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Set Theory


Recursion Theory


Recursion Theory


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Computational Prospects of Infinity
(20 Jun - 15 Aug 2005)

Organizing Committee · Confirmed Visitors · Overview · Activities · Membership Application

 Organizing Committee

  • Chi Tat Chong (National University of Singapore)
  • Qi Feng (Chinese Academy of Sciences, China and National University of Singapore)
  • Theodore A. Slaman (University of California at Berkeley)
  • W. Hugh Woodin (University of California at Berkeley)
  • Yue Yang (National University of Singapore)

 Confirmed Visitors

 Overview

This two-month program on Computational Prospects of Infinity will focus on recent developments in Set Theory and Recursion Theory, which are two main branches of mathematical logic.

Topics for Set Theory will include topics related to Cantor's Continuum Hypothesis (CH), with special attention paid to the importance of the Ω Conjecture. Cantor's Continuum Hypothesis states that there is no set whose size falls between those of the natural numbers and of the real numbers. In his famous lecture of 1900 in Paris, Hilbert placed the continuum hypothesis at the top of a list of the 23 most important mathematics problems of the 20th century. From the work of Gdel (1938) and Cohen (1963), we now know that it is impossible either to prove or to disprove the Continuum Hypothesis using the standard axioms. In the following decades, mathematicians tried to settle the Continuum Hypothesis one way or another by proposing new axioms. In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has linked the Continuum Hypothesis with the Ω Conjecture, which if true, would imply that every "elegant" axiom would make the Continuum Hypothesis false. If the Ω Conjecture is true, then in understanding why it is true, our understanding of sets will have advanced considerably. It will be possible to quantify the limits of forcing and to give an abstract definition of the hierarchy of Axioms of Infinity. One route to proving the Ω Conjecture is through the Ω Iteration Hypothesis. Understanding this latter hypothesis involves Inner Model Theory and Determinacy Axioms. The program on Set Theory will be devoted to these conjectures and the related problems in Inner Model Theory and determinacy.

Topics for Recursion Theory will include recursive enumerability and randomness. Recursive enumerability is one of the most important notions in Recursion Theory. The study of recursively enumerable (r.e.) degrees and global Turing degrees is arguably the center of Recursion Theory. Some of the main issues, such as whether or not r.e. degree has a nontrivial automorphism and whether all jump classes can be defined naturally, remain unsettled since 1960's. The computational aspect of randomness, also known as algorithmic information theory, has been revived after recent work by Slaman, Downey, Nies and many other people. A great deal of recent research has been done. But, fundamental problems remain. In particular, what is missing is the generalization of these results to other probability measures within the effective setting. This will require conceptual and technical advances. For example, the coincidence of Kolomogorov-Chaitin randomness and Martin-Lf randomness does not generalize from Lebesque measure to every effective measure (Reimann and Slaman).

Specifically, the program will focus on the following topics:

  1. Ω-conjecture
  2. Fine Structures
  3. Recursive Enumerability
  4. Effective Randomness

 Activities

The program activities consist of tutorials and seminars.

Tutorials

  • Two tutorials each consisting of 5 lectures. The primary purpose is to provide background material to the participants, especially the student participants, on the main topics of the program.
  • For the first month, the tutorials will be in Ω-logic (Woodin) and Fine Structure (to be determined).
  • During the second month, the tutorials will be in Recursive Enumerability (Slaman) and Effective Randomness (to be determined).

Seminars and Talks

  • Seminars and talks will be given by participants on recent results latest developments related to the core themes of the programs.

Public Lecture

  • Logic and Computation
    Date & Time
    : 1 Aug 2005, 6:30pm - 7:30pm
    Speaker: Ted Slaman, University of California, Berkeley, USA
    Venue: LT 33, School of Computing, NUS
    ...organized in conjunction with Department of Mathematics and Singapore Mathematical Society

Mathcamp

  • Date & Time: 22 Jun 2005, 09:00am - 03:00pm
    Speakers: W. Hugh Woodin, University of California, Berkeley, USA
                     Qi Feng, National University of Singapore and Chinese Academy of
                     Sciences
    Venue: IMS Auditorium, 3 Prince George's Park, Singapore 118402
    Participants: 35 students from NUS High School, Temasek Junior College and
                        Raffles Junior College

Schedule of Talks and Tutorials

IMS Membership is not required for participation in above activities. For attendance at these activities, please complete the registration form (MSWord|PDF|PS) and fax it to us at (65) 6873 8292 or email it to us at ims@nus.edu.sg.

If you are an IMS member or are applying for IMS membership, you do not need to register for these activities.

 Membership Application

The Institute for Mathematical Sciences invites applications for membership for participation in the above program. Limited funds to cover travel and living expenses are available to young scientists. Applications should be received at least three (3) months before the commencement of membership. Application form is available in (MSWord|PDF|PS) format for download.

More information is available by writing to:
Secretary
Institute for Mathematical Sciences
National University of Singapore
3 Prince George's Park
Singapore 118402
Republic of Singapore
or email to imssec@nus.edu.sg.

For enquiries on scientific aspects of the program, please email Qi Feng matfq@nus.edu.sg.

Organizing Committee · Confirmed Visitors · Overview · Activities · Membership Application