the classical segal--bargmann transform (sbt) is an isomorphism between a real gaussian hilbert space and a reproducing kernel hilbert space of holomorphic functions.  it arises in quantum field theory, as a concrete witness of wave-particle duality.  introduced originally in the 1960s, it has been generalized and extended to many contexts: lie groups (hall, driver, late 1980s and early 1990s), free probability (biane, early 2000s), and more recently $q$-gaussian factors (cébron, ho, 2018).

in this talk, i will discuss current work with charlesworth and ho on a version of the sbt in bifree probability, a "two faced" version of free probability introduced by voiculescu in 2014.  our work leads to some interesting new combinatorial structures ("stargazing partitions"), as well as a detailed analysis of the resultant family of reproducing kernels.  in the end, the bifree sbt has a surprising connection with the $q$-gaussian version for some $q\ne 0$.