This volume consists of edited lecture notes from two events at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, organized in honor of Melvin Hochster and Craig Huneke. The two events were the online International Graduate course on Tight Closure and Its Applications, spread over summer 2022 and the May 2023 School on Commutative Algebra and Algebraic Geometry in Prime Characteristics. The unifying basis of these events was the theory of tight closure, the brainchild of Melvin Hochster and Craig Huneke in the late 1980s, which has had a dramatic effect on the field of commutative algebra, giving unified proofs and strong generalizations of many major theorems in commutative algebra, and stimulating much research, including recent proofs of longstanding conjectures. The aim of the two events as well as of this volume is to provide training in the foundations of commutative algebra and algebraic geometry in prime characteristic and to present some of the exciting recent developments in and beyond tight closure. The lecture notes for the online school also come with exercises and solutions. The topics in the volume include characteristic p methods, test ideals, direct summands, singularities, Hilbert-Kunz multiplicity, Briancon-Skoda theorems, big Cohen-Macaulay algebras, the localization problem, uniform Artin-Rees results, vector bundles and tight closure, singularity invariants. The intended audience is graduate students learning the material as well as researchers wanting the latest advances in one reference.
