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Project

Reductions, deformations & resolutions in representation theory and noncommutative geometry (FWOTM885)

Quantum mechanics is a staple of 20th century science, and has led to the realisation that  physical quantities are governed by noncommutative algebra. More precisely, Werner Heisenberg replaced classical mechanics, where observable quantities pairwise commute, with matrix mechanics, where crucial observables like position and momentum no longer commute with each other. To study quantum mechanics, it is therefore natural to also try and extend the classical geometry of points, lines, planes etc. to the noncommutative world. This gives rise to the mathematical field of noncommutative geometry. Later on, the mathematician Hermann Weyl realised that the operators corresponding to position and momentum satisfied relations that occurred in another area of mathematics called representation theory, which studies the "symmetries" of abstract mathematical objects. 
In this project we analyse several spaces appearing in (noncommutative) geometry by looking at their symmetries and deformations, and use representation theory to say something new about them. The fundamental idea, which goes back to Alexander Grothendieck, is to associate to a possibly noncommutative space an algebraic invariant (its derived category), which is rich enough to capture a lot of the geometry of the space while at the same time being sufficiently flexible, moving the focus from geometry to a more algebraic point of view.
Date:1 Oct 2017  →  Today
Keywords:reductions, noncommutative geometry
Disciplines:General mathematics