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Andres Caicedo

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### Andres Caicedo

Shared publicly -A couple of days ago I received by mail a copy of what seems to be Hugh's first (unpublished) paper (this is [28] in the quotes below). The title page reads "Discontinuous homomorphisms from C(Ω) and the partially ordered set ω^ω", Hugh Woodin, Fall 1976.

Together with Paul Larson and two other colleagues, I am in the process of editing a volume of proceedings to appear in Contemporary Mathematics, to celebrate Hugh's 60th birthday. There was a meeting in Harvard last year in his honor.

https://plus.google.com/+AndresCaicedo0/posts/NmUBxppwoxy

https://andrescaicedo.wordpress.com/2015/04/01/woodin-meeting/

One of the papers is a lovely survey by Garth Dales on "Norming infinitesimals of large fields". Dales uses the opportunity to recall a couple of anecdotes regarding Hugh.

He says in the introduction: "I first met Hugh in the year 1983-84 when he was one of my TAs for a calculus class that I was teaching at Berkeley. (As a TA, Hugh was initially quite puzzled that there were students at Berkeley who did not understand concepts that were very obvious to him, but he quickly learned about reality.) Subsequently we worked together on two books, and this meant that Hugh came to our house in England quite a few times. This was great pleasure for us. It was something of a shock to my wife and myself to receive an invitation to come to Harvard for a conference to mark Hugh's 60th birthday: how could this charming young man have reached so mature a level?"

He adds some historical remarks at the end: "The paper [5] was mainly worked out in the year 1973-74, whilst I was at UCLA; I thank Phil. Curtis for inviting me to UCLA and for much support. I gave lectures on this at the inaugural conference on Banach algebras in July, 1974. The work was mostly written in the year 1974-75." He proceeds to explain his result (obtained independently at about the same time by Jean Esterle), namely that, under the continuum hypothesis (CH), given any infinite compact space Ω, there is a discontinuous homomorphism from C(Ω) into some Banach algebra.

He continues: "The paper [5] was submitted in May, 1976.

"I then thought how one could remove `(CH)' from the theorem, but could not do this; with some trepidation I wrote in June 1976 to Professor Robert Solovay at Berkeley, and asked for his help in this. (Young colleagues might like to know that one wrote by hand on paper in those distant days; but we did have airmail.) This letter eventually reached Solovay at Caltech, where he was on leave.

"The next part of the story is based on information from Frederick Dashiell, then a Bateman Research Instructor at Caltech; I am grateful to Fred for this.

"Hugh Woodin, then a junior at Caltech, approached Fred in January 1976 for a topic for his senior thesis. Fred suggested that Hugh organize what was known about Kaplansky's problem up to that point, and explain the heart of the open question on homomorphisms from C(βℕ). Hugh ignored the survey part of the suggestion, and immediately started trying to construct a model of NDH ["no discontinuous homomorphisms"]; he was learning about forcing at that time, and discussed the question with Solovay, the leading expert on forcing. Hugh produced a type-written document [28] that I have

in `Fall 1976'; by that time Hugh had seen the preprint of [5]. It seems that this document was not submitted as a senior thesis, and that it has not been published, but results from it are contained in Hugh's thesis [29] and in [8]. The paper [28] seems to be Hugh's first contribution to mathematics.

"In fact, Hugh proved in [28,§5] that it is consistent with ZFC that there is an ultrafilter U such that ℓ^∞/U does not admit a non-zero algebra semi-norm, and, in [28,§6], he gave a set-theoretic condition, now `Woodin's condition', which, if satisfied, implies NDH. Subsequently Solovay showed that this condition was consistent with ZFC, and lectured on this on 26 October 1976; this argument was never published because Hugh himself soon gave a shorter proof of the same result. ...

"After some time Solovay kindly replied to my letter, saying that `Woodin' had shown that there are models of set theory in which NDH is true. I wrote to the person that I took to be `Professor Woodin' at Caltech to ask about this, and received [28] in response."

A short while back there were a post or two here on Woodin's "cryptic marks",

https://plus.google.com/+DavidRoberts/posts/TtiW4dMiFcF

and

https://plus.google.com/+DavidRoberts/posts/P7oxqv8nhxL

Dales also has something to say in this regard: "It will be apparent that the original insights into the theorems of the second half of [9], including all theorems that involve forcing, were due to Hugh Woodin.

My comment on his method of proof is the following: first, there were discussions on the likelihood that one class was contained in another class, and possible counter-examples were considered; the questions were clarified and the undergrowth cleared away; then Hugh thought about the matter for some time, and drew a large number of squiggly lines on a sheet of white paper; then he opined, using remarkable insight on what would be true in certain models of set theory, that a particular containment would be relatively consistent with ZFC + GCH. These insights were almost always completely correct in the end; but it took many days and many pages to craft a careful proof."

The references mentioned above are

[5] H. G. Dales, A discontinuous homomorphism from C(X), American J. Math., 101 (1979), 647-734.

[8] H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series 115, Cambridge University Press, 1987.

[9] H. G. Dales and W. H. Woodin, Super-real fields: totally ordered fields with additional structure, London Mathematical Society Monographs, Volume 14, Clarendon Press, Oxford, 1996.

[29] W. H. Woodin, Discontinuous homomorphisms of C(Ω) and set theory, Thesis, University of California, Berkeley, 1984.

Together with Paul Larson and two other colleagues, I am in the process of editing a volume of proceedings to appear in Contemporary Mathematics, to celebrate Hugh's 60th birthday. There was a meeting in Harvard last year in his honor.

https://plus.google.com/+AndresCaicedo0/posts/NmUBxppwoxy

https://andrescaicedo.wordpress.com/2015/04/01/woodin-meeting/

One of the papers is a lovely survey by Garth Dales on "Norming infinitesimals of large fields". Dales uses the opportunity to recall a couple of anecdotes regarding Hugh.

He says in the introduction: "I first met Hugh in the year 1983-84 when he was one of my TAs for a calculus class that I was teaching at Berkeley. (As a TA, Hugh was initially quite puzzled that there were students at Berkeley who did not understand concepts that were very obvious to him, but he quickly learned about reality.) Subsequently we worked together on two books, and this meant that Hugh came to our house in England quite a few times. This was great pleasure for us. It was something of a shock to my wife and myself to receive an invitation to come to Harvard for a conference to mark Hugh's 60th birthday: how could this charming young man have reached so mature a level?"

He adds some historical remarks at the end: "The paper [5] was mainly worked out in the year 1973-74, whilst I was at UCLA; I thank Phil. Curtis for inviting me to UCLA and for much support. I gave lectures on this at the inaugural conference on Banach algebras in July, 1974. The work was mostly written in the year 1974-75." He proceeds to explain his result (obtained independently at about the same time by Jean Esterle), namely that, under the continuum hypothesis (CH), given any infinite compact space Ω, there is a discontinuous homomorphism from C(Ω) into some Banach algebra.

He continues: "The paper [5] was submitted in May, 1976.

"I then thought how one could remove `(CH)' from the theorem, but could not do this; with some trepidation I wrote in June 1976 to Professor Robert Solovay at Berkeley, and asked for his help in this. (Young colleagues might like to know that one wrote by hand on paper in those distant days; but we did have airmail.) This letter eventually reached Solovay at Caltech, where he was on leave.

"The next part of the story is based on information from Frederick Dashiell, then a Bateman Research Instructor at Caltech; I am grateful to Fred for this.

"Hugh Woodin, then a junior at Caltech, approached Fred in January 1976 for a topic for his senior thesis. Fred suggested that Hugh organize what was known about Kaplansky's problem up to that point, and explain the heart of the open question on homomorphisms from C(βℕ). Hugh ignored the survey part of the suggestion, and immediately started trying to construct a model of NDH ["no discontinuous homomorphisms"]; he was learning about forcing at that time, and discussed the question with Solovay, the leading expert on forcing. Hugh produced a type-written document [28] that I have

in `Fall 1976'; by that time Hugh had seen the preprint of [5]. It seems that this document was not submitted as a senior thesis, and that it has not been published, but results from it are contained in Hugh's thesis [29] and in [8]. The paper [28] seems to be Hugh's first contribution to mathematics.

"In fact, Hugh proved in [28,§5] that it is consistent with ZFC that there is an ultrafilter U such that ℓ^∞/U does not admit a non-zero algebra semi-norm, and, in [28,§6], he gave a set-theoretic condition, now `Woodin's condition', which, if satisfied, implies NDH. Subsequently Solovay showed that this condition was consistent with ZFC, and lectured on this on 26 October 1976; this argument was never published because Hugh himself soon gave a shorter proof of the same result. ...

"After some time Solovay kindly replied to my letter, saying that `Woodin' had shown that there are models of set theory in which NDH is true. I wrote to the person that I took to be `Professor Woodin' at Caltech to ask about this, and received [28] in response."

A short while back there were a post or two here on Woodin's "cryptic marks",

https://plus.google.com/+DavidRoberts/posts/TtiW4dMiFcF

and

https://plus.google.com/+DavidRoberts/posts/P7oxqv8nhxL

Dales also has something to say in this regard: "It will be apparent that the original insights into the theorems of the second half of [9], including all theorems that involve forcing, were due to Hugh Woodin.

My comment on his method of proof is the following: first, there were discussions on the likelihood that one class was contained in another class, and possible counter-examples were considered; the questions were clarified and the undergrowth cleared away; then Hugh thought about the matter for some time, and drew a large number of squiggly lines on a sheet of white paper; then he opined, using remarkable insight on what would be true in certain models of set theory, that a particular containment would be relatively consistent with ZFC + GCH. These insights were almost always completely correct in the end; but it took many days and many pages to craft a careful proof."

The references mentioned above are

[5] H. G. Dales, A discontinuous homomorphism from C(X), American J. Math., 101 (1979), 647-734.

[8] H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series 115, Cambridge University Press, 1987.

[9] H. G. Dales and W. H. Woodin, Super-real fields: totally ordered fields with additional structure, London Mathematical Society Monographs, Volume 14, Clarendon Press, Oxford, 1996.

[29] W. H. Woodin, Discontinuous homomorphisms of C(Ω) and set theory, Thesis, University of California, Berkeley, 1984.

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4 comments

+Andres Caicedo Oops. Well, I'm sure my "correction" was still in the vein of Dales...

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### Andres Caicedo

Shared publicly -The Twitter image of the email seems to have disappeared from the

(Thanks to http://elanormal.com/posts/28051-rumor-mill-heats-up-again-for-discovery-of-gravitational-waves for keeping a copy, where you can find the usual warnings that this is not confirmed or released etc.)

Personally, I would be totally :-D if the team put the paper on the arXiv, like all other sensible astrophysicists did. I don't think the glossy mags would pass up a chance to publish one of the major findings in physics merely because there was a preprint, no?

#gravitationalWaves

*Science*news article and from Twitter, so here it is.**edit:**hmm, I can see it on my phone, but not on my laptop. Could be a cache thing?(Thanks to http://elanormal.com/posts/28051-rumor-mill-heats-up-again-for-discovery-of-gravitational-waves for keeping a copy, where you can find the usual warnings that this is not confirmed or released etc.)

Personally, I would be totally :-D if the team put the paper on the arXiv, like all other sensible astrophysicists did. I don't think the glossy mags would pass up a chance to publish one of the major findings in physics merely because there was a preprint, no?

#gravitationalWaves

1

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### Andres Caicedo

Shared publicly -Alan Moore's Levitz Grid for all 12 issues of Big Numbers, continued on the next double page spread! Illustration from "Alan Moore: Storyteller" by Gary Spencer Millidge.

While it definitely is a Levitz Grid, it doesn't use the Levitz Paradigm. In those terms, the plot progression is:

DEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

Until issue 11, where he wanted to add yet another letter.

More on the Levitz Grid and Paradigm here: https://plus.google.com/116234966919830684742/posts/agG3z6KTJcW

While it definitely is a Levitz Grid, it doesn't use the Levitz Paradigm. In those terms, the plot progression is:

DEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

ABCDEFGHIJKLMNOPQRSTUVWXYZ...

Until issue 11, where he wanted to add yet another letter.

More on the Levitz Grid and Paradigm here: https://plus.google.com/116234966919830684742/posts/agG3z6KTJcW

1

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### Andres Caicedo

Shared publicly -Frankl's union-closed conjecture is the following easy sounding statement. Let A_1,...,A_n be n subsets of a set X and suppose that for each i and j the union of A_i and A_j is one of the subsets (so the collection of sets is closed under taking unions). Then there is an element x of X that belongs to at least n/2 of the sets.

Seems simple? Then why not help me prove it? But I should mention, by way of a mild warning, that it is not even known whether there must be an element of X that belongs to at least n/1,000,000 of the sets (or indeed cn for any fixed constant c>0). But in a way that makes it easier to make progress -- even a statement that falls quite some way short of the conjecture would be an advance.

Seems simple? Then why not help me prove it? But I should mention, by way of a mild warning, that it is not even known whether there must be an element of X that belongs to at least n/1,000,000 of the sets (or indeed cn for any fixed constant c>0). But in a way that makes it easier to make progress -- even a statement that falls quite some way short of the conjecture would be an advance.

1

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### Andres Caicedo

Shared publicly -A new journal seen via Prof. Mark Sapir on Facebook:

"Journal of Combinatorial Algebra is devoted to publication of research articles of the highest level. Its domain is the rich and deep area of interplay between combinatorics and algebra. Its scope includes combinatorial aspects of group, semigroup and ring theory, representation theory, commutative algebra, algebraic geometry and dynamical systems. Exceptionally strong research papers from all parts of mathematics related to these fields are also welcome."

"Journal of Combinatorial Algebra is devoted to publication of research articles of the highest level. Its domain is the rich and deep area of interplay between combinatorics and algebra. Its scope includes combinatorial aspects of group, semigroup and ring theory, representation theory, commutative algebra, algebraic geometry and dynamical systems. Exceptionally strong research papers from all parts of mathematics related to these fields are also welcome."

Journal of Combinatorial Algebra European Mathematical Society. Editorial Board · Aims and Scope · Instructions for Authors.

4

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### Andres Caicedo

Shared publicly -The Foundations of mathematics (FOM) mailing list, which I left a while ago due to unfair moderating practices, seems to have gone crazy with infinite discussions of "Set theory versus Homotopy Type Theory".

I find the whole activity ridiculous and based on the false premise that the two foundations are in opposition (as attested by the titles of discussion threads), or that they cannot co-exist, or that all foundations of mathematics need to be linearly ordered. These are all childish and false motivations.

The discussions are further hampered by the fact that no expert in homotopy type theory is taking part. This might partly be a consequence of the lessons learned in the 1990's when FOM was extremely hostile to category theorists, although I suspect (and can affirm for myself personally) that the questions asked and the issues raised by Harvey Friedman are simply not relevant to the HoTT crowd. There seems to be a very fundamental difference in perception of what foundations of mathematics are, ought to be, and what purpose they serve.

I find the whole activity ridiculous and based on the false premise that the two foundations are in opposition (as attested by the titles of discussion threads), or that they cannot co-exist, or that all foundations of mathematics need to be linearly ordered. These are all childish and false motivations.

The discussions are further hampered by the fact that no expert in homotopy type theory is taking part. This might partly be a consequence of the lessons learned in the 1990's when FOM was extremely hostile to category theorists, although I suspect (and can affirm for myself personally) that the questions asked and the issues raised by Harvey Friedman are simply not relevant to the HoTT crowd. There seems to be a very fundamental difference in perception of what foundations of mathematics are, ought to be, and what purpose they serve.

**Supplemental:**I have deleted the extremely long comments that appeared under this post without reading them and have disabled further comments. I also deleted one paragraph of my post which was arguably something that required a reply. I am simply not in the business of prolonging discussions with titles like "HoTT vs. ZFC" that present "challenges to HoTT" or "challenges to ZFC", etc. They are counter-productive and do not serve any purpose that would warrant them appearing on my posts. If anyone feels like producing yet more discussions, they are welcome to do so under their own G+ account. It is easy enough to reference this post. But do not expect me to participate, I simply wanted to express publicly my disappointment with the FOM contracting a very common Internet sickness.April 2016 Archives by thread. Messages sorted by: [ subject ] [ author ] [ date ] · More info on this list... Starting: Fri Apr 1 01:23:14 EDT 2016. Ending: Wed Apr 13 19:14:11 EDT 2016. Messages: 79. [FOM] Philosophical coherence and foundational role of set and type theory Astor, Eric P.

3

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### Andres Caicedoowner

Discussion - From Nam Trang and Martin Zeman:

"Dear colleagues,

the next Core Model Induction + Hod Mice meeting will be held at UC Irvine during July 18 -- July 29, 2016. The format of the meeting will be the same as that of previous meetings in Muenster and Palo Alto. That is, the second week is expected to be more informal with more emphasis on interaction between the participants."

For details, in particular regarding funding, please contact Martin or Nam at UC Irvine.

"Dear colleagues,

the next Core Model Induction + Hod Mice meeting will be held at UC Irvine during July 18 -- July 29, 2016. The format of the meeting will be the same as that of previous meetings in Muenster and Palo Alto. That is, the second week is expected to be more informal with more emphasis on interaction between the participants."

For details, in particular regarding funding, please contact Martin or Nam at UC Irvine.

1

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### Andres Caicedo

Shared publicly -Dear Colleagues: The Journal of Number Theory is delighted to publish a special issue in honor of the work of Winnie Li! The papers will be freely available till the end of May. For more information, please visit

http://www.journals.elsevier.com/journal-of-number-theory/news/honoring-of-the-lifelong-work-of-wen-ching-winnie-li/

http://www.journals.elsevier.com/journal-of-number-theory/news/honoring-of-the-lifelong-work-of-wen-ching-winnie-li/

Elsevier is a world-leading provider of scientific, technical and medical information products and services

1

2 comments

Thanks, Tim. Much more constructive than the comment I was tempted to make.

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### Andres Caicedo

Shared publicly -**Henstock's lectures on Henstock integration**

Of course, this notion of integral has many other names, but these notes (~130 page) look to be rather nice

http://arxiv.org/abs/1602.02993

Title: Ralph Henstock's Lectures on the Theory of Integration

Authors: Ralph Henstock, Pat Muldowney

Abstract: These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the Henstock-Kurzweil integral). They also introduce Henstock's abstract theory of integration.

Comments: Ralph Henstock (d. 2007) is the author of these notes. Pat Muldowney is the note-taker (1970-71) and editor (2015)

#arXiv

Abstract: These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the Henstock-Kurzweil integral).

1

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### Andres Caicedo

Shared publicly -Registration for the Twelfth Symposium on General Topology – Toposym 2016 is now open. The conference will take place in Prague, Czech Republic on July 25–29, 2016.

The following people have accepted our invitation to present plenary lectures: A. Arhangel'skii, L. Aurichi, D. Dikranjan, A. Dow, M. Hrušák, O. Kalenda, A. Kechris, P. Koszmider, M. Krupski, W. Kubis, A. Kwiatkowska, J. Lopez-Abad, V. Martinez de la Vega, J. Melleray, J. van Mill, A. Miller, J. Moore, C. Mouron, L. Nguyen Van Thé, L. Oversteegen, C. Rosendal, M. Sabok, S. Solecki, L. Soukup, M. Tkachenko, S. Todorcevic, T. Usuba, B. Weiss.

More info and registration at http://www.toposym.cz/

The following people have accepted our invitation to present plenary lectures: A. Arhangel'skii, L. Aurichi, D. Dikranjan, A. Dow, M. Hrušák, O. Kalenda, A. Kechris, P. Koszmider, M. Krupski, W. Kubis, A. Kwiatkowska, J. Lopez-Abad, V. Martinez de la Vega, J. Melleray, J. van Mill, A. Miller, J. Moore, C. Mouron, L. Nguyen Van Thé, L. Oversteegen, C. Rosendal, M. Sabok, S. Solecki, L. Soukup, M. Tkachenko, S. Todorcevic, T. Usuba, B. Weiss.

More info and registration at http://www.toposym.cz/

1

6 comments

Thank you, +David Chodounsky.

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### Andres Caicedo

Shared publicly - #mathematics accelerating the speed of convergence of a series , an example with zeta(2) http://goo.gl/4DzLwz

5

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Mathematician. Associate editor at Math Reviews. Father of two.

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