This project runs from May 2017 to April 2022, and is funded by the UK Engineering and Physical Sciences Research Council.
Principal Investigator: Sibylle Schroll
Research Associate 1: Hipolito Treffinger
Research Associate 2: Haibo Jin (starting 1 February 2021)
PhD student: Nick Williams (October 2018 - )
Project related working seminar Friday 10am click here for schedule
Project Visitors:
Aaron Chan, Nagoya, Japan (November 2019)
Elsa Fernandez, Puerto Madeiro, Argentina, (October 2019)
Sonia Trepode, Mar del Plata, Argentina (October 2019)
Marco Armenta, Guanajuato, Mexico (June 2019)
Javad Asadollahi, Ishfahan, Iran (June 2019)
Eduardo Marcos, Sao Paolo, Brazil, (February 2019)
Goran Malic, Boston, USA (January 2019)
Andrea Solotar, Buenos Aires, (January 2019)
Lleonard Rubio y Degrassi, Stuttgart, Germany (October 2018 - February 2019)
Jenny August, Glasgow (November 2018)
Claude Cibils, Montpellier, France, (October 2018)
Cristian Chaparro Acosta, Buenos Aires, Argentina (October 2018)
Goran Malic, Manchester, UK (May-June 2018)
Andrea Solotar (April 2018)
Yanki Leikili, (May 2018)
David Jordan, Edinburgh, UK (January 2018)
Joe Grant, East Anglia (December 2017)
Kaveh Moussand, Montreal, Canada (December 2017)
Lleonard Rubio y Degrassi, City, London (November 2017)
Andrea Solotar, Buenos Aires, Argentina (November 2017)
Jan Geuenich, Bielefeld, Germany (September 2017)
Karin Baur, Graz, Austria (June 2017, August 2017)
Joseph Kung, North Texas, USA (July 2017)
Ilke Canakci, Durham, UK (July 2017)
Ed Green, Virginia Tech, USA (June 2017)
Workshops partially supported by the project:
Workshop on the Interactions of Representation Theory and Mirror Symmetry, 6-10 May 2019, Leicester, organised by Pierre-Guy Plamondon and Sibylle Schroll.
LMS Regional Meeting and workshop, Galois covers, Grothendieck-Teichmüller theory and dessins d'enfants, 4-7 June 2018, Leicester, organized by Frank Neumann and Sibylle Schroll.
Publications:
On higher torsion classes, (with J. Asadollahi, P. Jorgensen and H. Treffinger), arXiv:arXiv:2101.01402, pdf.
A geometric realization of silting theory for gentle algebras, (with Wen Chang), arXiv:2012.12663, pdf.
Grassmannian categories of infinite rank, (with J. August, M.-W. Cheung, E. Faber and S. Gratz) arXiv:2007.14224, pdf.
Derived categories of skew-gentle algebras and orbifolds, (with D. Labardini-Fragoso and Y. Valdivieso), arXiv:2006.05836, pdf.
A tau-tilting approach to the first Brauer-Thrall conjecture, (with H. Treffinger), arXiv:2004.14221, pdf.
On band modules and \tau-tilting finiteness, (with H. Treffinger and Y. Valdivieso), arXiv:1911.09021, pdf.
Dessins d'enfants and Brauer configuration algebras, (with G. Malic), arXiv: 1908.05509, pdf.
Higher extensions for gentle algebras, (with K. Baur), preprint, arXiv:1906.0527, pdf.
A complete derived invariant for gentle algebras via winding numbers and Arf invariants, (with C. Amiot, P.-G. Plamondon), preprint, arXiv:1904.02555, pdf.
The first Hochschild (co)homology when adding arrows to a bound quiver algebra, (with C. Cibils, M. Lanzilotta, E. Marcos and A. Solotar), preprint, arXiv: 1904.03565, pdf.
The first Hochschild cohomology as a Lie algebra, (with L. Rubio y Degrassi and A. Solotar), preprint, arXiv:1903.12145, pdf.
Dessins d'enfants, Brauer graph algebras and Galois invariants, (with G. Marlic), preprint, arXiv: 1902.09876, pdf.
Lattice bijections for string modules, snake graphs and the weak Bruhat order (with I. Canakci), preprint, arXiv: 1811.06064, pdf.
On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras (with C Chaparro and A. Solotar), arXiv:1811.02211, pdf.
A geometric model of the derived category of a gentle algebra (with P.-G. Plamondon and S. Opper), arXiv:1801:09659, pdf.
On quasi-hereditary algebras, (with E. L. Green), arXiv:1710.06674, pdf.
Algebras and Varieties, (with E. L. Green and L. Hille), arXiv:1707.07877, pdf.
On extensions for gentle algebras, (with I. Canakci and D. Pauksztello), arXiv: 1707.06934, pdf.
On the representation dimension and finitistic dimension of special multiserial algebras, to appear in Proceedings of the AMS, arXiv:1704.00612, pdf.
Stratifying systems through τ-Tilting theory (with Octavio Mendoza). arXiv.
An algebraic approach to Harder-Narasimhan filtrations. arXiv.
Stability Conditions and Maximal Green Sequences in Abelian Categories (with Thomas Brüstle and David Smith). arXiv.
Wall and Chamber Structure for finite-dimensional Algebras (with Thomas Brüstle and David Smith). Advances in Mathematics, Volume 354, 1 October 2019, 106746. arXiv.
On sign coherence of c-vectors. Journal of Pure and Applied Algebra, Volume 223, Issue 6, June 2019, Pages 2382-2400. arXiv
Project summary (for non-specialists):
This intra-disciplinary proposal links algebra, combinatorics and number theory through the introduction of new geometric and combinatorial structures. These fields lie at the cutting edge of modern mathematics research and promise potential benefits in applications ranging from theoretical physics to computer science and optimization problems in a wide variety of contexts such as logistics, economics and machine learning.
By introducing special classes of graphs and their generalizations, such as ribbon graphs and hypergraphs, to algebras and their representations, I recently established an exciting link between geometry, combinatorics and non-commutative algebra. This proposal builds and expands upon this new knowledge. More precisely, through novel ideas it will introduce combinatorial objects, the so-called matroids, to catalyse the study of one of the most ubiquitous classes of algebras: wild algebras. Matroids and the hypergraphs that give rise to them are generalizations of graphs that find applications in optimization problems, image clustering and artificial intelligence.
Non-commutative algebra and representation theory in particular is the study of symmetries through the action of collections of linear transformations on vector spaces. A (finite) group is a collection of linear transformations that are invertible. Algebras are more general in that they also model non-invertible processes. Algebras can be divided into two classes: tame and wild. Tame algebras generally have a well-behaved representation theory and the majority of the work in representation theory to date has been devoted to their study. In contrast, there are currently very few tools available to study wild algebras and their representation theory. At the same time, most naturally occurring algebras are wild.
The proposed research introduces new tools for wild algebras in the form of geometric surface models based on novel applications of combinatorial structures including hypergraphs and matroids. Geometry is concerned with the configurations and spatial relations of geometric objects such as points, lines and circles. In modern geometry, such basic geometric objects and their arrangements in space encode complicated structures whose origins arise, for example, from models of the physical world such as string theory, a mathematical model describing the fundamental forces in nature and all forms of matter.
The most basic objects in number theory after integers are fractions of integers, also known as rational numbers. An important open question in number theory is the characterisation of the action of the absolute Galois group, a group based on the rational numbers, on a set of graphs introduced by Grothendieck, called dessins d'enfants. The central related open problem is to find invariants characterizing this action. This research aims to generate new such invariants through the application of the connections of algebra and combinatorics established in the proposed research.