Current Research Grants (Karin Baur)
Dimer algebras on
project P 30549, FWF, 2017 - 2020.
Researchers on the project:
Dr. Ch. Beil, Dr. A. Garcia Elsener.
This project studies interactions between geometrical objects such as curves on surfaces and categories of representations.
The main focus of the project is the development of a general theory of surface categories capturing properties of their geometry.
The project will provide a novel combinatorial
geometric approach to the study of module categories and new approaches to important problems
in plane and surface geometry from an algebraic perspective.
At the heart of all mathematical modelling is representation theory; and at the heart of representation theory lies quiver algebras.
These are algebras defined from oriented graphs, a key notion of
the proposal. A dimer model is an oriented graph embedded in a surface such that its complement is a union of disks.
This data naturally associates an algebra to a dimer model, the dimer algebra, with
basis all the paths in the graph (infinitely many), with multiplication arising from concatenation of
paths (we work over the complex numbers) and where the
relations are given by a so-called potential. The boundary algebra of a dimer model is the idempotent subalgebra of
the dimer algebra with respect to boundary vertices.
Sources for dimer algebras and their boundaries are abundant, most relevant for the proposal
are the dimers models arising from Postnikov's strand diagrams on disks.
More generally, dimer algebras arise from arbitrary tilings of surfaces.
With this proposal, we want to study categories which arise from dimers, i.e.
from graphs embedded on surfaces, capturing the essential properties of the surface.
We call such categories surface categories.
Examples for these are the Grassmannian cluster categories and the (higher) cluster categories.
The main aim of the proposal is to study dimer algebras on surfaces and the boundary algebras arising from them.
It is supported by five objectives.
Determine boundary algebras for surfaces with punctures, for surfaces with several boundary components, and for higher genus.
Explore module categories of boundary algebras and their stable parts. Study homological properties of algebras of infinite global dimension.
Determine boundary algebras for infinity-gons, for surfaces with asymptotic arcs.
Associate dimer algebras to rhombic tilings, study algebras for Grassmann permutations. Explore the exchange graph of Yang-Baxter moves.
(5) Explore the interactions between noncommutative resolutions, nonnoetherian geometry, and the homological properties of
dimer algebras on surfaces.
Mathematics and Arts: Towards a balance between artistic intuition and mathematical
complexity, KFU Graz, October 2014-September 2017.
Principal investigators: K. Baur, K. Fellner.
Gerhard Eckel, Tamara Friebel.
Project page: thecollaborativemind
Article in IMN, 2016
Geistesblitz (Standard) August 2014
Salzburger Nachrichten August 2014
Falter Heureka Nr. 2, 2015
Die Presse January 2016.
project P 25141, FWF, 2012 - 2015.
Researchers on the project: Dr. Mark Parsons, Hannah Vogel.
We propose to study a new class of algebras which we call surface algebras.
To a given surface we associate a collection of algebras, the so-called surface algebras.
For our project, we have Riemann surfaces in mind, over the complex numbers, with
In recent research, algebras
arising from surfaces have been studied in various instances.
A class of algebras (which we call boundary algebras) arising from surfaces, with interesting connections to cluster theory, were introduced in Jensen-King-Su and subsequently studied in detail by Baur-King-Marsh.
Our surface algebras provide a generalisation of these algebras.
We will approach our study of surface algebras from a number of different angles.
One is to study algebras associated to a
surface, incorporating the geometric properties of that surface.
A second approach is to closely examine the links of surface algebras to boundary algebras, thereby providing valuable insights into these algebras.
A third approach is to vary the set-up and restrict the generators of the algebras we study.
Symmetric groups and geometric representation
theory, project P 25647, FWF, 2013 - 2016.
Researcher on the project: Dr. Dusko Bogdanic.
The project is set in pure mathematics in the
areas of representation theory of associative algebras
and Lie theory. Representation theory is a branch of
mathematics that studies abstract algebraic structures
by representing their elements as linear transformations of vector spaces.
A representation makes an
abstract algebraic object concrete by describing its
elements by matrices and the algebraic operations
in terms of matrix addition and matrix multiplication.
Representation theory is a powerful tool because it
reduces problems in abstract algebra to problems
in linear algebra, which is well understood.
The algebraic objects that can be represented in such a way
include groups, associative algebras and Lie algebras.
One of the most important classes of associative
algebras are group algebras. Their structure depends
on the structure of the group involved. If the group
is finite, then the group algebra is of finite dimension.
This means that, as a vector space, this group algebra
has finite dimension. Such an algebra can be written
as a product of algebras that cannot be reduced
any further. These indecomposable summands are
The goal of our project is to contribute to the
structure theory of the blocks of group algebras of
symmetric groups with non-abelian defect groups.
Past Grants (Karin Baur & group)
2009-2013: Cluster categories and triangulations, SNSF (prodoc module, 3y + 1y extension)
2011-2013: Marie Curie Intra-European Fellowship with Dr. Dusko Bogdanic,
2007-2011: SNSF Professorship
2003-2004: SNSF, fellowship for prospective researchers
1999-2002: SNSF, Marie Heim Vögtlin fellowship for graduate studies