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# Lecture Series “Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry”

## July 13, 2022 at 12:15 – 13:45 CEST

There will be a 4-talk Lecture series on mixed characteristic algebraic geometry by Kevin Tucker (UIC) during July here in Mainz. The first talk is a Kolloquium style talk and should be interesting for (und largely understandable by) most people; and you can decide if you continue with the other 3, more demanding (but even more rewarding), talks.

Talk 1: Wednesday July 13: 12:15–13:45

Talk 2: Monday July 18: 12:15–14:45

Talk 3: Monday July 18: 16:15–17:45

Talk 4: Wednesday July 20: 12:15-13:35

All talks are taking place in the Hilbertraum 05-432, Staudingerweg 9, 55099 Mainz, or alternatively via Zoom:

https://zoom.us/j/91070632898?pwd=ODM2a1RlZ1RwdVhxVkg2dEk1Vy9CZz09 Meeting ID: 910 7063 2898 Passcode: 123123

Title: Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry

Speaker: Kevin Tucker (University of Illinois at Chicago)

Abstract: Standard “reduction to characteristic p” techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). For example, log terminal and F-regular singularities are known to correspond to one another via reduction to characteristic p >> 0. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well, drawing on the (conjectured) characterization of F-regular rings as splinters in positive characteristic. A ring is a splinter if it is a direct summand of every finite cover, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster’s verification of the conjecture in equal characteristic using Frobenius techniques. In these talks, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).