Periodic Reporting for period 2 - MMiMMa (MMP and Mirrors via Maximal Modification Algebras)
Okres sprawozdawczy: 2022-12-01 do 2024-05-31
This project involves fundamental problems in modern geometry. This includes a proposed classification of 3-dimensional "surgeries" in algebraic geometry called flops, perhaps the most basic of all higher dimensional surgeries, and it also includes many other related questions and structures, such as determining when curves contract (in other words, can be squished to a point), together with connections to more modern notions such as stability conditions and Frobenius manifolds.
The main novelty is that to solve all these questions it is essential to use noncommutative algebra, a structure which is very different from our geometric intuition. These are objects where order matters; it is this extra information that then gives the finer information needed to finally classify and prove various results.
There are a number of specific objectives in the proposal, and they are spread across both algebra and geometry. Most involves first both developing general noncommutative (and homological) machinery, and then second applying this machinery to different (mainly geometric) applications.
The main novelty is that to solve all these questions it is essential to use noncommutative algebra, a structure which is very different from our geometric intuition. These are objects where order matters; it is this extra information that then gives the finer information needed to finally classify and prove various results.
There are a number of specific objectives in the proposal, and they are spread across both algebra and geometry. Most involves first both developing general noncommutative (and homological) machinery, and then second applying this machinery to different (mainly geometric) applications.
There have been various results obtained by both the PI and the wider team. Highlights include:
-The paper arXiv:2111.05900 joint with Gavin Brown, which invents noncommutative singularity theory, and discovers various normal forms, with immediate applications to the first classification of Type D flops.
-The paper arXiv:2205.11552 joint with Wahei Hara, which fully classifies spherical objects in various contexts, where the main insight is that in fact much more general (and unexpected) homological statements are in fact true.
-The paper arXiv:2310.18062 by Namanya, which establishes the most general form of pure braid actions from algebraic flops, generalising and simplifying many previous constructions.
-The paper arXiv:2109.13289 with Nabijou, which gives a description of the pole locus of the quantum potential in terms of hyperplanes, establishing the Crepant Resolution Conjecture for flops, with algebraic applications to contraction algebras.
-The paper arXiv:2111.05900 joint with Gavin Brown, which invents noncommutative singularity theory, and discovers various normal forms, with immediate applications to the first classification of Type D flops.
-The paper arXiv:2205.11552 joint with Wahei Hara, which fully classifies spherical objects in various contexts, where the main insight is that in fact much more general (and unexpected) homological statements are in fact true.
-The paper arXiv:2310.18062 by Namanya, which establishes the most general form of pure braid actions from algebraic flops, generalising and simplifying many previous constructions.
-The paper arXiv:2109.13289 with Nabijou, which gives a description of the pole locus of the quantum potential in terms of hyperplanes, establishing the Crepant Resolution Conjecture for flops, with algebraic applications to contraction algebras.
The main progress beyond the state-of-the-art expected between now and the end of the project is that noncommutative singularity theory will produce a new ADE-style classification, complete with a full list of normal forms, and these normal forms will have fully control 3-fold flops.
There are various homological results too that are now expected to be within reach. The classification of spherical objects (and their generalisations) initiated in arXiv:2205.11552 is expected to give similar results in affine type; schobers are now expected for 3-fold flops; topological obstructions are expected to the existence of NCCRs; contractibility is now at least as complicated as noncommutative deformation theory.
There are various homological results too that are now expected to be within reach. The classification of spherical objects (and their generalisations) initiated in arXiv:2205.11552 is expected to give similar results in affine type; schobers are now expected for 3-fold flops; topological obstructions are expected to the existence of NCCRs; contractibility is now at least as complicated as noncommutative deformation theory.