The APDE seminar on Monday, 2/5, will be given by Haoren Xiong (UCLA) in-person in **Evans 740,** and will also be broadcasted online via Zoom from **4:10pm to 5:00pm PST**. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

**Title:** Toeplitz operators, semiclassical asymptotics for Bergman projections

**Abstract:** In the first part of the talk, we discuss boundedness conditions of Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with H. Xu), we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We shall review a direct approach to the construction of asymptotic Bergman projections, developed by A. Deleporte – M. Hitrik – J. Sj\”ostrand in the case of real analytic weights, and M. Hitrik – M. Stone in the case of smooth weights. We shall explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit $h \to 0+$.