Reductions & resolutions in representation theory and algebraic geometry

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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, in which observable quantities commute pairwise, 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 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 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.To give at least one concrete example of a problem we consider in this project, consider the Markoff equation, a diophantine equation given by x^2 +y^2 +z^2 = 3xyz,which was introduced by Markoff back in 1880 while investigating minimal values taken up by integral quadratic forms. Markoff showed that all solutions to this equation could be obtained from a simple inductive process. An obvious unicity question was formulated by Frobenius in 1913: given a triple (a, b, c), with c as largest value, satisfying the equation, does c uniquely determine this triple? As is often the case in number theory, elementary questions can give rise to deep theories in diverse areas of mathematics, at first glance unrelated to the problem. One of the objectives in the current project is to investigate a connection between the representation theory of the noncommutative symmetry group of the projective plane and the solutions of Markoff's equation.