Minicourses

Holger Brenner - Asymptotic properties of differential operators around a singularity

For a local algebra R over a field, we study the decomposition of the module of principal parts. A free summand of the n-th module of principal parts is essentially the same as a differential operator E of order greater or equal than n with the property that the partial differential equation E(f) =1 has a solution. The asymptotic behavior of the size of the free part gives a measure for the singularity represented by R. We compute this invariant, called the differential signature, for invariant rings, toric monoid rings, determinantal rings, quadrics, and we compare it with the F-signature, which is an invariant in positive characteristic defined by looking at the asymptotic decomposition of the Frobenius. This is joint work with Jack Jeffries and Luis Nuñez Betancourt.

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Linquan Ma - Homological conjectures/theorems in commutative algebra

The homological conjectures have been a focus of research in commutative algebra since 1960s. They concern a number of interrelated conjectures concerning various homological properties of commutative rings to their internal ring structures. These conjectures had largely been resolved for rings that contain a field, but several remained open in mixed characteristic---until recently Yves Andre proved Hochster's direct summand conjecture, which lie in the heart of the homological conjectures. This lecture will focus on the recent breakthroughs. In particular, we will talk about the direct summand conjecture and the existence of big Cohen-Macaulay algebras in positive and mixed characteristic. The main new ingredient in the solution in the mixed characteristic case is the use of perfectoid spaces, and we will give a brief introduction to this subject. We will also discuss some interesting open questions.