Abstracts

Ted C.K. Chinburg: “Mel, Blues Piano, and Automorphisms of Power-Series Rings”

As a sophomore at Harvey Mudd, I remember Mel pounding the blackboard like a Delta river blues piano player at his keyboard. In this talk I'll play a few pieces of music that remind me of Mel. I'll then play some abstract algebra tunes having to do with automorphisms of power-series rings that I hope Mel might have liked.

W. Wistar Comfort: “Remembering Mel Henriksen and (Some of) His Theorems”

Following introductory comments I will cite theorems, with proofs where time permits, from three of Mel's important papers in “pure set-theoretic topology”. I will discuss several intriguing, attractive questions which remain open today, some of them raised by Mel and his co-authors and some by subsequent workers building on these groundbreaking papers.

The following paragraphs capture or summarize the general flavor of my remarks:

1. With Isbell, 1958 The development of perfect functions. For many properties $\mathcal{P}$, the preimage under a perfect function of a space with $\mathcal{P}$ again has $\mathcal{P}$. Many applications and consequences arise.
2. With Isbell and Johnson, 1961 For many properties $\mathcal{P}$ and $\mathcal{Q}$, every Baire set in a space with $\mathcal{P}$ has $\mathcal{Q}$. For many spaces $X$, every closed Baire set in $X$ is a zero-set of $X$.
3. With a committee, appointed, convened, coordinated and chaired by Mel, 2000
When is $|C(X\times Y)| = |C(X)|\cdot|C(Y)|$? For which $X$ does this occur for every $Y$? Is the class of such $X$ closed under (finite, countable, arbitrary) products? [Answer: Yes, Yes, No].

How large can $|X|$, or $|C(X)|$, or $w(X)$ be for such $X$?

Garth Dales: “Large Fields from Algebras of Continuous Functions—Ramifications of a Seminal Paper of Erdős Gillman, and Henriksen”

On 6 July, 1954, a paper of Erdős, Gillman, and Henriksen called “An Isomorphism Theorem for Real-Closed Fields” was submitted to Annals of Mathematics; it appeared in 1955. The main theorem is that two real-closed, $\eta_{\alpha}$-sets of cardinality $\aleph_{\alpha}$ are isomorphic. The main example involves quotients of the algebra $C(X)$ of all real-valued, continuous functions on a completely regular topological space $X$. Thus we see striking and subtle connections between topological spaces, continuous functions, and algebraic structure, topics that lay at the heart of much of Mel's work. This paper was the source of two chapters in Gillman and Jerison's great and influential book.

We shall recall the main results of the paper, and then look at some subsequent work.

First, we now know more (but not everything that we would like to know) about the structure of large real-closed fields; some of this material will be taken from the book Super-Real Fields (OUP, 1996) by myself and W.H. Woodin.

Second, I shall talk about the insights we now have on the structure of larger copies of the real numbers, and the role of these large fields in Banach algebra theory.

Finally, I shall mention some of Mel's recent work, including a paper submitted on July 14, 2008, 54 years after the earlier paper.

Ralph Kopperman: “How Mel Drove Me to Asymmetry”

I first met Mel when I spoke at the Claremont Colleges colloquium. Mel then arranged for me to speak at his seminar. After I spoke for three sessions, he asked me for a generalized metric structure that would give rise to the hull-kernel topology on the space of prime ideals, Spec($R$).

The answer: For any collection $X$ of subsets of a set $R$, define the subset-valued $d$ on $XxX$, by
$d(I,J) = J–I$, the set difference. To get a topology so the balls of “positive” radius form a neighborhood base about each $I$, we use the collection of cofinite sets as the “positive” elements.

This structure yields the right topology, as well as some seemingly algebraic notions, such as “annihilator”.

Suzanne Larson: “SV Spaces: A Case Study in Mel's Enthusiasm for Mathematics”

Mel was enthusiastic about all things mathematical—intriguing definitions, theorems, examples, and, of course, the people who appreciate and create mathematics. Many episodes in the development of the study of topological spaces $X$ and the corresponding ring of continuous functions $C(X)$ for which $C(X)$ mod $a$ prime ideal is a valuation ring were influenced by Mel's enthusiasm and work. I will give a brief accounting of this through mathematics and reminiscences.

Frank Smith: “Mel and His Regular Ideals”

In January, 2005, I arrived in Claremont for a sabbatical. Mel gave me several things to read, one of which was a partial article involving regular rings, a work in progress with two faculty from a Jordanian University. As things progressed, we resolved that a successful effort required solving two questions.

The first involved the ring $C(X)$, which was easily solved; the second, the ring $_{Z}n[i]$. This ring's locality depends on whether there exist nontrivial idempotents, which is only a problem for $n = p^{k}$. Finally, by brute force, we found there were no nontrivial idempotents when $n = 9$ but that one was found for $n = 25$. At that point I set a computer to work and found idempotents for powers of 5, 13, and 29 but none for powers of 11, 19, or 23. Going back to theory, I managed to prove that there were idempotents if and only if you could solve the congruence $c^{2}$ congruent to $-1 \pmod{p^{k}}$. From there Mel's memory kicked in and the result was found.

Joanne Walters-Wayland: “Framing ‘Mel’ ”

As a tribute to Mel Henriksen and in thanks for his ability to inspire and motivate, I would like to share a few snapshots of work he has done over the years, taken through a localic lens. Because of time constraints, I will only be able to give a very brief introduction to Frame Theory and will hopefully be able to give some idea of how Mel's work, albeit inadvertently, influenced this field—starting with his work presented to the AMS in 1954 on finitely generated ideals, the “raison d'etre” of $F$-frames, $P$-frames, and so on, including his work on prime ideals (1965), quasi-$F$-covers (1987), and pretty bases (1991), concluding with some results about cozero complemented spaces (2003). I would also like to take this opportunity to spend a few minutes introducing OCCTAL and explaining Mel's very important role in its existence.

R. Grant Woods: “Mel Henriksen in Winnipeg: 25 Years of Anecdotes and Theorems”

Mel Henriksen visited Winnipeg almost annually for nearly 25 years. In this talk I discuss the effect that he had on Winnipeg and that Winnipeg had on him, as well as some of the mathematics that we did together during his visits. I will emphasize our generalizations of the “absolute” of a topological space, and our work relating the topology of a Tychonoff space $X$ to various algebras of real-valued functions with domain $X$.