Reductions & resolutions in representation theory and algebraic geometry
Lead Research Organisation:
University of Glasgow
Department Name: School of Mathematics & Statistics
Abstract
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.
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.
Planned Impact
The main impact of this research proposal will be to add to the existing expertise of pure mathematics in the UK. My intradisciplinary research programme connects representation theory, noncommutative algebra and algebraic geometry and will build on the great experience and knowledge in geometry and representation theory of mathematicians from Glasgow and the whole UK. Moreover, this proposal will strengthen other, less represented areas in UK mathematics, for example (positive characteristic) commutative algebra and the related homological structures, and finite dimensional algebras. These synergies will enhance the influence of UK mathematics in general and strengthen existing links between these fields, in particular.
Establishing lasting collaborations with UK researchers is an important aim for my fellowship and I will build new links and strengthen existing connections with the mathematical community in Belgium and Germany and also with researchers working in the United States and Canada. My close connections with mainland Europe through my background, education and collaboration would help to maintain and strengthen mathematical links between the UK and the rest of Europe in coming years.
The transfer of methods and ideas between representation theory, noncommutative algebra and algebraic geometry will enhance the understanding of each of them. Moreover, abstraction will lead to new insights into homological questions and noncommutative geometry in general. An ideal framework to faciliate and aid this process is an intradisciplinary workshop with international experts from the different fields, which I will organise during my fellowship in Glasgow. Certainly, this workshop will have a high impact on the international and UK mathematical communities.
Establishing lasting collaborations with UK researchers is an important aim for my fellowship and I will build new links and strengthen existing connections with the mathematical community in Belgium and Germany and also with researchers working in the United States and Canada. My close connections with mainland Europe through my background, education and collaboration would help to maintain and strengthen mathematical links between the UK and the rest of Europe in coming years.
The transfer of methods and ideas between representation theory, noncommutative algebra and algebraic geometry will enhance the understanding of each of them. Moreover, abstraction will lead to new insights into homological questions and noncommutative geometry in general. An ideal framework to faciliate and aid this process is an intradisciplinary workshop with international experts from the different fields, which I will organise during my fellowship in Glasgow. Certainly, this workshop will have a high impact on the international and UK mathematical communities.
Publications
Belmans P
(2020)
Derived Categories of Noncommutative Quadrics and Hilbert Squares
in International Mathematics Research Notices
Belmans P
(2019)
Hilbert squares: derived categories and deformations
in Selecta Mathematica
Eisele F
(2018)
A reduction theorem for $$\tau $$ t -rigid modules
in Mathematische Zeitschrift
Eisele F
(2020)
On solvability of the first Hochschild cohomology of a finite-dimensional algebra
in Transactions of the American Mathematical Society
Manin Yuri I.
(2018)
Quantum Groups and Noncommutative Geometry
Raedschelders T
(2022)
The Frobenius morphism in invariant theory II
in Advances in Mathematics
Raedschelders T
(2019)
The Frobenius morphism in invariant theory
in Advances in Mathematics
Raedschelders T
(2018)
The Tannaka-Krein formalism and (re)presentations of universal quantum groups
Raedschelders T
(2019)
The Frobenius morphism in invariant theory
in Advances in Mathematics