This award supports participation in the workshop "Theory and Applications of Elliptic Partial Differential Equations" held August 17-21, 2026 at University of California, Irvine. The workshop consists of three short courses given by experts in elliptic partial differential equations (PDE). Elliptic PDE play an important role in problems from physics, biology, and geometry, and the field has seen exciting advances in recent years. The overall goals of the courses are to bring graduate students up to speed in the most active areas of elliptic PDE, to promote a sense of community within the field, and to set the course for possible future research directions. At a technical level, the courses present recent advances made on regularity and stability in fluid dynamics, fully nonlinear geometric PDE, and homogenization. All of these areas have experienced spectacular developments in the past few years, some examples being classification results for steady solutions to the Euler equations in two dimensions, Bernstein type theorems for the Monge-Ampère equation with periodic right-hand side, and optimal convergence rates for periodic homogenization of Hamilton-Jacobi equations. The event provides students an opportunity to forge new research directions and to make connections with fellow students as well as established mathematicians. The participants also have an opportunity to give short talks and poster presentations on their research. This serves as training in scientific communication, and as a catalyst for networking. Products such as sets of lecture notes and lecture videos will be made accessible to the broader mathematical community. A website for the workshop is available at https://ucipde2025.github.io/2026-PDE-Summer-School-Website-Public-keep-public/. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.