H2020Индивидуална стипендия2018–2020

SINGREP · Linking singularity theory and representation theory with homological methods

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2018-08-01 → 2020-07-31
Финансиране от ЕС: 183 455 €
Участници: 1
Схема: MSCA-IF-EF-ST

Линиите свързват координатора с партньорите.

Накратко на български

Математическите връзки между теорията на сингулярностите и теорията на представянията се изследват чрез методи като т.нар. съответствие на Маккей. Това помага за по-доброто разбиране на сингулярните алгебрични многообразия и свойствата на алгебрите.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Linking singularity theory and representation theory with homological methods

This project lies at the crossroads of singularity theory, algebraic geometry, commutative algebra, and representation theory. The main goal was to develop homological methods to find novel links between singularity theory and representation theory. Further, the experienced researcher (in the following referred to as the PI) wanted to exploit these connections in both directions to increase the understanding of singular algebraic varieties as well as the representation theory of algebras. During the fellowship, the PI focussed on three work packages: (1) the construction and study of noncommutative (crepant) resolutions of singularities (NC(C)Rs), (2) a McKay correspondence for reflection groups, (3) the study of friezes and singularity categories. (1) The aim was to develop new methods for finding Noncommutative (Crepant) Resolutions of singularities (NC(C)Rs) and to determine their properties in important cases (in particular for toric rings). (2) The aim was to establish a McKay correspondence for reflection groups. The construction proposed by Buchweitz, the PI, and Ingalls, is both natural and surprising, since reflection groups have not been studied yet in context of the McKay correspondence. During the fellowship, the PI proposed to work on several algebraic aspects emerging from this correspondence. (3) The first objective was to find a categorical interpretation of higher SL_k-friezes. Further, the PI proposed to find new links between friezes and singularities by studying categories of maximal Cohen-Macaulay modules over coordinate rings of singular varieties to obtain friezes and other combinatorial data.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

In algebraic geometry one tries to understand and explain geometric phenomena of zerosets of polynomial equations (algebraic varieties) with algebraic techniques. Singularities of algebraic varieties are, roughly speaking, points of indeterminacy, where most analytical methods collapse. Geometrically, this corresponds e.g. to cusps or crossing points. In a practical example, the arm of a robot can break if it passes through a singular point, which could result in a complete breakdown of the system. Such a situation should be avoided by theoretical considerations.This project lies at the intersection of singularity theory, (non-commutative) algebraic geometry, commutative algebra, and representation theory. The main goal is to develop homological methods to understand geometric phenomena of algebraic varieties in the presence of singularities and use them to study representation theoretic concepts such as cluster categories and friezes. The project will provide a bridge between these seemingly distant areas that can be exploited in both directions.The specific research objectives: (1) Construction of noncommutative (crepant) resolutions of singularities (NC(C)Rs), in particular for not necessarily normal varieties/rings: computation of global dimension, application to positive characteristic (global dimension of ring of differential operators)(2) McKay correspondence for reflection groups: study of the geometry of discriminants of pseudo-reflection groups and their relation to the representation theory of the groups, characterization of McKay quivers(3) Friezes and singularities: show how (higher) integral friezes can be constructed from cluster categories and categories of maximal Cohen-Macaulay modulesThe project will be carried out by Eleonore Faber, supervised by Robert Marsh at the University of Leeds. Apart from the scientific value, this project should serve to integrate Faber in the algebra research group and to establish her as a research leader.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITY OF LEEDS · LeedsКоординаторОбединеното кралство