PFA implies all automorphisms are trivial

Trivial Automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, d, is equal t...

Strongly regular graphs with non-trivial automorphisms

All Automorphisms of the Calkin Algebra Are Inner

We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips–Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown–Douglas–Fillmore. We also give a simpler and self-contained proof of the Phillips–Weaver result. ...

Membership Query Removal Implies Non-trivial Cryptographic Primitives

Understanding learning and discovering to what extent it can be automated is one of the major problems in computer science. This is of obvious relevance when trying to create intelligent robots, for example, and fields as diverse as biology and economics use learning algorithms to help manage and understand their extremely large data sets. There has even been work devoted to analyzing computati...

When automorphisms of P ( κ ) / [ κ ] < א 0 are trivial off a

It is shown that if κ > 2א0 and κ is less than the first inaccessible cardinal then every automorphism of P(κ)/[κ]<א0 is trivial outside of a set of cardinality 2א0 .