In the first hour, we will discuss the relationship between the geometry of a domain in Euclidean space and the properties of canonical solutions to Laplace's equation in that domain. We will emphasize the extent to which the behavior of the harmonic measure of a domain determines the structure of its boundary.
In the second hour, we address related questions for certain divergence form elliptic operators. In particular we will describe some recent results concerning the relationship between the behavior of the elliptic measure for these operators and the geometry of the boundary of the domain where the operators are defined.
Professor Toro earned her PhD from Stanford University in 1992. Following postdoctoral positions at the Institute for Advanced Study, University of California, Berkeley, and University of Chicago, she joined the faculty at University of Washington in 1996. Toro is a leading expert working at the interface of geometric measure theory, harmonic analysis, and partial differential equations. Professor Toro was an invited speaker in 2010 International Congress of Mathematicians and invited plenary speaker at the Joint Mathematics Meetings in 2011, 2016, and 2020. Toro has held the NSF postdoctoral fellowship, A.P. Sloan fellowship, and Guggenheim fellowship and is a corresponding member of the Academia Colombiana de Ciencias Exactas, Fiscas y Naturales. She is an editor of the Transactions and Memoirs of the AMS and Analysis & PDE among other journals. At the national service level, Professor Toro currently serves on the IPAM Board of Trustees, the MSRI Scientific Advisory Board, and the AMS Editorial Boards Committee. Her notable outreach activities include organizing the Lat@math conferences in 2015 and 2018, which showcased the achievements of latinx mathematicians in the United States. In June 2019, Professor Toro was awarded the Marsha L. Landolt Distinguished Graduate Mentor Award at the University of Washington. Toro has advised 7 PhDs at the University of Washington from 2007-2019.