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SUMMARY:Symmetries - Mota Gaytn\, M A (Instituto Tecnolgico Autnomo de Mxi
 co)
DTSTART:20150825T140000Z
DTEND:20150825T150000Z
UID:TALK60445@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:Abstract: In the last years there has been a second boom of th
 e technique of forcing with side conditions (see for instance the recent w
 orks of Asper'{o}-Mota\, Krueger and Neeman describing three different per
 spectives of this technique). The first boom took place in the 1980s when 
 Todorcevic discovered a method of forcing in which elementary substructure
 s are included in the conditions of a forcing poset to ensure that the for
 cing poset preserves cardinals. More than twenty years later\, Friedman an
 d Mitchell independently took the first step in generalizing the method fr
 om adding small (of size at most the first uncountable cardinal) generic o
 bjects to adding larger objects by defining forcing posets with finite con
 ditions for adding a club subset on the second uncountable cardinal. Howev
 er\, neither of these results show how to force (with side conditions toge
 ther with another finite set of objects) the existence of such a large obj
 ect together with the continuum being small. In the first part of this tal
 k I will discuss new results in this area. This is joint work with John Kr
 ueger improving the symmetric CH preservation  argument previously made by
  Asper'{o} and Mota. In the second part of this talk I will use generalize
 d symmetric systems in order to prove that\, for each regular cardinal k\,
  there is a poset $P_k$ forcing the existence of a (k\,k++)-superatomic bo
 olean algebra. This is joint work with William Weiss inspired in an unpubl
 ished note from September 2009 where Asper'{o} and Bagaria introduced the 
 forcing $P_{omega}$.\n
LOCATION:Seminar Room 1\, Newton Institute
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