Woodin conference

March 11, 2015

Woodin_Poster

The conference in honor of Hugh Woodin’s 60th birthday will take place at Harvard University, on March 27-29, 2015. The meeting is partially supported by the Mid-Atlantic Mathematical Logic Seminar and the National Science Foundation. Funding is available to support participant travel. Please write to woodinbirthdayconference@gmail.com to apply for support, and to notify the organizers if you are planning to attend.

The list of speakers is as follows:

  • H. Garth Dales
  • Qi Feng
  • Matthew D. Foreman
  • Ronald Jensen
  • Alexander S. Kechris
  • Menachem Magidor
  • Donald A. Martin
  • Grigor Sargsyan
  • Theodore A. Slaman
  • John R. Steel.

We expect to publish proceedings of the conference, together with select additional research and survey papers, through the series Contemporary Mathematics, of the AMS. The editors of the proceedings are myself, James Cummings, Peter Koellner, and Paul Larson. Please contact me for information regarding the proceedings.

Additional information can be found at the conference website.

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Cryptic marks

February 5, 2015

New scientist recently ran a series on articles on “How to think about…” One of them, by Richard Webb and published December 13, 2014, was about infinity. It contains this quote:

Woodin’s notepads consist mainly of cryptic marks he uses to focus his attention, to the occasional consternation of fellow plane passengers. “If they don’t try to change seats they ask me if I’m an artist,” he says.

David Roberts wondered on Google+ what these cryptic marks look like. This reminded me of some pictures I took of them at the Conference on inner model theory at UC Berkeley last year.

2014-06-10 17.59.20

2014-06-10 17.37.00

2014-06-10 17.36.08

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580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that {\omega\rightarrow(\omega)^n_m} (Ramsey) and {\omega\rightarrow[\omega]^n_\omega} ({\mbox{Erd\H os}}-Rado) for any {n,m<\omega.} On the other hand, we also have that {2^\kappa\not\rightarrow(3)^2_\kappa} ({\mbox{Sierpi\'nski}}) and {2^\kappa\not\rightarrow(\kappa^+)^2} ({\mbox{Erd\H os}}-Kakutani) for any infinite {\kappa.}

Positive results can be obtained for larger cardinals than {\omega} if we relax the requirements in some of the colors. A different extension, the {\mbox{Erd\H os}}-Rado theorem, will be discussed later.

Theorem 1 ({\mbox{Erd\H os}}-Dushnik-Miller) For all infinite cardinals {\lambda,} {\lambda\rightarrow(\lambda,\omega)^2.}

This was originally shown by Dushnik and Miller in 1941 for {\lambda} regular, with {\mbox{Erd\H os}} providing the singular case. For {\lambda} regular one can in fact show something stronger:

Theorem 2 ({\mbox{Erd\H os}}-Rado) Suppose {\kappa} is regular and uncountable. Then
\displaystyle \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2, which means: If {f:[\kappa]^2\rightarrow2} then either there is a stationary {H\subseteq\kappa} that is {0}-homogeneous for {f}, or else there is a closed subset of {\kappa} of order type {\omega+1} that is {1}-homogeneous for {f}.

(Above, top stands for “topological.”)

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