Abstracts Week 1
Lecture series

Linquan Ma
Title: Perfectoid algebras and singularities in mixed characteristic
Abstract: In this lecture series, we give an introduction to recent applications ofp-adic methods to commutative algebra. We first prove a p-adic version of Kunz’s theorem
characterizing regular local rings via perfectoid algebras. We will discuss André’s flatness
lemma and use it to sketch a proof of the direct summand theorem and the existenceof big Cohen-Macaulay algebras. Finally, we will use perfectoid big Cohen-Macaulay
algebras to define and study singularities in mixed characteristic, and we will introduce
a numerical invariant called perfectoid signature, which is defined using Bhatt-Scholze’s
perfectoidization functor and Faltings’ normalized length function that measures thesesingularities.

Karl Schwede
Title: Global and local applications of perfectoid algebras in mixed characteristic
Abstract: In the first lecture, we will discuss some motivation related to Kodairavanishing in characteristic zero, Frobenius in positive characteristic, and how finite covers
and perfectoid methods can be applied in mixed characteristic to obtain similar results.
In the second lecture we will discuss lifting sections via such methods in global and local settings (for instance inversion of adjunction), and how that can be used in variousapplications. In the third lecture, we will discuss global generation properties and applications that these methods can be used to deduce. Finally, in the fourth lecture we will
discuss forthcoming work unifying various notions of mixed characteristic test ideals withmultiplier ideals in characteristic zero.

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Research talks

Emelie Arvidsson
Title: Properties of log canonical singularities in positive characteristic
Abstract: We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic
and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see thatthe pathological behavior of these singularities in positive characteristic is closely linked
to the failure of Kodaira-type vanishing theorems in positive characteristic. Additionally,we will explore how these questions are related to the moduli theory of varieties of general
type.
This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint
work with Q. Posva.

Manuel Blickle
Title: F -finite schemes have a dualizing complex
Abstract: I will present joint work with Bhatt, Schwede and Tucker where we showthat by surprisingly elementary means one can equip the category of F -finite schemes witha canonical dualizing complex ω • such that for all essentially finite type maps f : X → Y• ∼there is a canonical isomorphism ωX= f ! ωY• . This is in particular true for the finite•• ∼.
Frobenius morphism yielding a canonical isomorphism ωX= F ! ωX
Kevin Tucker
Title: On F -pure Inversion of Adjunction
Abstract: Suppose R is a local ring of prime characteristic p > 0 and f is aregular element. Viewed through the lens of reduction to characteristic p > 0, Kawakita’s
inversion of adjunction for log canonical singularities roughly predicts that R/f R is F17→F∗ f p−1

pure only if the pair (R, div(f )) is as well, i.e., R −−−−−−→ F∗ R is pure. In this talk, Iwill discuss joint work with Thomas Polstra and Austyn Simpson showing this equivalence
provided R is Q-Gorenstein with arbitrary index. In particular, we recover previous resultsof Schwede (who showed this equivalence provided the index is prime to p) and PolstraSimpson (who showed F-purity deforms in Q-Gorenstein rings). Time permitting, we will
also discuss additional results relating the F -purity of divisor pairs (R, S + B) and thatof (S, Diff S (B)).