Seshadri constants with a view toward commutative algebra
Description: Seshadri constants were introduced by J.-P. Demailly in 1990, inspired by an ampleness criterion of C. S. Seshadri from the 1960s. These constants are defined for line bundles on projective varieties and measure local positivity of a line bundle at points on the variety. Let X be a projective variety, and let L be an ample line bundle on X. For a point xin X, the Seshadri constant of L at x is defined as the infimum, taken over all curves C passing through x, of the ratios \frac{L.C}{m}, where L.C denotes the intersection product of L and C, and m is the multiplicity of C at x. Seshadri constants provide insights into both the local behavior of L at x and certain global properties of X. They are connected to several fundamental problems in algebraic geometry.
Seshadri constants also have significant relevance from the point of view of commutative algebra. In recent years, there has been considerable interest in questions related to symbolic powers of homogeneous ideals in polynomial rings. Of particular interest is the case of ideals of points
in the projective space. If I is the ideal of a finite set Z of points, then the m-th symbolic power of I is the ideal of forms vanishing on Z to order at least m. An important question concerns containment relations between ordinary and symbolic powers of I. From a geometric perspective, it is important to study the Hilbert function of the symbolic powers of I. Many key concepts, such as Waldschmidt constants, are defined in this context. The famous Nagata conjecture can also be interpreted in terms of the symbolic powers of suitable ideals of points in the projective plane. It turns out that Seshadri constants are closely related to all these notions. In these lectures we will introduce Seshadri constants, emphasizing their connections with these ideas from commutative algebra.

An introduction to invariant theory
Description: Given a group action on a polynomial ring, one may ask which polynomials are fixed by every element of the group. The set of all such polynomials forms a ring, called the ring of invariants of the action. The study of such rings has motivated many key developments in Commutative Algebra—for example, Hilbert proved his famous Basis Theorem, Syzygy Theorem, and Nullstellensatz all in pursuit of finiteness properties of rings of invariants. In the aftermath of Hilbert, many beautiful results have established that rings of invariants have good ring-theoretic properties, or related favorable ring-theoretic properties to properties of the group action. In this series we will discuss some classical results on rings of invariants, as well as modern developments and questions in this field.