.
ABSTRACT: There are three papers
under discussion here. The first is my up-dating of my 1962 representation theorem
for reduced archimedean f-rings, which I presented here two years ago
(more-or-less). I will review this, taking it as our definition of such rings.
The second paper in question, joint with Tony Hager, discussed the canonical extension
of such a ring, A, to one with identity, \uA, and the class \bf U of
u-extendable maps (i.e., homomorphisms which lift over the u's to
identity preserving homomorphisms). We showed that \bf U is a category
and u becomes a functor from \bf U which is a monoreflection; the maps
in \bf U were characterized. Some of this was presented at the earlier
talk, but I do not recall how much.
I will review all of the above, supplying any details
that seem relevant or desireable. In the third paper, Tony and I address the interaction
between our functor u, and v\,, the vector lattice monoreflection in
archimedean \ell-groups (due to Conrad and Bleier). In short, v
restricts to a monoreflection of reduced archimedean f-rings into
reduced archimedean f-algebras, \psi\in\bf U if and only if
v\psi\in\bf U, and vu is a monoreflection into reduced archimedean
f-algebras with identity.
This work was motivated by the question put to us by G. Buskes:
what maps are o-extendable; i.e., extend over the orthomorphism rings? (The
orthomorphism ring oA is a unital extension of uA, and any
o-extendable map lies in \bf U.) While a complete answer seems quite
complicated (if not hopelessly out of reach), here we shall identify a
class of objects D for which oD=vuD and all maps from D
lie in \bf U, hence any map from D to a reduced archimedean
f-algebra is o-extendable.
I find this work to be unified and quite beautiful.
My goal here is to spend an enjoyable hour with you, answering any questions (that I can)
and getting only as far as this allows. There seems to be no end of this
project in sight \cdot\cdot\cdot. I am sure that I'll be back.
MARCH 15 : Scott Williams, SUNY at Buffalo, "The Topology of Metric
spaces". at C. W. Post Center of LIU. Talk 4:00, tea 5:15, Pell Hall
240. For parking and more information, contact Susan Andima,
sandima@liu.edu.
ABSTRACT: We are interested in
topological properties that affect, or are the consequence of metric constructions
such as the products, sums and quotients as favorable to metric spaces as opposed to topology.
We consider connected, compact, complete, totally bounded, zero
dimensional, metric spaces, as well as cardinal and ordinal functions.
We isolate a notion we call taut, which in the presence
of complete implies connected. For example, each metric space is contained in a
taut metric space, and is the minimal image of a taut metric space.
Yet the process of tautification is not only uncountable, but without
bound.
Related event MARCH 15, 12-2pm : CCNY Best finals luncheon and awards
ceremony (NAC 8/133), followed by talk: Scott Williams, SUNY at
Buffalo, "A Theory of Metric Spaces". For parking, more information,
contact Ralph Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Metric spaces were introduced
by Froet (1906) to illustrate the consequences of the notion of "distance." However, soon
they became an after thought to the study of topology which includes
n-dimension Euclidean space. With some help from Cantor and Hausdorff,
we consider a theory of metric spaces.
MARCH 22 : Vladimir Tkachuk, Autonomous Metropolitan
Univerisity (UAM), Mexico City, "DUALITY WITH RESPECT
TO NEIGHBOURHOOD ASSIGNMENTS: A YEAR LATER." At CCNY; tea 3:15pm, NAC 8/133, talk 4pm,
NAC 6/114. For parking and more information, Ralph Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT: A family
{\bf O}=\{Ox:x\in X\} is a neighbourhood
assignment in a space X if Ox is an open neighbourhood of
the point x for every x\in X. A set K is a kernel of
\bf O if \bigcup\{Ox:x\in K\}=X. The space X is dually
\bf P if every neighbourhood assignment in X has a kernel
with the property \bf P.
We continue the study of spaces which are
dually \bf P for some widely known properties \bf P (a report
on our previous progress was presented at this Seminar on the 30th of March of
2006). It turned out that dually discrete spaces constitute
quite an interesting class. We will show that even if we ask
whether every space is dually \bf P for a given property
\bf P, the answer is often far from being trivial.
APRIL 10 : Ilya Kofman, College of Staten Island, "Spanning trees and
Khovanov homology". At CCNY; tea 3:15pm, NAC 8/133, talk 4pm, NAC
6/114. For parking and more information, contact Ralph Kopperman,
rdkcc@ccny.cuny.edu.
ABSTRACT: Khovanov constructed
a bigraded complex whose bigraded Euler characteristic is the Jones polynomial of a link.
Khovanov homology is a stronger knot invariant than the Jones polynomial,and has received a
lot of attention recently. We will review the Kauffman bracket, Jones
polynomial, and Viro's elementary construction of Khovanov homology.
The Jones polynomial can be expressed in terms of spanning
trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists
a complex generated by these spanning trees whose homology is the reduced Khovanov
homology. The spanning trees provide a filtration on the reduced Khovanov
complex and a spectral sequence that converges to its homology. This is joint
work with A.Champanerkar.
MAY 10 : Aaron Todd, Baruch College, CUNY, "The Frechet-Urysohn Property
in Topological Vector Spaces." At College of Staten Island, tea 3:15 p.m. 1S/215, talk
4:00 p.m. 1S/112. For parking contact P. R. Misra, (718) 982-3626,
prmisra@netzero.net.
ABSTRACT: A topological space X is Frechet-Urysohn (F-U) if,
for each subset A of X and each point x in the closure of A, there is a sequence (an)
in A convergent to x. Of course, {an} together with {x} is compact, and, in case
X is a topological vector space (TVS) E (over the real or complex field), the set {an}
is contained in some multiple of each neighborhood of zero in E. Therefore, an F-U TVS E is of
bounded tightness, that is, for each subset A of E and each point x in the closure of A, there is
a bounded subset B of A with x in the closure of B. We show the converse, thereby giving a new
characterization of the F-U property in this context and answering a question of P. Nykos.
[This reports on joint work with J. Kakol and M. Lopez Pellicer.]
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