Warwick EPSRC Symposium on Derived Categories and Applications
Lead Research Organisation:
CARDIFF UNIVERSITY
Department Name: Sch of Mathematics
Abstract
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Organisations
People |
ORCID iD |
| Timothy Logvinenko (Principal Investigator) |
Publications
Anno R
(2016)
Orthogonally spherical objects and spherical fibrations
in Advances in Mathematics
Anno R
(2017)
Spherical DG-functors
in The Journal of the European Mathematical Society
Anno Rina
(2016)
Bar category of modules and homotopy adjunction for tensor functors
in arXiv e-prints
Cautis S
(2017)
Derived Reid's recipe for abelian subgroups of SL 3 (C)
in Journal für die reine und angewandte Mathematik (Crelles Journal)
| Description | This grant funded a year-long EPSRC Symposium at Warwick, featuring a large number of schools, workshops, seminars and conferences in the area of Derived Categories and Applications. The main overall organisers were Miles Reid, Timothy Logvinenko, Dmitriy Rumynin, Toby Stafford, but individual events were co-organised by (among others) Arend Bayer, Tom Bridgeland Alexander Efimov, Dmitry Kaledin, Peter Newstead and the VBAC committee, Iain Gordon, Kai Behrend, Alexander Kuznetsov, Mark Gross, Ludmil Katzarkov. The activities ranged from beginner-level graduate student schools on orbifolds and representation theory through to specialist workshops on McKay correspondence and orbifolds, mirror symmetry, stability conditions, to larger, wider scope conferences on birational geometry, on derived algebraic geometry and on geometric representation theory. |
| Exploitation Route | Several hundred people have attended the workshops of the EPSRC Warwick Symposium 2015-16 on Derived Categories and Applications. They've gained a lot of new knowledge and they will put it to use in their own research. |
| Sectors | Education,Other |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/index.html |
| Title | Bar category of modules |
| Description | During the EPSRC Symposium on Derived Categories and Applications 2014-15 funded by this award, my collaborator Rina Anno and I have had a number of discussions with leading authorities on DG-enhancements: Alexander Efimov, Alexei Bondal, Mikhail Kapranov, Valery Lunts, Olaf Schnurer, etc. As result we've developed a new method for DG-enhancing the derived categories of coherent sheaves on algebraic varieties, and more generally the derived category of modules over a small DG-category A. It is called the bar category of modules \modbar(A), and it is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically in the language of DG-categories and requires no complex machinery or sign conventions of A-infinity categories. We defined for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. |
| Type Of Material | Improvements to research infrastructure |
| Year Produced | 2016 |
| Provided To Others? | Yes |
| Impact | Immediately, as a demonstration of the power of our new method we've applied it to write down a theory of homotopy adjunction for tensor functors (R. Anno, T. Logvinenko, "Bar category of modules and homotopy adjunction for tensor functors", 2016, arXiv:1612.09530). We have then applied to write down the notion of non-split P-functors, which was a long-standing open problem (R.Anno, T.Logvinenko, "P-functors", in progress). We are also in process of applying it to write down the notion of DG-enhanced monads and their categories of algebras (S.Arkhipov, T.Logvinenko, "On DG-enhanced monads", in progress). |
| URL | https://arxiv.org/abs/1612.09530 |
| Description | Introductory School on Derived Categories (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | The workshop aimed to highlight a currently thriving area of research, including its links with representation theory, dimer models and symplectic geometry, and to map out new directions of research, such as derived geometry applied to the case of GL(3), SL(4) and higher dimensions. |
| Year(s) Of Engagement Activity | 2015 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/02-mck.html |
| Description | School "Derived categories, Weyl algebras and Hodge theory" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | This school aims to introduce broad audience, including PhD students, early postdocs and specialists in neighbouring areas, to the category of constructible sheaves on an algebraic variety. It will also feature several related topics in D-modules, Hodge theory and algebraic topology, with the ultimate goal of explaining the proof of Beilinson-Bernstein-Deligne decomposition theorem. The school leads up to the "Derived categories, algebra and representation theory" conference the following week. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/05-waho.html |
| Description | Workshop "DG-enhancements and higher category methods" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The conference will focus on recent developments arising from higher category theory, including DG enhancements of derived categories and Kontsevich's program of noncommutative geometry that treats abstract DG categories geometrically. Scope: DG, A_infty and higher categories. Enhancements. Derived Morita theory. DG categories of matrix factorisations. Moduli of objects in a DG category. Noncommutative geometry and noncommutative Hodge theory. Noncommutative motives. |
| Year(s) Of Engagement Activity | 2015 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/03-hcat.html |
| Description | Workshop "Derived Algebraic Geometry, with a focus on derived symplectic techniques" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | During the 2014-15 Warwick EPSRC Symposium on Derived Categories and Applications, there was a two week activity on symplectic techniques in derived algebraic geometry. In the first week there were a series of lectures introducing the topic, accompanied by informal talks, then the conference took place in the second week. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.math.ubc.ca/~behrend/warwick/ |
| Description | Workshop "Derived categories and birational geometry" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The goal of this workshop is to bring together specialists in birational geometry and derived categories to develop and capitalise on some of the advances made in the last decade in these subjects. These include Kawamata's description of Minimal Model Program for derived categories of toric stacks, Van den Berg and Iyama-Wemyss' constructions of noncommutative resolutions of singularities and Kuznetsov-Lunts' construction of a categorical resolution of irrational singularities. Scope: Minimal model program via derived categories. Mori contractions and exceptional objects, flops and spherical objects. Categorical resolution of singularities. Birational invariants via derived categories. Homological projective duality. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/08-birg.html |
| Description | Workshop "Derived categories, algebra and representation theory" |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This conference aims to explores recent advances on classes of algebras whose representation theory mirrors that of simple finite dimensional Lie algebras, starting from rational Cherednik algebras and finite W-algebras. The methods used includes Lie, including D-modules, localisations, quantisations and derived categories. Two recent applications of these ideas are the Alday-Gaiotto- Tachikawa conjecture and the fusion of the geometric Langlands program with string theory. Scope: Simple Lie algebras, W-algebras, Cherednik algebras, symplectic reflection algebras, category O, D-modules, Hilbert schemes, AGT conjecture, quantum cohomology, derived categories. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/06-algrt.html |
| Description | Workshop "Geometry from stability conditions" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The topic of this workshop will be applications of the theory of stability conditions to results in algebraic geometry. Scope: Birational geometry of moduli spaces. Stability conditions on threefolds. Potential connections to the Gross-Siebert programme via wall-crossing and mirror symmetry. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.maths.ed.ac.uk/~abayer/2014-wrsym/04-stability.html |
| Description | Workshop "McKay correspondence, orbifolds, quivers" (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The workshop aims to highlight a currently thriving area of research, including its links with representation theory, dimer models and symplectic geometry, and to map out new directions of research, such as derived geometry applied to the case of GL(3), SL(4) and higher dimensions. Scope: Derived category techniques linking orbifold geometry, quiver representations, dimer models, etc. The derived category of a Deligne-Mumford stack and the McKay correspondence. Quiver varieties. Dimer models. Mutations. Bipartite field theories. |
| Year(s) Of Engagement Activity | 2015 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/02-mck.html |
| Description | Workshop VBAC2015 (EPSRC Symposium 2015-16) |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Fourier-Mukai transforms were introduced in a celebrated paper of Mukai in 1981. His motivation was to obtain new information about certain moduli spaces. Since then, the theory has been massively developed and applied. The VBAC 2015 workshop will review these developments and draw attention to potential new applications. The focus of the workshop will be the interaction between the study of derived categories and more classical problems in algebraic geometry. Topics to be covered include enumerative geometry, the geometry of moduli spaces of sheaves and Bridgeland stability. We shall also look at the relation to Morita theory and to the conjectural interpretation of the geometric Langlands correspondence. |
| Year(s) Of Engagement Activity | 2016 |
| URL | http://www.cantab.net/users/t.logvinenko/2014-wrwsym/09-vbac.html |