FEBRUARY 10: Mel Henriksen, Harvey Mudd College. "Residue
class fields of rings of real analytic and real entire functions". At City
College. Tea 3:00, NAC 8/133; Mel has just returned from a trip to Iran, and
might be willing to talk about it. Talk 4:00 NAC 6/114. For parking, contact R.
Kopperman, rdkcc@cunyvm.cuny.edu.
ABSTRACT:
More than 50 years ago, I showed that if M is a maximal ideal of the ring E(C)
of entire functions, then E(C)/M is always isomorphic as a ring the complex
field C, even though this residue class field is sometimes infinite dimensional
as an algebra over the real field R. In this talk residue class fields of the
ring A(R) of real analytic functions and the ring E(R) of real entire functions
are described. They turn out to be either R, C, or a real closed field F that is
an eta-one set. That is: whenever A and B are countable (possibly empty) subsets
of F such that a in A and b in B imply a < b, there is an x in F such that a
< x < b. By a theorem of A. Dow, any two fields of this latter kind are
isomorphic iff the continuum hypothesis holds. This is part of joint research
with Marek Golasinski. Unfamiliar definitions and techniques will be
reviewed.
MARCH 3: Don Johnson, "A representation
theorem revisited". At City College. Tea 3:15, NAC 8/133; talk 4:00 NAC 4/113.
For parking, contact R. Kopperman, rdkcc@cunyvm.cuny.edu.
ABSTRACT:
In 1962, I represented the members of a certain class of lattice-ordered rings
(the reduced Archimedean f-rings) as rings of extended continuous functions on
locally compact spaces. The existence of this representation has proved
useful.However, the opacity* of the proof obscured the identity of the
representing functions, thus limiting this usefulness. Herewith, a new, more
transparent proof: one which finds ready application. First, a very strong
statement of the uniqueness of the representation, something that had been
missed the first time around. Second,the relationship between homomorphisms and
continuous functions is explored: the homomorphisms are, indeed, defined by
composition with continuous functions. This produces some insight with regard to
the question that prodded me to return to this topic: "What can one say about
the functoriality of this representation?"
Further applications abound,most
of which are joint work with Tony Hager.
*From
the Random House unabridged: opaque, impenetrable, dull, stupid, or
unintelligent
MARCH 10: Venu Menon, Univ. of Conn,
Stamford. "Intrinsic topologies on posets". At City College. Tea 3:15, NAC
8/133; talk 4:00 NAC 4/113. For parking, contact R. Kopperman, rdkcc@cunyvm.cuny.edu.
ABSTRACT:
A topology on a partially ordered set is called an intrinsic topology if it is
defined directly from the order. So, it is only natural to look for purely order
theoretic characterizations of topological properties like compactness and
separation properties. In this talk we look at Scott topology, Lawson topology,
and interval topology and search for order theoretic characterizations for some
topological properties. We will review several well known results, present few
new results, and pose some questions for future research.
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
Marymount College
(914-631-3200): M. Hastings
Queens College, Math. (718-997-5849): G.
Itzkowitz
Queens College, Comp. Sci. (718-997-3478): T. Y.
Kong