Schedule, Notes/Handouts & Videos

We will present a self-contained introduction to the Castelnuovo–Mumford regularity for standard graded rings over general Noetherian base rings. In particular, we will show that it can be defined in terms of the vanishing of local cohomology modules, vanishing of Koszul homology and the shifts in a minimal free resolution. We will present a proof of a classical result on regularity of the powers of an ideal due originally to Cutkosky, Herzog and Trung and, independently, to Kodiyalam. Variants for products of powers and generalizations will be discussed as well. Examples of determinantal or combinatorial nature will be presented.

This series of lectures aims to introduce symbolic powers of ideals and geometric reasons for studying them. The first lecture will present the early origins of symbolic powers in commutative algebra and the geometric characterization for symbolic powers of radical ideals as higher order vanishing ideals due to Zariski and Nagata. The second lecture will survey 21st century developments in the study of symbolic powers, with a focus on the containment problem between the symbolic and ordinary powers of an ideal. The final lecture will discuss higher order polynomial interpolation problems in algebraic geometry, some asymptotic invariants associated to them, and their connection to symbolic power ideals.

Linkage, or liaison, is a tool for classifying and studying varieties and ideals that has its origins in 19th century algebraic geometry. Its generalization, residual intersection, has broad applications in enumerative geometry, intersection theory, the study of Rees rings, and multiplicity theory. After surveying basic properties of linkage, we will focus on the computation of Picard groups and divisor class groups and on the structure of rigid algebras in the linkage class of a complete intersection. We will describe applications of residual intersections and explain the techniques used to determine their Cohen-Macaulayness, canonical modules, duality properties, and defining equations. An emphasis will be on weakening the hypotheses classically required in this subject.

In this lecture series we will explore the use of symmetry in the study of cohomological invariants that appear naturally in commutative algebra. Symmetries occur frequently in nature and have important structural implications, which is why it is critical to identify and take advantage of them whenever possible. Grassmannians and flag varieties, spaces of matrices or of more general tensors, binary forms, or monomial ideals, are among the objects that we understand quite well, and our understanding relies in an essential way on our ability to exploit their symmetries. The goal of my lectures is to discuss key examples and highlight a variety of open questions.

In this lecture series, we give an introduction to recent applications of p-adic methods to commutative algebra. We start by proving a p-adic version of Kunz's theorem characterizing regular local rings via perfectoid algebras. We then focus on the direct summand theorem and the existence of big Cohen-Macaulay algebras, and we will sketch a proof of them via Andre's flatness lemma. Finally, we use perfectoid big Cohen-Macaulay algebras to define and study singularities in mixed characteristic, and we will discuss some recent work studying various notions of mixed characteristic test ideals via the p-adic Riemann-Hilbert functor. 

Maximal Cohen Macaulay modules are studied in homological algebra and singularity theory. They tell us a lot about the structure of a commutative ring, but in general it is hard to write down such a module explicitly. However, for hypersurface rings, Eisenbud's matrix factorizations provide a good framework to work with them.  Motivated by Physics, Maximal Cohen Macaulay modules have been established as a useful tool in construction of noncommutative desingularizations and recently also have been studied in the context of representation theory of algebras, in particular cluster structures on categories of MCM modules. These lectures should serve as an overview of the field to acquaint the participants with important results and techniques, with a particular focus on motivating examples like Auslander's algebraic version of the McKay correspondence.

In this lecture series, we give an introduction to recent applications of p-adic methods to commutative algebra. We start by proving a p-adic version of Kunz's theorem characterizing regular local rings via perfectoid algebras. We then focus on the direct summand theorem and the existence of big Cohen-Macaulay algebras, and we will sketch a proof of them via Andre's flatness lemma. Finally, we use perfectoid big Cohen-Macaulay algebras to define and study singularities in mixed characteristic, and we will discuss some recent work studying various notions of mixed characteristic test ideals via the p-adic Riemann-Hilbert functor. 

Maximal Cohen Macaulay modules are studied in homological algebra and singularity theory. They tell us a lot about the structure of a commutative ring, but in general it is hard to write down such a module explicitly. However, for hypersurface rings, Eisenbud's matrix factorizations provide a good framework to work with them.  Motivated by Physics, Maximal Cohen Macaulay modules have been established as a useful tool in construction of noncommutative desingularizations and recently also have been studied in the context of representation theory of algebras, in particular cluster structures on categories of MCM modules. These lectures should serve as an overview of the field to acquaint the participants with important results and techniques, with a particular focus on motivating examples like Auslander's algebraic version of the McKay correspondence.

Linkage, or liaison, is a tool for classifying and studying varieties and ideals that has its origins in 19th century algebraic geometry. Its generalization, residual intersection, has broad applications in enumerative geometry, intersection theory, the study of Rees rings, and multiplicity theory. After surveying basic properties of linkage, we will focus on the computation of Picard groups and divisor class groups and on the structure of rigid algebras in the linkage class of a complete intersection. We will describe applications of residual intersections and explain the techniques used to determine their Cohen-Macaulayness, canonical modules, duality properties, and defining equations. An emphasis will be on weakening the hypotheses classically required in this subject.

In this lecture series we will explore the use of symmetry in the study of cohomological invariants that appear naturally in commutative algebra. Symmetries occur frequently in nature and have important structural implications, which is why it is critical to identify and take advantage of them whenever possible. Grassmannians and flag varieties, spaces of matrices or of more general tensors, binary forms, or monomial ideals, are among the objects that we understand quite well, and our understanding relies in an essential way on our ability to exploit their symmetries. The goal of my lectures is to discuss key examples and highlight a variety of open questions.