NEW YORK SEMINAR ON GENERAL TOPOLOGY AND TOPOLOGICAL ALGEBRA
September - December 2009
CHECK FOR CAMPUS AND LOCATION
SEPTEMBER 24: Philip Ording, Medgar Evers College, CUNY, "Heegaard Knot
Diagrams", at CCNY. Tea 3pm, NAC 8/133, talk 4pm, room NAC 4/205. For more Information, parking:
R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: A successful approach to studying 3 manifolds
has been to take a manifold M and divide it along a closed orientable surface. Heegaard showed
for closed orientable connected 3 manifolds that this can be done so that the two halves are
handlebodies of the same genus as the splitting surface. Heegaard splittings have many desirable features,
one of which is that the splitting is characterized completely by a surface diagram consisting of a set of
simple closed curves in the splitting surface. This means, to paraphrase Rolfsen, that some
problems in 3 manifold topology boil down to problems in 2 manifold topology.
For instance, given any Heegaard (surface) diagram for M,
we can uniquely specify any knot in M by placing pairs of basepoints in the diagram away from
the curves. This view suggests several questions: How can knot invariants be computed from
Heegaard knot diagrams? What is the simplest (lowest genus, fewest intersections) Heegaard knot
diagram for a given knot? Which knots have the simplest diagrams? This talk will connect these
motivating questions to the knot Floer homology developed by Ozsvath and Szabo. Recent
computational results will be discussed as time permits.
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OCTOBER 1: Ralph Kopperman, City College, CUNY, "Issues in finite
Approximation", at CCNY. Tea 3pm, NAC 8/133, talk 4pm, NAC 4/205. For more Information, parking:
R. Kopperman, rdkcc@ccny.cuny.edu.
ABSTRACT: Each compact Hausdorff space is the subspace of
closed points in an inverse limit of finite T0-spaces and special continuous maps. But finite
T1-spaces are discrete, so their inverse limits are zero dimensional; thus all points are
closed, and therefore the finite spaces that are involved rarely have stronger separation
properties.
We discuss how these representations determine various
separation and other properties of the original compact Hausdorff space, including connectedness.
As time allows, we discuss compactification theory and other issues.
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OCTOBER 22: Aaron R. Todd, Baruch College, CUNY, "Baire Category Type
Properties of Cc(X) Characterized by Properties of X", at Baruch College,
Lexington Avenue and 24th Street, Math. Dept., Vertical Campus, Room VC6-215, Tea at 3:15pm,
Talk at 4:00 pm. For more Information contact Aaron Todd,
artodd@panix.com or
aaron.todd@Bruch.cuny.edu.
ABSTRACT: By giving the space, C(X), of real-valued
functions continuous on a Tychonov space X its compact-open topology c or its finite-open
topology p, we obtain a natural laboratory of functional analytic properties on Ct(X)
(t=c or p) and topological properties of X. In this talk, we picture properties on
Cc(X) of a Baire category nature characterized by properties of X. We
also discuss two properties, one of finite subsets of X developed independently by several
investigators for characterizing the Baire space property for Cp(X), and, another,
of compact subsets of X, developed by Ma and Gruenhage for Cc(X). Appropriate
concrete examples will also be described as time allows.
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NOVEMBER 12: Gerry Itzkowitz (Queens College, CUNY), "A NOTE ON FINITE
PRODUCTS OF BANACH SPACES". At Baruch College, Tea 3:15pm, Talk 4:00pm, Conference Room,
Mathematics Department, 6th floor, Vertical Campus, Lexington Avenue and 24th Street. Contact
A. R. Todd, (646) 312-4136,
Aaron.Todd@Baruch.cuny.edu or artodd@panix.com.
ABSTRACT: This is an outgrowth of a homework problem I
assigned my Functional Analysis class. The second author was a student who came up with a novel
proof of the assignment and which I realized could be used to generalize a famous classical
theorem. First, a definition. Notation: Let ( Ei, || . ||i ) ,
i=1,...n, be Banach spaces and let E = prod{i=1}^n Ei. Let || || be a
norm on E. Let Ei denote the space ( Ei, || . ||i ) , let
Ei* = {0}x...x{Ei}x{0}x ...{0} as a normed subspace of E, and let
pi be the projection of E onto Ei*.
Definition: A norm on E is said to be compatible with the Banach
spaces E_i, i=1,...,n if the projection maps (p)i:E into Ej are all
continuous and Ej* is topologically isomorphic to Ej.
THEOREM: Let ( Ei, || . ||i ) , i=1,...n,
be Banach spaces and let E = prod{i=1}^n Ei. Let || ||a and
|| ||b be norms compatible with the Banach spaces Ei, i=1,...,n, then
(E, || ||a) and (E, || ||b) are topologically isomorphic.
Now if one defines the l^p norms on a product of Banach spaces
E the way it is done on R^n and defines E^(p) as E with the l^p norm then one gets:
COROLLARY: All the spaces E^(p) are topologically isomorphic.
(Joint with S.M. Loh)
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NOVEMBER 19: Lawrence Narici (St. John's University, New York),
"On the Hahn-Banach Theorem". At Baruch College, (For times, location and contact, see Nov. 12.)
ABSTRACT: I love the Hahn-Banach theorem. I love it the way
I love "Casablanca" and the Fontana di Trevi. It is something not so much to be read as fondled.
What is the Hahn-Banach theorem? Let f be a continuous linear functional defined on a subspace M
of a normed space X. Take as the Hahn-Banach theorem the property that f can be extended to a
continuous linear functional on X without changing its norm. Innocent enough, but the
ramifications of the theorem pervade functional analysis and other disciplines (even
thermodynamics!) as well. Where did it come from? Were Hahn and Banach the discoverers? The
axiom of choice implies it, but what about the converse? Is Hahn-Banach equivalent to the axiom
of choice? (No.) Are Hahn-Banach extensions ever unique? They are in more cases than you might
think, when the unit ball of the dual is round, as for lp with 11 or l\infty. Instead of a linear functional,
suppose we substitute a normed space Y for the scalar field and consider a continuous linear map
A: M\rightarrow Y. Can A be continuously extended to X with the same norm? Well, sometimes.
Unsurprisingly, it depends on Y, more specifically, on the geometry of Y. If the unit ball of Y
is a cube, as for Y = ( R^n, || \cdot ||\infty ) or Y= l\infty, for
example, then for any subspace M of any X, any bounded linear map A : M \rightarrow Y can be
extended to X with the same norm. This is not true if Y = ( R^n, || \cdot ||p ),
n>1, for 10, the extendibility dies. This talk traces the
evolution of the analytic form as well as subsequent developments up to 2004.
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For more information, contact:
CCNY (212-650-5346): R. Kopperman (on leave)
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
Queens College, Comp. Sci. (718-997-3478): T. Y. Kong
Medgar Evers College, (718-270-3478), H. Pajoohesh
Queensboro College, (718-281-5291), F. Jordan