Explore the intricate relationship between differential operators and the geometry of domains in Euclidean space in this inaugural AMS Maryam Mirzakhani Lecture delivered at the Joint Mathematics Meetings 2020 in Denver, CO. Delve into advanced mathematical concepts as the speaker presents groundbreaking insights on this complex topic. Gain a deeper understanding of how differential operators interact with geometric structures and their implications in Euclidean space. This 51-minute lecture offers a unique opportunity to engage with cutting-edge research in mathematics and honors the legacy of Maryam Mirzakhani.