Interactions between Moduli Spaces, Non-Commutative Algebra, and Deformation Theory.
Lead Research Organisation:
University of Sheffield
Department Name: Mathematics and Statistics
Abstract
Many great successes within mathematics arise from linking between seemingly disjoint fields of research, allowing techniques and insights developed in one area to shine a new light on problems in another. One example of this is the use of non-commutative algebra to study geometry. Combining both algebraic and geometric insight often allows results to be extended to more natural levels of generalisation, breaking out of restrictions imposed by geometric settings and producing interesting algebraic structures from the geometry. This approach has been particularly successful in the study of resolutions of singularities.
An example is provided by minimal resolutions of rational surface singularities having a non-commutative interpretation as reconstruction algebras. Another feature that these minimal resolutions of rational surface singularities possess is that they have a particularly fascinating and beautiful geometric deformation theory, however currently this is not understood from a non-commutative viewpoint. The deformation theory of the reconstruction algebras is expected to be intrinsically linked to the geometric case and so should mirror its interesting features while offering new insights from a non-commutative viewpoint.
This research seeks to understand examples such as this by building a bridge between the geometric and non-commutative deformation theory. This will involve developing techniques to construct deformations of non-commutative algebras and producing methods of recovering geometric deformations from non-commutative ones as moduli spaces. It will also encompass general situations, such as moving outside the setting of smooth varieties, which will generate a wide range of new applications in areas such as the construction of 3-folds in the minimal model program.
An example is provided by minimal resolutions of rational surface singularities having a non-commutative interpretation as reconstruction algebras. Another feature that these minimal resolutions of rational surface singularities possess is that they have a particularly fascinating and beautiful geometric deformation theory, however currently this is not understood from a non-commutative viewpoint. The deformation theory of the reconstruction algebras is expected to be intrinsically linked to the geometric case and so should mirror its interesting features while offering new insights from a non-commutative viewpoint.
This research seeks to understand examples such as this by building a bridge between the geometric and non-commutative deformation theory. This will involve developing techniques to construct deformations of non-commutative algebras and producing methods of recovering geometric deformations from non-commutative ones as moduli spaces. It will also encompass general situations, such as moving outside the setting of smooth varieties, which will generate a wide range of new applications in areas such as the construction of 3-folds in the minimal model program.
Planned Impact
This proposal is intradisciplinary, and it will deepen links between different disciplines within mathematics. It will combine algebraic geometry, non-commutative algebra, and deformation theory, and it will achieve impact by drawing together results and researchers from these different areas. The main way this work will create impact is by the widespread dissemination of the results, aided by speaking at conferences within the UK and internationally, and by collaborative work with the mathematical community worldwide. In order to do this the proposal includes funding to attend conferences internationally and within the UK, and also to make overseas research visits.
People |
ORCID iD |
| Joseph Karmazyn (Principal Investigator / Fellow) |
Publications
Craw A
(2017)
Multigraded linear series and recollement
in Mathematische Zeitschrift
Kalck M
(2018)
Ringel duality for certain strongly quasi-hereditary algebras
in European Journal of Mathematics
Karmazyn J
(2018)
Deformations of algebras defined by tilting bundles
in Journal of Algebra
Karmazyn J
(2017)
Quiver GIT for varieties with tilting bundles
in manuscripta mathematica
Karmazyn J
(2017)
The length classification of threefold flops via noncommutative algebras
Karmazyn J
(2019)
The length classification of threefold flops via noncommutative algebras
in Advances in Mathematics
Karmazyn J
(2018)
Derived categories of singular surfaces
Karmazyn J
(2021)
Derived categories of singular surfaces
in Journal of the European Mathematical Society
| Description | The broad description of the aims of this grant "Interations between Moduli Spaces, Non-Commutative Algebra, and Deformation Theory" was to understand and generalise a range of phenomena in algebraic geometry using techniques arising in noncommutative algebra; two different areas of mathematics. One aim was to develop techniques to: take problems, examples and phenomena in algebraic geometry, translate them into questions in noncommutative algebra, attack them with the techniques available there, and finally use moduli techniques to carry the algebraic results back to geometric ones. This was focused on deformation related phenomena, i.e. more complex/higher dimensional examples that can be built from less complex/lower dimensional examples, and in particular in understanding how deformations of surfaces singularities are used to create and understand higher dimensional examples, and studying moduli constructions. This work of this grant contributed to the development of several techniques in this area, and these results were published in the 4 papers: The length classification of threefold flops via noncommutative algebras, Deformations of Algebras Defined by Tilting Bundles, Quiver GIT for Varieties with Tilting Bundles, and Multigraded linear series and recollement (joint with Alastair Craw and Yukari Ito). As a highlight, one particular problem that was studied in this manner was to understand flops in the dimension 3 minimal model program. The minimal model program classifies geometric spaces into certain families, and builds a recipe to move from any starting space towards a simple representative member of the family. Flops are a construction that occur during this procedure, and describe how to pass from one example to another. It was abstractly understood in algebraic geometry that flops in dimension 3 all arise as certain deformations of dimension 2 examples, and this provides an invariant to classify flops in dimension 3. While it was known that examples realising each of these invariants had to exist, explicit examples had not been constructed for many of them. Using the noncommutative techniques outlined above, one finding of this grant was to find a method - via noncommutative algebra - to construct explicit examples of three dimensional flops that realise all of these invariants. Findings culminating from the techniques developed throughout this work include: building new examples of 3-fold flops that realise all values of the length invariants; positively answering mathematical-physics-motivated conjectures raised by Curto and Morrison that proposed that three-fold flops can be understood via universal objects interpreted as matrix factorisations; and generalising the moduli of quiver representations aspects of the 2D McKay correspondence from the Kleinian singularity case to also include general rational surface singularities. A second idea underlying this grant was that noncommutative algebra offers powerful techniques to generalise geometric results about singularities. In particular, an aim was to extend geometric results known for particular singularities to also encompass more general singularities via re-interpretation as noncommutative algebras. There are two collaborative papers awaiting publication that address problems in this area, and currently one paper published: Ringel duality for certain strongly quasi-hereditary algebras (joint with Martin Kalck). Findings from this area of work include: a generalisation of classical Kn\"{o}rrer periodicity to relate non-Gorenstein cyclic surface singularities to particular finite dimensional algebras, Knorrer invariant algebras; proving structural results about Knorrer invariant algebras; and a method to break down (derived categories of) singular projective surfaces into collections of individual (derived categories of) Knorrer invariant algebras. |
| Exploitation Route | The outputs of this grant will be of interest to academic audiences. The main output of this grant was published in 5 research papers associated to this grant. There are also several further papers being worked on currently, that will be published over the next few years. Throughout the grant talks on the outputs were given to a range of different audiences; 16 invited talks were given during the grant. These were to both geometric and algebraic audiences, and at seminars, workshops, and conferences in both the UK and abroad. As such, the results of the this research have been communicated to the wider research community. It is hoped that the results of this grant will contribute to future research in the mathematical community; indeed, the some earlier results of the grant have already prompted future collaborative work, not yet published, continuing the ideas studied here. |
| Sectors | Other |