The lectures will start with elementary techniques relating generic ultrapowers, ideals and generic elementary embeddings. The "three parameters" will be introduced and the ideas of "natural" and "induced" ideals will be discussed. We will then move to applications of generic large cardinal embeddings to some classical problems in set theory and some applications in algebra and topology. After this we will consider some special cases, including natural ideals such as the nonstationary ideal on the first uncountable cardinal.
The second part of the lectures will deal with the existence of generic elementary embeddings. The comments will have two directions: outright proofs of the existence of generic elementary embeddings from large cardinals and relative consistency results.
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