Silver trees and Cohen reals
Wednesday, 3.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will sketch the main ideas of my recent result that the\nmeager ideal is Tukey reducible to the Mycielski ideal. The latter one is\nthe ideal associated with Silver forcing. This implies that every\nreasonable amoeba forcing for Silver adds a Cohen real. This has been open\nfor some years.\n
Schanuel's Conjecture and Exponential Fields
Wednesday, 10.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
Schanuel's Conjecture states that for a collection of n complex\nnumbers z1, ..., zn, linearly independent over the field of\nrational numbers, the transcendence degree of z1, ..., zn,\nexp(z1), ..., exp(zn) is at least n.\n\nZilber constructs in [Zilber, Pseudo-exponentiation on\nalgebraically closed fields of characteristic zero] a sentence\nwhose models are structures called strongly\nexponentially-algebraically closed fields with\npseudo-exponentiation, which are unique in every uncountable\ncardinality. One of their main properties is that Schanuel's\nConjecture holds in those fields.\n\nFirstly, I will outline the properties of Zilber's fields.\nSecondly, I will sketch the proof given in [Marker,\nA Remark on Zilber's Pseudoexponentiation] showing that, if one\nassumes Schanuel's Conjecture, the simplest case of one of the\naxioms of Zilber's fields holds in the complex exponential field.\n
Amoeba and tree ideals
Wednesday, 17.12.14, 16:30-17:30, Raum 404, Eckerstr. 1
I will talk about what I asked to Spinas in the end of his talk, i.e., whether an\namoeba for Silver might add Cohen reals. Two weeks ago he proved that add(J(Silver)) is at most\nadd(M). However this is not strictly sufficient to infer that any proper amoeba for Silver does\nadd Cohen reals, but only that it does not have the Laver property. I will clarify this\nissue. If there will be any time left I will also present some results about other tree ideals,\nwhich are part of a joint work, still in preparation, with Yurii Khomskii and Wolfgang\nWohofsky.\n