Lieven Le Bruyn emeritus with assignment

Research Lieven Le Bruyn

Research team

Expertise

Non-commutative geometry and representation theory.

Toposes of monoid actions and noncommutative geometry. 01/10/2020 - 30/09/2023

Abstract

To each monoid we can naturally associate a topos, consisting of sets with a right action of this monoid. This opens the door to many geometrical invariants associated to the monoid, following the philosophy of toposes as generalized topological spaces. For example, toposes have points, and for the toposes associated to monoids, calculating the points can give surprising results. A simple example is the monoid of nonzero natural numbers under multiplication. Alain Connes and Caterina Consani showed that the points of the associated topos are up to isomorphism given by a double quotient featuring the finite adeles. They then constructed a structure sheaf on the topos, and showed that this combination of topos and structure sheaf, their Arithmetic Site, is related to the noncommutative geometry approach to the Riemann Hypothesis. In this research project, we will systematically study the toposes associated to monoids from a geometric point of view. In certain cases, we will construct structure sheaves on these toposes, leading to generalized Connes-Consani arithmetic sites.

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An algebraic approach to Connes--Consani arithmetic sites. 01/10/2019 - 30/09/2020

Abstract

For a large number N, the Riemann Hypothesis would give a very precise estimate of the amount of prime numbers smaller than N. Because prime numbers are the foundation of number theory, many mathematical problems depend on it, which is why the Riemann Hypothesis is considered to be one of the most important unresolved problems in mathematics. In the 1940's, André Weil, a famous mathematician and brother of philosopher Simone Weil, has proved a variant of the Riemann Hypothesis regarding estimation problems for a very specific type of polynomials. His proof was related to geometry, more specifically to the study of curves. In a recent series of papers, Alain Connes and Caterina Consani have described an approach to the Riemann Hypothesis by introducing and studying the Arithmetic Site. This is a new geometric object describing the distribution of the prime numbers, constructed with contemporary mathematical techniques. It has properties similar to that of a curve in geometry, so the hope is that eventually Weil's proof can be translated to a proof of the original Riemann Hypothesis. While Connes and Consani focus on a tropical geometry point of view, we will construct alternatives that allow for a more algebraic point of view. In particular, we want to relate their approach to the study of noncommutative algebras, a subject for which the University of Antwerp is well-known.

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Azumaya representation varieties and stacks 01/10/2017 - 30/09/2019

Abstract

A lot has changed in geometry since the study of shapes (like triangles) in ancient Greece. The shapes studied nowadays very often have more than three dimensions and are curved, and can be extremely complicated. This is useful for fields like physics or engineering, which have also evolved drastically since Pythagoras and need geometrical techniques that become more and more involved. In this project we want to investigate a way to stretch the abstraction of geometry still further. To this extent, we will need concepts from algebra, like coordinates and equations, but further elaborated and more abstract. More precisely, we will study 'rings', collections of values that you can multiply or add. One particular kind of rings that will be important are 'Azumaya algebras'. We will need to solve some specific questions about these algebras to get a better understanding of the geometry in question. The study of Azumaya algebras or rings in general is also interesting on its own because they appear everywhere in mathematics, and there are still a lot of unresolved questions about them. Additionally, the proposed research will have implications in physics, more precisely string theory. From a string theorist's perspective, the smallest building blocks of the universe are vibrating strings (like guitar strings). Endpoints of these strings are called 'D-branes', and they are described accurately by the geometry we propose to explore.

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Constructing superpotential algebras from finite group actions. 01/10/2017 - 30/09/2018

Abstract

To a superpotential \Phi is associated an associative algebra J_{Q,\Phi} by taking cyclic derivatives of \Phi. If C\Phi is a 1-dimensional G-subrepresentation of CQ/[CQ,CQ], then G acts on J_{Q,\Phi} as algebra automorphisms. The point of this research is to study J_{Q,\Phi} in the hope of finding Cayley-smooth orders (that is, algebras finite over their center such that their trace preserving representation varieties are smooth) on which G acts. The starting point would be to take a superpotential \Phi_0 which gives an algebra J_{Q,\Phi_0} that is PI and consider other superpotentials \Phi such that C\Phi \cong C\Phi_0 as G-representation, such that \Phi degenerates to \Phi_0 in a controlled way.

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Azumaya representation varieties and stacks. 01/10/2015 - 30/09/2017

Abstract

A lot has changed in geometry since the study of shapes (like triangles) in ancient Greece. The shapes studied nowadays very often have more than three dimensions and are curved, and can be extremely complicated. This is useful for fields like physics or engineering, which have also evolved drastically since Pythagoras and need geometrical techniques that become more and more involved. In this project we want to investigate a way to stretch the abstraction of geometry still further. To this extent, we will need concepts from algebra, like coordinates and equations, but further elaborated and more abstract. More precisely, we will study 'rings', collections of values that you can multiply or add. One particular kind of rings that will be important are 'Azumaya algebras'. We will need to solve some specific questions about these algebras to get a better understanding of the geometry in question. The study of Azumaya algebras or rings in general is also interesting on its own because they appear everywhere in mathematics, and there are still a lot of unresolved questions about them. Additionally, the proposed research will have implications in physics, more precisely string theory. From a string theorist's perspective, the smallest building blocks of the universe are vibrating strings (like guitar strings). Endpoints of these strings are called 'D-branes', and they are described accurately by the geometry we propose to explore.

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Geometric and algebraic aspects of the representation and invariant theory of quivers with relations and other combinatorial objects. 01/10/2006 - 31/12/2008

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Cayley-Hamilton Algebras in Noncommutative Geometry. 01/10/2004 - 30/09/2007

Abstract

One of the main results obtained in earlier research is a proof for the Popov conjecture [1] for quiver representations, i.e. the quotient maps are not equidimensional if the quotient variety has singularities. This study is to expand to the more general situation of Brauer-Severi varieties (introduced by M. Van den Bergh) over (smooth) Cayley-Hamilton orders. The dimension of the fiber of the Brauer-Severi fibration over an arbitrary point in the variety of isomorphism classes of trace preserving representations of a smooth Cayley-Hamilton order will be computed. In the end, this should lead to determining when such a Brauer-Severi fibration is a flat morphism.

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Quiver Singularities and their applications in Algebraic Geometry, Invariant Theory and Theoretical Physics. 01/10/2003 - 30/09/2006

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Non commutative geometry and commutative singularities. 01/10/2003 - 30/09/2005

Abstract

In our work we look for methods to desingularise quotient varieties of the variety of n-dimensional representations of an algebra A, under the natural action of GLn. In particular in the case of isolated singularities, we hope to make progress.

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Geometry of matrixinvariants and arithmetic geometry. 01/10/2000 - 30/09/2005

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Rationality problem for quotients of PFLn-varieties. Connection between ringtheoretical properties of Sklyanin algebras and arithmetic of elliptic curves.

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Geometry of matrixinvariants and arithmetic geometry. 01/10/1997 - 30/09/2000

Abstract

Rationality problem for quotients of PFLn-varieties. Connection between ringtheoretical properties of Sklyanin algebras and arithmetic of elliptic curves.

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    Fundamental Mathematics. 01/01/1997 - 31/12/1997

    Abstract

    The library is, in all its aspects, the research lakoratory in mathematica. This is certainly the case for fundamental mathematica. With the promised money books will be purchased. As such the research in algebra end analysis/stochastics will be enhanced.

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      Geometric methods in algebraic classification problems. 01/10/1996 - 31/12/1997

      Abstract

      Invariant theoretic methods are applied to the classification of finite dimensional algebras, Hopf algebras and Lie stacks.

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        Quantized algebras: representations and weight modules. 30/09/1994 - 31/12/1996

        Abstract

        We study certain classes of quantized algebras and their representations in connection with their non-communative geometry. Specific classes of modules are treated in detail. We try to apply "braided techniques" to this theory.

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          Graphical study of quantum spaces. 01/01/1993 - 31/12/1993

          Abstract

          Quantum deformations of the enveloping algebra of Su (2) lead to non-commutative projective 3-spaces. The local structure of these spaces is studied by graphical methods implemented in "Mathematica".

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            Hopf algebra actions and the ring of invariants or semi-invariants related to quantum spaces and quantum groups. 30/09/1989 - 29/09/1992

            Abstract

            Hopf algebra actions on algebra extensions may be viewed as an extension of classical Galois theory. Induction and coinduction from invariants is being studied.

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              Geometry of matrixinvariants and arithmetic geometry. 30/09/1986 - 30/09/1997

              Abstract

              Rationality problem for quotients of PFLn-varieties. Connection between ringtheoretical properties of Sklyanin algebras and arithmetic of elliptic curves.

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                • Research Project