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## Project

# Noncommutative crepant resolutions for three- and higher- dimensional singularities (FWOTM976)

It is very natural for humans to want to classify similar things, or likewise to want to make distinctions between things which have different properties. In the domain where my research lies, 'algebraic geometry', one of the aims is to classify certain geometric figures called 'varieties'. Most of the geometric figures that you can imagine are of this type, think for instance of a line, circle or sphere.

Within algebraic geometry lies the field of 'birational geometry'. Here one classifies varieties up to so called 'birational equivalence', which means that two varieties which roughly look the same are considered to belong to the same (equivalence) class. However, not all varieties which are birationally equivalent are equally 'nice'. Therefore, it is an interesting question whether or not one can find the "nicest" one.

This leads to so called 'resolution of singularities'. Moreover, ideally one would like a resolution to have as little discrepancies with the original variety as possible, which leads to the notion of a 'crepant resolution' (crepant is a shortening of non-discrepant).

This proposal studies a 'noncommutative' version of these types of resolutions, namely 'noncommutative crepant resolutions'. The main objectives of this proposal are to complete our understanding of these types of resolutions in the three-dimensional case and to investigate what can be said in higher dimensions.

Within algebraic geometry lies the field of 'birational geometry'. Here one classifies varieties up to so called 'birational equivalence', which means that two varieties which roughly look the same are considered to belong to the same (equivalence) class. However, not all varieties which are birationally equivalent are equally 'nice'. Therefore, it is an interesting question whether or not one can find the "nicest" one.

This leads to so called 'resolution of singularities'. Moreover, ideally one would like a resolution to have as little discrepancies with the original variety as possible, which leads to the notion of a 'crepant resolution' (crepant is a shortening of non-discrepant).

This proposal studies a 'noncommutative' version of these types of resolutions, namely 'noncommutative crepant resolutions'. The main objectives of this proposal are to complete our understanding of these types of resolutions in the three-dimensional case and to investigate what can be said in higher dimensions.

Date:1 Nov 2019
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Keywords:noncommutative geometry, categorical geometry

Disciplines:Algebraic geometry , Associative rings and algebras , Category theory, homological algebra