Recorded 01 May 2025. Ralph Sabbagh of the University of California, Irvine, presents "On the Weyl symbols of Gaussian semigroups" at IPAM's Dynamics of Density Operators Workshop. Abstract: Inspired by a recent interpretation of the Husimi function in terms of Gaussian semigroups and its connection to the Wigner distribution, we begin the talk by presenting a generalized approach for constructing positive representations of quantum states with respect to a set of quantum observables. The main focus of the talk is to then view each representation as a smearing of its corresponding Wigner distribution, in complete analogy with the Gaussian smearing that gives rise to the Husimi function. A key step towards this is studying the existence of a Weyl symbol for the Gaussian semigroup that would give rise to such a smearing, hence the title of the talk. Relevant concepts that will be covered include the Weyl functional calculus for unbounded self-adjoint operators and infinitesimal representations of Lie groups. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-dynamics-of-density-operators/