Raf Bocklandt School of Mathematics and Statistics Newcastle University Newcastle upon Tyne NE1 7RU United Kingdom office phone: +44 (0)191 222 5370 fax: +44 (0)191 222 8020 e-mail: |

Research interests

- Noncommutative geometry
- Resolutions of singularities
- Geometric invariant theory
- Representation theory of quivers
- Calabi Yau Manifolds and Algebras
- Applications of quivers in theoretical physics and topology

Publications of Raf Bocklandt

Bocklandt, Raf; Smooth quiver representation spaces. J. Algebra 253 (2002), no. 2, 296--313.

Bocklandt, Raf;
The Local structure of Calabi Yau Algebras.

preprint: arXiv:0711.0179

Bocklandt, Raf;
Calabi Yau algebras and Quiver Polyhedra

preprint: arXiv:0905.0232

Bocklandt, Raf;
Consistency Conditions for dimer models.
preprint: arXiv:1104.1592

Bocklandt, Raf;
Generating toric noncommutative crepant resolutions

preprint: arXiv:1104.1597

Bocklandt, Raf;

The geometry of quotient Varieties of
quivers,

PhD thesis

Overview of my research

My main research area is the representation theory of noncommutative algebras and its connections to various other parts of mathematics, such as Algebraic geometry, Non-commutative geometry, algebraic topology, Knot theory and theoretical physics.I started out studying the geometry of quotient varieties of quivers, classifying all quivers and dimension vectors such that the quotient space classifying all isomorphism classes of semi-simple representations is a smooth variety.

Together with Lieven Le Bruyn, Geert Van de Weyer, Jan Adriaensens and Stijn Symens I extended these methods to investigate singularities that can occur in these varieties, the properties of the fibers of the quotient map and other geometric objects associated to noncommutative algebras like character varieties, moduli spaces, graded representation spaces and Brauer-Severi varieties.

Together with Lieven Le Bruyn also studied the connection between noncommutative symplectic geometry, preprojective algebras and a type of infinite dimensional Lie algebras called Necklace Lie Algebras.

I also have been investigating Calabi-Yau algebras. These algebras appear in the homological study of 3-dimensional gorenstein singularities and also in the study of superstring theory on calabi yau manifolds. I showed that graded 3-dimensional Calabi-Yau algebras derive from superpotentials and together with Michael Wemyss and Travis Schedler we extended the superpotential method to arbitrary dimensions in the case that the algebra is Koszul. In the case of toric singularities I proved that noncommutative toric crepant resolutions in dimension 3 alway come from dimer models and I studied the connection between the different different consistency conditions for dimer models.

Recently I have been working on a program that can construct all possible dimer models for a given 3-dimensional Gorenstein singularity. This program can be tried out here.

Talks

Lecture notes

Here are some lecture notes for undergraduate and graduate courses that I taught.
Advanced Master Course on Kleinian Singularities

Coding theory and Cryptography

Capita Selecta on noncommutative resolutions of singularities