Join us in celebrating...

Conference on Abelian Groups and on Constructive Mathematics

in honor of Ray Mines's and Fred Richman's 70th birthdays
May 9-11, 2008, at Florida Atlantic University in Boca Raton

> Registered Participants:

1. Basak Ay, Florida Atlantic University
   
2. Douglas Bridges, University of Canterbury, Christchurch
   Constructive reverse mathematics
   Abstract: The Julian-Richman (1983) paper on positive-valued continuous functions can be regarded as the inaugural work (at least in the modern era) in what has become known as constructive reverse mathematics.In this talk I shall discuss recent work that has been inspired by their paper.
   
3. Marcela Chiorescu, Florida Atlantic University
   Minimal zero-dimensional algebras
   Abstract: Miroslav Arapovic showed that a unique minimal zero-dimensional extension exists within any zero-dimensional extension (The minimal 0-dimensional overrings of commutative rings, Glasnik Mat. 18 (1983), 47-52). We will characterize the minimal zero-dimensional R-algebras with finitely many idempotents for any ring R and the minimal zero-dimensional extensions of a one-dimensional ring R with Noetherian spectrum and zero a primary ideal. These rings include Dedekind domains but need not be Noetherian or integrally closed.
   
4. Don Cook, Albany, Georgia
   
5. Audrey Doughty, Florida Atlantic University
   
6. Ted Faticoni, Fordham University
   Class number of an abelian group
   Abstract: A group G has the power cancellation property if for any H and integer n > 0, Gⁿ isomorphic to Hⁿ implies G isomorphic to H. Let k be an algebraic number field and let O(k) be the set of rtffr integrally groups G such that QEnd(G) = k.
   Theorem: The class number of k is 1 iff each G in O(k) has the power cancellation property.
   
7. Mary Flagg, University of Houston
   A radical revisit with isomorphism theorems
   Abstract: A class of modules satisfies an isomorphism theorem if an isomorphism between the endomorphism algebras of two modules in that class implies that the modules are isomorphic. Does the presence of an isomorphism theorem require the properties of the whole endomorphism ring, or is the information about the underlying module contained in an ideal of the endomorphism ring? I will "visit" the isomorphism theorems for modules over a complete discrete valuation domain and explain the role of the Jacobson radical of the endomorphism ring in the isomorphism theorems.
   
8. Laszlo Fuchs, Tulane University
   Weak-injective modules over integral domains
   Abstract: This is a survey talk on recent results on weak-injective modules. M is weak-injective if Ext^1(A,M)=0 for all modules A of weak dimension at most 1.
   
9. Anthony J. Giovannitti, Clayton State University
   
10. Mary E. Hopkins, Florida Atlantic University
    Weakly integrally closed numerical monoids
    Abstract: Brewer and Richman defined a weakly integrally closed domain as a domain D such that if x is in its quotient field, and xJ is contained in J for some nonzero finitely generated ideal J, then x is in J. Weakly integrally closed monoids are defined analogously. If a monoid is not weakly integrally closed, then neither is its monoid algebra.
    We say that a numerical monoid M contains a forbidden pattern if there exists a natural number x not in M and a finite subset F of M such that x+F is contained in F+F. A numerical monoid is weakly integrally closed if and only if it contains no forbidden patterns. We will talk about a Javascript program which finds all minimal forbidden patterns, if any exist.
    
11. Katherine Humphreys, FAU Ph.D., Greensboro, NC
    
12. Hajime Ishihara, Japan Advanced Institute of Science and Technology
    A continuity principle, a version of Baire's theorem and a boundedness principle
    Abstract: We deal with a restricted form WC-N' of the weak continuity principle, a version BT' of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N', BT'+~LPO and BD-N+~LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.
    
13. Lee Klingler, Florida Atlantic University
    Distribution of elements in products in a group
    Abstract: A little-known theorem of Fred's about finite (possibly non-abelian) groups.
    
14. Vladik Kreinovich, University of Texas at El Paso
    Metrization theorem for space-times: A constructive solution to Urysohn's Problem
    Abstract: In the early 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as "first non-trivial result of point set topology"). Among other applications, this lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized.
    A few years before that, in 1919, a complex mathematical theory was experimentally proven to be extremely useful in the description of real world phenomena: namely, during a solar eclipse, General Relativity theory -- that uses pseudo-Riemann spaces to describe space-time -- has been (spectacularly) experimentally confirmed. Motivated by this success, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. After Urysohn's early death in 1924, this activity was continued in Russia by his student Vadim Efremovich, Efremovich's student Revolt Pimenov, and by Pimenov's students (and also by H. Busemann in the US and by E. Kronheimer and R. Penrose in the UK). By the 1970s, reasonably general space-time versions of Uryson's lemma and metrization theorem have been proven.
    However, these 1970s results are not constructive. Since one of the main objectives of this activity is to come up with useful applications to physics, we definitely need constructive versions of these theorems -- versions in which we not only claim the theoretical existence of a pseudo-metric, but we also provide an algorithm enabling the physicist to generate such a metric based on empirical data about the causality relation. An additional difficulty here is that for this algorithm to be useful, we need a physically relevant constructive description of a causality-type ordering relation.
    In this talk, we propose such a description and show that for this description, a combination of D. Bridges' constructive ideas with the known (non-constructive) proof leads to successful constructive space-time versions of the Uryson's lemma and of the metrization theorem.
    A big remaining challenge is related to the fact that in modern physics, one of the most important and fruitful notion is the notion of symmetry. There exist (non-constructive) versions of space-time metrization theorems which show that often, for symmetric causality relations, invariant pseudo-metrics are possible. It is still not clear how to constructivize these results. We hope that the constructive groups ideas of R. Mines and F. Richman -- and their further development in the work of other participants -- will help in this space-time-related challenge.
    
15. Alice Loth, University of Massachusetts Boston
    
16. Peter Loth, Sacred Heart University
    
17. Robert Lubarsky, Florida Atlantic University
    Representations of Riesz Spaces
    Abstract: Classically, Riesz spaces can be represented via a Stone-like embedding as function spaces. The same holds constructively, as shown recently by Coquand-Spitters, but might (depending on the context) use Dependent Choice. In joint work with Fred, we have shown that (an appropriate weakening of) DC is necessary for (indeed, equivalent to) some of these results.
    
18. Aaron Meyerowitz, Florida Atlantic University
    Integrality of group rings: A constructive result
    Abstract: Let G be a group, F a field and A(G) the group algebra of G over F. Irving Kaplansky asked the following question: If G has no elements of finite order, Does A(G) have no divisors of zero? One could replace F by an integral domain and G by a semigroup. There is a nice non-constructive proof in the case that G is abelian. We give a constructive proof in this case and discuss related questions.
    
19. Ray Mines, New Mexico State University
    Constructive mathematics at NMSU
20. Joan Moschovakis, Santa Monica, CA
    Unavoidable choice sequences
    Abstract: An unavoidable choice sequence is one whose existence is established only classically. No nonrecursive sequence is unavoidable in Kleene's axiomatization FIM of intuitionistic analysis; but if one adds strong Markov's Principle to a minimal subtheory of FIM without countable or continuous choice, every classically Delta-1-1 sequence is unavoidable. We show that strong MP is independent of FIM together with axioms asserting that (i) the characteristic function of every arithmetical predicate (with sequence parameters allowed) is unavoidable, (ii) every purely arithmetical predicate is decidable, and (iii) no sequence can fail to be classically Sigma-1-1 (hence Delta-1-1).
    
21. Bruce Olberding, New Mexico State University
    Factorization of ideals in commutative rings
    Abstract: We describe the class of rings in which proper ideals factor into prime and invertible ideals, and discuss how examples that are not Dedekind domains can arise (arguably) in "nature". We relate this to some early work of Fred Richman and some recent work of Ray Mines.
    
22. Nicola Pace, Florida Atlantic University
    
23. Kulumani M. Rangaswamy, University of Colorado at Colorado Springs
    Infinite rank Butler groups which are pure subgroups of completely decomposable groups
    Abstract: This is a joint work with Dave Arnold and deals with B-2 groups which arise as pure subgroups of completely decomposable groups
    
24. Fred Richman, Florida Atlantic University
    Deconstructing Elbert
    Abstract: In a seminal paper, Elbert Walker showed that finitely generated abelian groups have the cancellation property under direct sum. Does this theorem have a constructive proof? If we restrict ourselves to finitely presented abelian groups, it comes down to proving that Z has the cancellation property. That looks like it might be an interesting problem. There are also related problems with bounded finitely generated groups and with modules over k[X] where k is a discrete field.
    
25. Wim Ruitenburg, Marquette University
    Constructive mathematics with excluded middle
26. Carol Walker, New Mexico State University
    Some comments about Fred and Ray
27. Elbert Walker, New Mexico State University
    Some remarks about Fred
28. Bill Wickless, University of Connecticut