William (=Bill) Crawley-Boevey

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Main duties for 2013/2014

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Research interests

My research has mainly been on the representation theory of finite-dimensional associative algebras (see fdlist), and related questions in linear algebra, ring and module theory, and algebraic geometry.

A quiver of Dynkin type E6 In recent years I have concentrated on representations of quivers and preprojective algebras. A quiver is essentially the same thing as a directed graph, and a representation associates a vector space to each vertex and a linear map to each arrow. The subject was started by P. Gabriel in 1972, when he discovered that the quivers with only finitely many indecomposable representations are exactly the ADE Dynkin diagrams which occur in Lie theory (for example a quiver of type E6 is illustrated on the left). Quivers and their representations now appear in all sorts of areas of mathematics and physics, including representation theory, cluster algebras, geometry (algebraic, differential, symplectic), noncommutative geometry, quantum groups, string theory, and more.

A Kleinian singularity and some deformations The preprojective algebra associated to a quiver was invented by I. M. Gelfand and V. A. Ponomarev. Its modules are intimately related to representations of the quiver, but it is often the modules for the preprojective algebra which are of relevance in other parts of mathematics. There is beautiful geometry linked to the preprojective algebra, including Kleinian singularities and H. Nakajima's quiver varieties. (The illustration on the right shows the real-valued points of varieties associated to a quiver of extended Dynkin, that is, affine, type D4.)

Paths on the Rieman Sphere There are also links between the preprojective algebra and the classification of differential equations on the Riemann sphere. They are used in work on the Deligne-Simpson problem, which concerns the existence of matrices in prescribed conjugacy classes whose product is the identity matrix, or whose sum is the zero matrix. (The picture on the left shows loops on the punctured Riemann sphere which generate its fundamental group. Consideration of the monodromy around such loops links the classification of differential equations on the Riemann sphere to the Deligne-Simpson problem.)

In earlier times I was interested in tame algebras, matrix problems, and infinite-dimensional modules. Finite dimensional associative algebras naturally divide into three classes: algebras finite representation type with only finitely many indecomposable modules, wild algebras for which the indecomposable modules are unclassifiable (in a suitable sense), and those on the boundary between these classes, the tame algebras. There are many interesting classes of tame algebras, and it is often a major problem to actually give the classification of the indecomposable modules.

A partitioned matrix One way to study tame algebras is to convert the problem of classifying their modules into a matrix problem: putting a partitioned matrix into canonical form using not all elementary operations, but a subset defined by the partition. (The illustration on the right shows what an arbitrary matrix can be reduced to if you allow all row and column operations; it also shows an example of a partition of a matrix.) Using advanced methods based on this idea, Yu. A. Drozd proved his wonderful Tame and Wild Theorem showing that there is a wide gulf between the behaviour of tame and wild algebras. The same methods can be used to show that tame algebras are characterized by the behaviour of their infinite-dimensional modules. In fact, the behaviour of infinite-dimensional modules for tame algebras is extremely interesting, and not at all understood.

List of publications

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  1. Polycyclic-by-finite affine group schemes, Proc. London Math. Soc., 52 (1986), 495-516.
  2. Locally finite representations of groups of finite p-rank, J. London Math. Soc., 34 (1986), 17-25.
  3. (With P. H. Kropholler and P. A. Linnell) Torsion-free soluble groups and the zero-divisor conjecture, J. Pure and Appl. Algebra, 54 (1988), 181-196.
  4. On tame algebras and bocses, Proc. London Math. Soc., 56 (1988), 451-483.
  5. Functorial filtrations and the problem of an idempotent and a square zero matrix, J. London Math. Soc., 38 (1988), 385-402.
  6. Functorial filtrations II: clans and the Gelfand problem, J. London Math. Soc., 40 (1989), 9-30.
  7. Functorial filtrations III: semidihedral algebras, J. London Math. Soc., 40 (1989), 31-39.
  8. Maps between representations of zero-relation algebras, J. Algebra, 126 (1989), 259-263.
  9. (With L. Unger) Dimensions of Auslander-Reiten translates for representation-finite algebras, Comm. Algebra, 17 (1989), 837-842.
  10. Matrix problems and Drozd's theorem, in 'Topics in Algebra', eds S. Balcerzyk et al., Banach Center publications, vol. 26 part 1 (PWN-Polish Scientific Publishers, Warsaw, 1990), 199-222.
  11. Regular modules for tame hereditary algebras, Proc. London Math. Soc., 62 (1991), 490-508.
  12. Tame algebras and generic modules, Proc. London Math. Soc., 63 (1991), 241-265.
  13. Lectures on representation theory and invariant theory, Ergänzungsreihe Sonderforschungsbereich 343 'Diskrete Strukturen in der Mathematik', 90-004, Bielefeld University, 1990, 74pp.
  14. Matrix reductions for artinian rings, and an application to rings of finite representation type, J. Algebra, 157 (1993), 1-25.
  15. (With C. M. Ringel) Algebras whose Auslander-Reiten quiver has a large regular component, J. Algebra, 153 (1992), 494-516.
  16. (With D. Happel and C. M. Ringel) A bypass of an arrow is sectional, Arch. Math. (Basel), 58 (1992), 525-528.
  17. Modules of finite length over their endomorphism rings, in 'Representations of algebras and related topics', eds H. Tachikawa and S. Brenner, London Math. Soc. Lec. Note Series 168, (Cambridge University Press, 1992), 127-184.
  18. Additive functions on locally finitely presented Grothendieck categories, Comm. Algebra, 22 (1994), 1629-1639
  19. Locally finitely presented additive categories, Comm. Algebra, 22 (1994), 1641-1674.
  20. Exceptional sequences of representations of quivers, in 'Representations of algebras', Proc. Ottawa 1992, eds V. Dlab and H. Lenzing, Canadian Math. Soc. Conf. Proc. 14 (Amer. Math. Soc., 1993), 117-124.
  21. (With O. Kerner) A functor between categories of regular modules for wild hereditary algebras, Math. Ann., 298 (1994), 481-487.
  22. (With D. J. Benson) A ramification formula for Poincaré series, and a hyperplane formula for modular invariants, Bull. London Math. Soc., 27 (1995), 435-440.
  23. Subrepresentations of general representations of quivers, Bull. London Math. Soc., 28 (1996), 363-366.
  24. (With R. Vila-Freyer) The structure of biserial algebras, J. London Math. Soc., 57 (1998), 41-54.
  25. Rigid integral representations of quivers, in 'Representations of algebras', Proc. Cocoyoc 1994, eds R. Bautista et al., Canad. Math. Soc. Conf. Proc., 18 (Amer. Math. Soc., 1996), 155-163.
  26. Tameness of biserial algebras, Arch. Math. (Basel), 65 (1995), 399-407.
  27. On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Amer. Math. Soc., 348 (1996), 1909-1919.
  28. (With M. P. Holland) Noncommutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605-635.
  29. Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Canadian Math. Soc. Conf. Proc., 23 (1998), 29-54 (pdf).
  30. Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv., 74 (1999), 548-574 (pdf).
  31. (With R. Bautista, T. Lei and Y. Zhang) On Homogeneous Exact Categories, J. Algebra, 230 (2000), 665-675.
  32. On the exceptional fibres of Kleinian singularities, Amer. J. Math., 122 (2000), 1027-1037 (pdf).
  33. Geometry of the moment map for representations of quivers, Compositio Math., 126 (2001), 257-293 (pdf).
  34. Decomposition of Marsden-Weinstein reductions for representations of quivers, Compositio Math., 130 (2002), 225-239 (math.AG/0007191).
  35. (with Christof Geiß) Horn's problem and semi-stability for quiver representations, in 'Representations of Algebras, Vol 1', Proceedings of the Ninth International Conference, Beijing, August 21-September 1, 2000, eds. D. Happel and Y. B. Zhang (Beijing Normal University Press, 2002), 40-48 (pdf).
  36. (with Jan Schröer) Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201-220 (math.AG/0103100).
  37. Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), 55-79 (math.AG/0105247).
  38. On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J. 118 (2003), 339-352 (math.RA/0103101).
  39. (with Michel Van den Bergh) Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima), Invent. Math. 155 (2004), 537-559. (pdf).
  40. Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity, Publ. Math. Inst. Hautes Etudes Sci. 100 (2004), 171-207. (math.AG/0307246).
  41. (with Peter Shaw) Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math. 201 (2006), 180-208. (math.RA/0404186).
  42. (With Bernt Tore Jensen) A note on sub-bundles of vector bundles, Glasgow Math. J. 48 (2006), 459-462. (math.AG/0505149).
  43. Quiver algebras, weighted projective lines, and the Deligne-Simpson problem, in: 'Proceedings of the International Congress of Mathematicians', vol. 2, Madrid 2006, eds M. Sanz-Solé et al. (European Mathematical Society, January 2007), 117-129. (pdf | math.RA/0604273).
  44. (With Pavel Etingof and Victor Ginzburg) Noncommutative Geometry and Quiver algebras, Adv. Math. 209 (2007), 274-336. (math.AG/0502301 ).
  45. General sheaves over weighted projective lines, Colloq. Math. 113 (2008), 119-149. (pdf).
  46. Kac's Theorem for weighted projective lines, Journal of the European Mathematical Society, 12 (2010), 1331-1345. (math.AG/0512078).
  47. Connections for weighted projective lines, J. Pure Appl. Algebra, 215 (2011), 35-43. (arXiv:0904.3430v1 [math.AG]).
  48. Poisson structures on moduli spaces of representations, J. Algebra 325 (2011), 205-215. (pdf).
  49. Monodromy for systems of vector bundles and multiplicative preprojective algebras, Bulletin of the London Mathematical Society 45 (2013), 309-317 (free access: abstract, pdf).
  50. Kac's Theorem for equipped graphs and for maximal rank representations, Proceedings of the Edinburgh Mathematical Society 56 (2013), 443-447. (pdf).

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