Prof. Dr. William (=Bill) CrawleyBoevey
Alexander von Humboldt Professor
Research group: BIREP
EMail: wcrawley
Telephone: +49 (0)521 106 5033
Office: V5221
Here is a link to my former homepage at University of Leeds, UK.
Teaching
I will teach a Masters' sequence starting in April 2019.

Summer Semester 2019. "Noncommutative Algebra 1".
My aim is to cover projective and injective modules, Morita equivalence, homological algebra, and some important examples of algebras, such as free algebras and path algebras, preprojective algebras, etc.

Winter Semester 2019/20. "Noncommutative Algebra 2".
I plan to discuss the representation theory of finite dimensional algebras and quivers,
including AuslanderReiten theory.

Summer Semester 2020. "Noncommutative Algebra 3".
I plan to discuss geometric methods for studying representations of algebras and quivers; constructions such as quiver varieties and character varieties. The course will include an introduction to the type of algebraic geometry needed for these applications.
I taught a similar sequence starting in October 2016. You can find information about the sequence
here,
here and
here.
My new masters sequence will be based on the previous one, but there will be significant changes.
Humboldt Award
The research group is supported by the Alexander von Humboldt Stiftung/Foundation
in the framework of the Alexander von Humboldt Professorship
endowed by the Federal Ministry of Education and Research.
The grant runs for 5 years, from October 2016 until October 2021 (but my position in Bielefeld is permanent).
Research Plan: Representations of Algebras and Classification Problems in Linear Algebra
Summary.
Linear algebra is one of the most successful tools in mathematics, but within linear algebra there are difficult problems,
especially the classification of configurations of matrices, perhaps satisfying algebraic equations,
up to the natural operation of simultaneous conjugacy. In abstract algebra this is formulated in terms of
'associative algebras' and their 'representations', and using this language one can bring to bear modern mathematical tools,
such as category theory and homological algebra. The aim of this project is to make progress in three areas of
the representation theory of algebras: the first area is to improve our understanding of algebras where it is
possible, or should be possible, to classify representations diagrammatically in terms of certain 'strings' and
'bands'; the next area concerns the
algebraicgeometric structures arising from representations of algebras; and the third relates to the
'Deligne Simpson Problem', a problem in linear algebra which arises when one attempts to classify linear
ordinary differential equations in terms of their singular points.
Miscellaneous
Distinctions
Editorial work
Research Students
 Roberto Vila Freyer completed an Oxford D. Phil.
on the topic 'Biserial algebras' in 1994.
 Nicola Richmond completed a Leeds Ph. D. on the topic 'The geometry of
modules over finite dimensional algebras' in 1999.
 Andrew Hubery
completed a Leeds Ph. D. on the topic 'Representations of quivers
respecting a quiver automorphism and a theorem of Kac' in 2002.
 Peter Shaw
completed a Leeds Ph. D. on the topic 'Generalisations of Preprojective algebras'
in 2005. pdf.
 Marcel Wiedemann
completed a Leeds Ph. D. on the topic 'On real root representations of quivers'
in 2008. pdf.
 Daniel Kirk completed a Leeds Ph. D. on the topic
'Representations of Quivers with Applications to Collections of Matrices
with Fixed Similarity Types and Sum Zero' in 2013. pdf.
 Raphael BennettTennenhaus completed a Leeds Ph. D.
on the topic
'Functorial Filtrations for Semiperfect generalisations of Gentle Algebras' in 2017.
pdf.
 Ulrike Hansper started a Bielefeld Ph. D. in September 2016.
 Sebastian Eckert started a Bielefeld Ph. D. in October 2017.
Research Interests
My research has mainly been on the representation theory of finitedimensional associative algebras
(see fdlist),
and related questions in linear algebra, ring and module theory, and algebraic geometry.
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
E_{6} 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.
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 realvalued points of
varieties associated to a quiver of extended Dynkin, that is,
affine, type D_{4}.)
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 DeligneSimpson 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 DeligneSimpson problem.)
In earlier times I was interested in tame algebras, matrix problems, and
infinitedimensional 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.
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 infinitedimensional modules. In fact, the behaviour of
infinitedimensional modules for tame algebras is extremely interesting, and
not at all understood.
Publications
Google
Scholar page  ResearcherID page

Polycyclicbyfinite affine group schemes,
Proc. London Math. Soc., 52 (1986), 495516.
(doi).

Locally finite representations of groups of finite prank,
J. London Math. Soc., 34 (1986), 1725.
(doi).

(With P. H. Kropholler and P. A. Linnell) Torsionfree soluble groups and the zerodivisor conjecture,
J. Pure and Appl. Algebra, 54 (1988), 181196.
(doi).

On tame algebras and bocses,
Proc. London Math. Soc., 56 (1988), 451483.
(doi).

Functorial filtrations and the problem of an idempotent and a square zero matrix,
J. London Math. Soc., 38 (1988), 385402.
(journal).

Functorial filtrations II: clans and the Gelfand problem,
J. London Math. Soc., 40 (1989), 930.
(doi).

Functorial filtrations III: semidihedral algebras,
J. London Math. Soc., 40 (1989), 3139.
(doi).

Maps between representations of zerorelation algebras,
J. Algebra, 126 (1989), 259263.
(doi).

(With L. Unger) Dimensions of AuslanderReiten translates for representationfinite algebras,
Comm. Algebra, 17 (1989), 837842.
(doi).

Matrix problems and Drozd's theorem, in 'Topics in Algebra',
eds S. Balcerzyk et al., Banach Center publications, vol. 26 part 1 (PWNPolish Scientific Publishers, Warsaw, 1990), 199222.
(pdf).

Regular modules for tame hereditary algebras,
Proc. London Math. Soc., 62 (1991), 490508.
(doi).

Tame algebras and generic modules,
Proc. London Math. Soc., 63 (1991), 241265.
(doi).

Lectures on representation theory and invariant theory,
Ergänzungsreihe Sonderforschungsbereich 343 'Diskrete Strukturen in der Mathematik', 90004, Bielefeld University, 1990, 74pp.
(pdf).

Matrix reductions for artinian rings, and an application to rings of finite representation type,
J. Algebra, 157 (1993), 125.
(doi).

(With C. M. Ringel) Algebras whose AuslanderReiten quiver has a large regular component,
J. Algebra, 153 (1992), 494516.
(doi).

(With D. Happel and C. M. Ringel) A bypass of an arrow is sectional,
Arch. Math. (Basel), 58 (1992), 525528.
(doi).

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), 127184.
(pdf).

Additive functions on locally finitely presented Grothendieck categories,
Comm. Algebra, 22 (1994), 16291639
(doi).

Locally finitely presented additive categories,
Comm. Algebra, 22 (1994), 16411674.
(doi).

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), 117124.
(pdf).

(With O. Kerner) A functor between categories of regular modules for wild hereditary algebras,
Math. Ann., 298 (1994), 481487.
(doi,
GDZ).

(With D. J. Benson) A ramification formula for Poincaré series, and a hyperplane formula for modular invariants,
Bull. London Math. Soc., 27 (1995), 435440.
(doi).

Subrepresentations of general representations of quivers,
Bull. London Math. Soc., 28 (1996), 363366.
(doi).

(With R. VilaFreyer) The structure of biserial algebras,
J. London Math. Soc., 57 (1998), 4154.
(doi).

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), 155163.
(pdf).

Tameness of biserial algebras,
Arch. Math. (Basel), 65 (1995), 399407.
(doi).

On homomorphisms from a fixed representation to a general representation of a quiver,
Trans. Amer. Math. Soc., 348 (1996), 19091919.
(doi).

(With M. P. Holland)
Noncommutative deformations of Kleinian singularities,
Duke Math. J., 92 (1998), 605635.
(doi).

Infinitedimensional modules in the representation theory of
finitedimensional algebras,
Canadian Math. Soc. Conf. Proc., 23 (1998), 2954.
(pdf).

Preprojective algebras, differential operators
and a Conze embedding for deformations of
Kleinian singularities,
Comment. Math. Helv., 74 (1999), 548574.
(doi,
pdf).

(With R. Bautista, T. Lei and Y. Zhang)
On Homogeneous Exact Categories,
J. Algebra, 230 (2000), 665675.
(doi).

On the exceptional fibres of Kleinian singularities,
Amer. J. Math., 122 (2000), 10271037.
(doi,
pdf).

Geometry of the moment map for representations of quivers,
Compositio Math., 126 (2001), 257293.
(doi,
pdf).

Decomposition of MarsdenWeinstein reductions for representations of
quivers,
Compositio Math., 130 (2002), 225239.
(doi,
math.AG/0007191).

(with Christof Geiß) Horn's problem and semistability for
quiver representations, in 'Representations of Algebras, Vol 1',
Proceedings of the Ninth International Conference, Beijing,
August 21September 1, 2000, eds. D. Happel and Y. B. Zhang
(Beijing Normal University Press, 2002), 4048.
(pdf).

(with Jan Schröer)
Irreducible components of varieties of modules,
J. Reine Angew. Math. 553 (2002), 201220.
(doi,
math.AG/0103100).

Normality of MarsdenWeinstein reductions for representations
of quivers,
Math. Ann. 325 (2003), 5579.
(doi,
math.AG/0105247).

On matrices in prescribed conjugacy classes with no common invariant
subspace and sum zero,
Duke Math. J. 118 (2003), 339352.
(doi,
math.RA/0103101).

(with Michel Van den Bergh) Absolutely indecomposable representations and
KacMoody Lie algebras (with an appendix by Hiraku Nakajima),
Invent. Math. 155 (2004), 537559.
(doi,
pdf).

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), 171207.
(doi,
NUMDAM).

(with Peter Shaw)
Multiplicative preprojective algebras, middle convolution
and the DeligneSimpson problem, Adv. Math. 201 (2006), 180208.
(doi).

(With Bernt Tore Jensen)
A note on subbundles of vector bundles, Glasgow Math. J. 48 (2006), 459462.
(doi,
pdf).

Quiver algebras, weighted projective lines, and the DeligneSimpson
problem, in: 'Proceedings of the International Congress of Mathematicians', vol. 2,
Madrid 2006, eds M. SanzSolé et al.
(European Mathematical Society, January 2007), 117129.
(pdf).

(With Pavel Etingof and Victor Ginzburg)
Noncommutative Geometry and Quiver algebras, Adv. Math. 209 (2007), 274336.
(doi).

General sheaves over weighted projective lines,
Colloq. Math. 113 (2008), 119149.
(doi,
EUDML).

Kac's Theorem for weighted projective lines,
Journal of the European Mathematical Society, 12 (2010), 13311345.
(doi, math.AG/0512078).

Connections for weighted projective lines,
J. Pure Appl. Algebra, 215 (2011), 3543.
(doi).

Poisson structures on moduli spaces of representations,
J. Algebra 325 (2011), 205215
(doi).

Monodromy for systems of vector bundles and multiplicative preprojective algebras,
Bulletin of the London Mathematical Society 45 (2013), 309317.
(doi)

Kac's Theorem for equipped graphs and for maximal rank representations,
Proceedings of the Edinburgh Mathematical Society 56 (2013), 443447.
(doi,
pdf).

Decomposition of pointwise finitedimensional persistence modules,
Journal of Algebra and Its Applications Vol. 14, No. 5 (2015), 1550066 (8 pages).
(doi,
arXiv:1210.0819 [math.RT]).

(With Frédéric Chazal and Vin de Silva)
The observable structure of persistence modules,
Homology, Homotopy and Applications 18 (2016), 247265.
(doi, pdf)

(With Julia Sauter)
On quiver Grassmannians and orbit closures for representationfinite algebras,
Mathematische Zeitschrift 285 (2017), 367–395.
(open access doi)

Classification of modules for infinitedimensional string algebras,
Transactions of the American Mathematical Society,
Trans. Amer. Math. Soc. 370 (2018), 32893313.
(doi, pdf)

Representations of equipped graphs: AuslanderReiten theory,
Proceedings of the 50th Symposium on Ring Theory and Representation Theory (Yamanashi, Japan, 2017).
(pdf,
entire proceedings)

(With Raphael BennettTennenhaus)
Σpureinjective modules for string algebras and linear relations,
Journal of Algebra 513 (2018), 177189.
(share link provided by Elsevier,
doi,
pdf.)
Preprints
ArXiv author page
Archived materials
 Lectures on representation theory and invariant theory
(pdf).
A graduate course given in 1989/90 at Bielefeld University.
This is an introduction to the representation theory of the symmetric
and general linear groups (in characteristic zero), and to classical
invariant theory.
List of corrections.
 Lectures on representations of quivers
(pdf 
scanned pdf  includes one extra diagram).
A graduate course given in 1992 at Oxford University.
This is an introduction to the representation
theory of quivers, and in particular the representation theory of extended
Dynkin quivers.
List of corrections.
 More lectures on representations of quivers
(scanned pdf).
Another graduate course from 1992 at Oxford University.
More about representations of quivers, including AuslanderReiten
theory, results of Kerner and of Schofield.
 Geometry of representations of algebras
(pdf).
A graduate course given in 1993 at Oxford University.
This is a survey of how algebraic
geometry has been used to study representations of algebras (and quivers
in particular).
 Cohomology and central simple algebras
(pdf).
An MSc course given in 1996 at Leeds University.
An introduction to homological algebra
and applications to central simple algebras.
 Representations of quivers, preprojective algebras and
deformations of quotient singularities
(pdf).
Lectures from a DMV Seminar in May 1999 on "Quantizations
of Kleinian singularities" organized by R. Buchweitz, P. Slodowy
and myself at Oberwolfach.
Here is the group photo from the meeting.
(There are mistakes in Lemma 4.5 and the proof of Theorem 5.9.
The first of these has been sorted out independently by P. Etingof
and V. Ginzburg in math.AG/0011114.
For the second, the proof elsewhere in the literature is correct.)
 The website
from a Summer School on "Geometry of Quiverrepresentations and Preprojective
Algebras" (Isle of Thorns/UK, September 10  17, 2000)
 An unfinished and abandoned paper
on generic deformed preprojective algebras
(pdf),
dating from the late 1990s,
and mentioned by P. Etingof and E. Rains in
math.RT/0503393.
 The website
for a conference in honour of John McConnell and Chris Robson
(Leeds, May 56, 2006)
 The slides of a talk in Glasgow in November 2014 on
"Two applications of the functorial filtration method".
Recent and forthcoming meetings

International AsiaLink conference on Algebras and Representations
(Beijing Normal University, May 2328, 2005)

Journées Solstice d'été 2005 : Groupes
(Paris, June 2325, 2005)

Interactions
between noncommutative algebra and algebraic geometry (Banff,
September 1015, 2005)

Workshop
in NonCommutative Geometry (Copenhagen, November 710, 2005)

Sklyanin Algebras and Beyond
(Leeds, December 1617, 2005)

Ring Theory: recent progress and applications
(Leeds, May 56, 2006)

Interactions
between Algebraic Geometry and Noncommutative Algebra
(Oberwolfach, May 713, 2006)

Workshop
on algebraic vector bundles (Münster, June 2630, 2006)

Workshop
on Triangulated Categories (Leeds, August 1319, 2006)

International Congress of Mathematicians
(Madrid, August 2230, 2006)

Representations
of Quivers, Singularities and Lie Theory
(Beijing, September 1317, 2006)

Representations
of Algebras and their Geometry (Paderborn, November 1011, 2006)

Recent
developments in the theory of Hall algebras (CIRM, Luminy, France, November 2024, 2006)

Trends in Noncommutative Geometry
(Newton Institute, Cambridge, December 1822, 2006)

Perspectives in AuslanderReiten Theory.
On the occasion of the 65th birthday of Idun Reiten
(Trondheim, May 1012, 2007)

Arithmetic
harmonic analysis on character and quiver varieties
(American Institute of Mathematics, Palo Alto, June 48, 2007)

XII International Conference on Representations of Algebras and Workshop
(Torun, August 1524, 2007)

Representation
Theory of Finite Dimensional Algebras (Oberwolfach, February 1723, 2008)

Maurice
Auslander Distinguished Lectures and International Conference (Woods
Hole, Cape Cod, April 2527, 2008)

Symmetries in Mathematics and
Physics, in honor of Victor Kac (Cortona, June 2228, 2008)

XIII International
Conference on Representations of Algebras and Workshop
(Sao Paulo, July 30August 8, 2008)

Minisymposium on Algebras
(Uppsala, February 20, 2009)

Combinatorial
Geometric Structures in Representation Theory
(Durham, July 616, 2009)

Summer school on Geometry of representations
(Cologne, July 2631, 2009)

Workshop on Noncommutative Algebraic Geometry and Related Topics
(Manchester, August 37, 2009)

Quiver
varieties, DonaldsonThomas invariants and instantons
(CIRM, Luminy, September 1418, 2009)

Interplay
between representation theory and geometry (Beijing, May 37, 2010)

Interactions
between Algebraic Geometry and Noncommutative Algebra (Oberwolfach, May 915, 2010)

XIV International Conference on Representations of Algebras and Workshop
(Tokyo, August 615, 2010)

Representation
Theory of Quivers and Finite Dimensional Algebras (Oberwolfach, February 2026, 2011)

New developments
in noncommutative algebra and its applications (Isle of Skye, June 26July 2, 2011)

Cluster
categories and cluster tilting. A conference honoring Idun
Reiten on the occasion of her 70th birthday (Trondheim, March 2830, 2012)

Representation
Theory and Geometry (Zurich, April 1014, 2012)

Workshop
and International Conference on Representations of Algebras (ICRA 2012) (Bielefeld, August 817, 2012)

Introductory Workshop: Noncommutative Algebraic Geometry and Representation Theory
(MSRI, Berkeley, January 28February 1, 2013)

Representation Theory, Homological Algebra, and Free Resolutions
(MSRI, Berkeley, February 1117, 2013)

Interactions between Noncommutative Algebra, Representation Theory, and Algebraic Geometry
(MSRI, Berkeley, April 812, 2013)

Representation Theory of Quivers and Finite Dimensional Algebras
(Oberwolfach, February 1622, 2014)

Interactions between Algebraic Geometry and Noncommutative Algebra
(Oberwolfach, May 1824, 2014)

XVI International Conference on Representations of Algebras (ICRA 2014)
(Sanya, Hainan Province, China, August 2029, 2014)

Representation Theory (MittagLeffler Institute, Sweden, Spring 2015)

Summer School on Koszul Duality (Bad Driburg, Germany, August 1014, 2015)

Derived structures in geometry and representation theory (Oxford, August 31 September 4, 2015)

Infinitedimensional Representations of Finitedimensional Algebras (Manchester, September 917, 2015)

Representations of Algebraic Groups, in honour of Stephen Donkin (York, July, 1315, 2016)

XVII International Conference on Representations of Algebras (ICRA 2016) (Syracuse, USA, August 1019, 2016)

Representation Theory of Quivers and Finite Dimensional Algebras (Oberwolfach, February 1925, 2017)

Triangulated categories and geometry  a conference in honour of Amnon Neeman
(Bielefeld, May 1519, 2017)

BIREP Summer School on Gentle Algebras
(Bad Driburg, Germany, August 14–18, 2017)

Noncommutative Algebraic Geometry and Related Topics
(RIMS, Kyoto, Japan, September 2529, 2017)

50th Symposium on Ring Theory and Representation Theory (Yamanashi, Japan, October 710, 2017)
[another link]

Workshop "Discrete Categories in Representation Theory" (Bielefeld, April 2021, 2018)

Maurice Auslander Distinguished Lectures and International Conference (Woods Hole, Massachusetts, USA,
April 25  30, 2018)

Interactions between Algebraic Geometry and Noncommutative Algebra
(Oberwolfach, May 27June 2, 2018)

XVIII International Conference on Representations of Algebras (ICRA 2018)
(Charles University, Prague, Czech Republic, August 817, 2018)

Interactions
between commutative algebra, representation theory, and algebraic geometry;
A conference in memoriam RagnarOlaf Buchweitz
(Münster, March 1923, 2019)