From Tampa Bay Times, October 1, 2009, page T10.

A student’s letter to tbt*

Editor’s note: On Monday, tbt* published a story describing President Obama’s desire to increase the number of days U.S. children go to school each year. A junior at a Pasco high school e-mailed the following response — with the subject line “why dose nobody ask the students’’ — to tbt*. It is printed here, unedited:

Dear “tbt” editor of which it may concern I’m a student of pasco coun ty a junior to be precise and pleas do not mistake this for a Dear Abby seg ment I am not a 40 year old women concerned about her felines. I am just appalled as a member of the student body that in ever y in other words the only article I could find of president Obahmas plan to extend the school year, school days, and e ven rumored around school of mak ing us work weekends that they have not a single students opinion on the subject.

more so that they believe that the school boards “research” should stand as good evidence of the im provement in our “higher achieve ment”. a very intelligent man once said if you obser ve you have therefor changed the out come (now notice I didn’t abbreviate that phrase as I did the other stated phrases that’s because it’s not exact not as an in sult to any of the readers but some would use this against the students to further there pointless war against other country’s grade average).

at any rate not only do I belie ve it to be unreliable research but I also believe it dose not properly address to how our mind will adjust to the added stress. now personally I don’t ge t what in heavens name these peo ple they put in these articles have to do with the students spending more time in school it’s not like there stuck in the classroom ding the e xtra work the teachers prepare to compensate for the e xtra time. are they siting first row in this teenage bomb they’re building.

the plan reducing summer well first why don’t they come clean that they all ready reduced it we’re only on vacation for two months give or take a few days and for years they lied to our faces saying we have three months vaca.

to sum it all up I just wish he pub lic the pollutions the school board the reporters would address us on things that affect US not them now granted there may have been a seg ment on the news some were that addressed students about it but if there was did you even pay attention. and let me end this by saying the subduction of certain rights in order to maintain the learning enviorment making your usage of the 1 st amend ment right lesser then you would have in normal society is the same as removing them at the front step!

(I would like to stay anonymous)

Recently, I gave the talk Undeterminacy and choice (Indeterminación y elección), at the XVII Colombian Mathematical Congress, in Cali. Slides can be found at my talks page.

The talk addressed the results of my recent paper on with Richard Ketchersid, mentioned in my previous entry, and some extensions, about which I expect to be posting soon. Afterwards, somebody asked me how much of the theory of determinacy can be extended to three-player (or more) perfect information games. Not much.

The following easy example was suggested by Richard Ketchersid: There is an undetermined one-move game where players I, II, III play 0 or 1, with I playing first, then II, and finally III. To see this, say that and are the numbers played, and that:

• Player I wins iff
• Player II wins iff and
• Player III wins if

(One may think of this game as a perfect information version of paper-rock-scissors.) I imagine this observation is ancient, and would be grateful for a reference.

[Updated: July 24, 2009]

Richard Ketchersid and I have submitted the paper A trichotomy theorem in natural models of  to the Proceedings of BEST. The preprint is available at my papers page. In the paper we provide references and background for the results we discuss, so here I will only mention briefly what the paper is about.

is a strengthening, due to Woodin, of the more familiar axiom of determinacy. In all known models of determinacy, it is the case that in fact holds. Since is an axiom about sets of reals, its natural models are those of the form although there are models of not of this form. 

In this paper, we prove the following result:

Theorem. Assume that and that holds. Let be any partially ordered set. Then either there is an injection of the full binary tree into such that no two points in its image are -comparable, or else can be written as a well-ordered union of -chains.

This statement should be reminiscent of the Harrington-Marker-Shelah theorem on Borel orderings, and in a sense our argument is a generalization of this result. 

Two corollaries are worth pointing out: Suppose first that is simply the diagonal on Then the theorem gives us:

Corollary. Assume that and that holds. Let be a set. Then either injects into or else is well-orderable.

This can be seen as a generalization of Silver’s theorem on co-analytic equivalence relation. In particular, we have the following basis result:

Corollary. Assume that and that holds. Then injects into every infinite set, and if is uncountable, then either or injects into

Our arguments make use of technology developed by Woodin. First, any model of of the form either satisfies or else it has the form for some set of ordinals.

In the second case, one argues via an analysis of the -Borel sets. Essentially, one uses what is sometimes called code compression to obtain, given an -Borel code for a set local versions of this code, that are sufficiently absolute in that they compute traces of correctly both in small inner models of choice, and their forcing extensions. Once this is obtained, the result essentially follows from soft forcing arguments as if the original sets under consideration were Borel. 

In the first case, one uses the argument above to see that a set is expressible as a well-ordered union of smaller sets, for which the result applies. This uses that, under models of the form are of the form where denotes the family of countable bounded subsets of One then uses the uniqueness of the supercompactness measures on for to “paste together” the smaller pieces that make up in a coherent way. The idea in this case was suggested by Woodin, and it is surprisingly flexible. 

As an application, we consider the countable-finite game due to Scheepers. In this game, one fixes a set and two players, I and II, alternate for -many moves, with I moving first, so that each move of I is a countable subset of and each move of II is a finite subset of Player II wins if and only if the union of the finite sets covers the union of the countable sets. If choice holds, it is obvious that player II has a winning strategy. The same argument shows that, without choice, player II has a winning strategy when is countable. In contrast, we prove:

Corollary. Assume that and that holds. Then the countable-finite game on is undetermined for all uncountable sets

For a brief presentation of these results, see the talks that Richard and I gave at BEST 18, available here or at my talks page.

(The “trichotomy” in the title refers to an additional clause in the main theorem, related to the Glimm-Effros dichotomy. I expect to post about an extension of this part of the result soon.)

Announcement 3: Call for papers, Lodging Deadlines.

The 18-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Friday) – March 29 (Sunday), 2009. It is organized by Liljana Babinkostova, Andres Caicedo, Stefan Geschke, Richard Ketchersid, and Marion Scheepers. Contributed and invited talks will be held on Friday, Saturday and Sunday at the Department of Mathematics, Boise State University. The four invited speakers are:

Steve Jackson (University of North Texas)

Ljubisa Kocinac (University of Nis, Republic of Serbia)

Assaf Rinot (Tel Aviv University, Israel)

Grigor Sargsyan (University of California, Berkeley)

Please consult the conference webpage at URL

http://math.boisestate.edu/~best/best18

There are three remaining important deadlines regarding the conference. CRITICAL DEADLINE: LODGING: The Hampton Inn & Suites is providing rooms at a reduced rate for BEST participants. To take advantage of the reduced rate, reservations must be made online by MARCH 12. Please follow this link to the Hampton Inn’s online reservation site for BEST. Anyone interested in participating should contact the organizers as soon as possible by sending an email to

best@math.boisestate.edu

DEADLINE 2: Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 18. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Links are available at the BEST 18 web site. DEADLINE 3: Call for papers: The organizers will be editors for a volume in the Contemporary Mathematics series. Research papers on topics related to Set Theory and its Applications will be considered for publication in this volume. All papers will go through a thorough referee process. Former and current participants of the BEST conferences or their collaborators are especially encouraged to consider submitting a research paper. Anyone interested in submitting a paper should contact Marion Scheepers as soon as possible at

marion@math.boisestate.edu

with this information. Subsequently information regarding preparation of papers will be sent to contributing authors by Contemporary Mathematics. The deadline for submitting a paper is JULY 21. On Saturday, March 28, Billy and Kris Hudson will host a social gathering at their home. All participants are cordially invited to ths social event. Kindly inform Billy at e-mail address

billyhudson@boisestate.edu

of plans to attend. More information is available at the conference web site. The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Contemporary Mathematics is published by the American Mathematical Society. Reduced lodging rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

John Steel was one of my advisors at UC Berkeley. The fifteenth Very Informal Gathering of Logicians at UCLA (VIG) was in honor of John Steel on his 60th birthday. 

The meeting was excellent, with some very interesting talks and nice shared memories. Plus, Benjamin Miller worked miracles and secured the photo above without John finding out.

The 18-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Friday) – March 29 (Sunday), 2009.

It is organized by Liljana Babinkostova, Andres Caicedo, Stefan Geschke, Richard Ketchersid and Marion Scheepers.

Contributed and invited talks will be held on Friday, Saturday and Sunday at the Department of Mathematics, Boise State University. The four invited speakers are:

Steve Jackson (University of North Texas)

Ljubisa Kocinac (University of Nis, Republic of Serbia)

Assaf Rinot (Tel Aviv University, Israel)

Grigor Sargsyan (University of California, Berkeley)

The conference webpage is available at URL

http://math.boisestate.edu/~best/best18

There are four important deadlines regarding the conference:

Lodging: The Hampton Inn & Suites is providing rooms at a reduced rate for BEST participants. To take advantage of the reduced rate, reservations must be made online by MARCH 12. Follow this link for the Hampton Inn’s online reservation site for BEST.

Financial support: Limited financial support is available to partially offset travel expenses of some participants. The criteria for granting support include whether the participant has alternative financial support for the conference, and whether the participant is presenting a talk at the conference. Preference is given to graduate students and early career researchers. The amount of support is contingent on the budget constraints. University accounting regulations require completing certain forms online BEFORE the conference, and submitting original receipts of expenses. Reimbursements will be sent after the conference. The deadline for applying for financial support is MARCH 3.

To apply for support, email the organizers at

best@diamond.boisestate.edu

Applications from graduate students must be supported by a separate email from their thesis advisor. Anyone interested in participating should contact the organizers as soon as possible by sending an email to

best@math.boisestate.edu

Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 18. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Links are available at the BEST 18 web site.

Call for papers: The organizers will be editors for a volume in the Contemporary Mathematics series. Research papers on topics related to Set Theory and its Applications will be considered for publication in this volume.

All papers will go through a thorough referee process. Former and current participants of the BEST conferences or their collaborators are especially encouraged to consider submitting a research paper. Anyone interested in submitting a paper should contact Marion Scheepers as soon as possible at 

marion@math.boisestate.edu 

with this information. Subsequently information regarding preparation of papers will be sent to contributing authors by Contemporary Mathematics. The deadline for submitting a paper is JULY 21.

The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Contemporary Mathematics is published by the American Mathematical Society. Reduced lodging rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

Youth without youth is director Francis Ford Coppola’s most recent film (2007), the first since The Rainmaker (1997).

What a disappointment! The story is impossibly meandering and most of it ends up not going anywhere. For example, for an hour or so, we are in the middle of a Nazi conspiracy, only to have it forgotten as we move twenty or so years into the future. The way this is handled, most of what happened with the Nazis turns out to be irrelevant. The film looks beautiful, but there is no narrative to guide it. There are some rather pedestrian conventions as well, which made me wince whenever they were used.

The main character, Dominic, is a philosopher of language interested in the origin of language, in some mythical protolanguage from which all others sprang. This idea of a single universal origin for language is already bothersome, but let’s say he really only means a proto-Indoeuropean language, which still fails to be convincing anyway. Unsatisfied with his life, lonely and sad for a love lost many years ago (not that the film tries to emotionally connect the audience), he decides to kill himself.

Before going through with his plan, light strikes and he ends up in a hospital, basically reborn. It is Easter, just in case we miss the subtlety. Instead of dying, the lightning somehow induces a rejuvenation process. It also gives him superpowers.

Sigh.

Not only superpowers, actually. It also gives him a “double,” that we see in mirrors, and who may or may not be evil, not that it matters or we care. Nazis blah blah OSS blah. At the end, he kills the double by breaking a mirror. There is shrieking.

His double is not his first victim. Before, he kills a Nazi bad guy with his magic mental powers. Before, any hope one might have had for this film had died as well.

Oh, there are dreams throughout the film. For no good reason, dreams are shown upside down. Whenever we see an upside down sequence (much more often than one would like), we are witnessing a dream. It is a plot device. At the end of the film, there is a scene which we would have clearly identified as a dream except that it is not upside down. So we know it is not a dream but New Age mystical magic. There is a lot of silly New Age mystical magic throughout the film.

After the Nazi nonsense, the movie turns into an exploration of Indian mysticism. Dominic finds his long lost love reincarnated. The woman starts to regress, and he moves in with her, and uses her regressions to explore his protolanguage theory. Oh, but she will die if he continues, so he doesn’t. Another plot that ends up going nowhere.

Tim Roth plays Dominic. Roth is in his mid-forties. People throughout the film, even people noticeably younger, call him a young man. I don’t know, maybe they see his soul, this movie is that silly.

The Arbiter, the student paper at BSU, had this Monday a short note on new faculty, including excerpts of a brief interview with myself.

I recently gave a talk at the Claremont Colleges Algebra/Number Theory/Combinatorics Seminar on the topic of this paper, which can be found in my papers page.

For a set let . A function is regressive iff for all in with . A set is min-homogeneous for iff whenever and .

Theorem. For all there exists such that if and is regressive, then there is of size at least and min-homogeneous for .

The theorem (due to Kanamori and McAloon) states a version of the classical Ramsey theorem for regressive functions. We cannot expect to be homogeneous, i.e., in general will not be constant. For example, consider . Notice also that without loss of generality , since . It is natural to try to establish the rate of growth of the function that to each assigns the least as in the theorem. Using tools of mathematical logic, as part of a more general result about regressive functions of variables, Kanamori and McAloon showed:

Theorem. The function grows faster than any primitive recursive function.

In my paper I show using finite combinatorics methods that grows precisely as fast as Ackermann’s function. This is obtained as part of an analysis of a more general function of two variables, defined as but with the additional requirement that . Obviously,  and . The situation for is less clear, although it is of exponential growth.

Theorem. We have:

1. , , , .
2. , so for .
3. .

Question. Does hold?

Although I have not been able to prove this, I do not expect it to be particularly difficult.