Homepage for Laurence Barker
Group representation theorist, Associate Professor in the Department of Mathematics at Bilkent University.
Index:
Postgraduate courses:
MATH 500: Mathematical AnalysisMATH 523: Algebra 1
Course archives: arch523fall20.pdf,
arch523fall16.pdf,
arch523fall12.pdf.
MATH 524: Algebra 2
Course archives: arch524spr23.pdf,
arch524spr15.pdf.
MATH 525: Group Representations
Course archives: arch525fall23.pdf,
arch525spr21.pdf,
arch525spr18.pdf,
arch525spr12.pdf.
MATH 527: Topics in Representation Theory
Course archive: arch527spr16.pdf.
MATH 616: Topics in Group Theory
Finite symmetries, which is to say, finite group theory, with a particular interest in p-local representation theory of finite groups.
A group is what we get when we express the symmetries of a thing, and then discard the thing whose symmetries have been expressed. For example the group S4 can be viewed as a the rotational symmetries of a cube, without the cube. The first theorem in p-local group theory, Sylow's Theorem, 1872, expresses how, given a prime p and a finite group G then, modulo dull objections, G has plenty of subgroups whose orders are powers of p, furthermore, G nicely expresses some symmetry of the creature arising from the way those subgroups fit together. (Thus, the group is still expressing symmetry, but now only of something constructed from itself.)
One of the aims of p-local group theory is to find some animal such that the p-local properties of G, whatever that might turn out to mean, would be exactly the features determined by the animal. Then we could understand the animal to be nothing more nor less than the p-local structure of G.
In the representation theory of groups, we study groups by looking at how they express symmetries of vector spaces and other linear things. (Thus, to some extent, we decide that it was not such a good idea, after all, to try to study groups without letting them express symmetries of things beyond themselves.) In p-local representation theory of finite groups, we get an angle on the p-local structure of G by taking the linear things to have some kind of p-local structure of their own. For instance, the linear thing might be a vector space over a field of characteristic p.
Pointed fusion systems of blocks.
PointedFusion.pdf, also
http://arxiv.org/abs/2305.05446
The pointed p-groups on a block algebra.
PointedGroups.pdf, also
http://arxiv.org/abs/2303.09159
with İsmail Alperen Öğüt, Semisimplicity of some deformations
of the subgroup
category and the biset category, Journal of Pure and Applied Algebra,
(to appear).
deformation.pdf, also
http://arxiv.org/abs/2001.02608
with Matthew Gelvin, Conjectural invariance with
respect to the fusion system of an
almost-source algebra, Journal of Group Theory, 25, 973-995 (2022).
ConjAlmostSource29Dec20.pdf,
also http://arxiv.org/abs/2103.02426
with İsmail Alperen Öğüt, Some deformations of the fibred
biset category.
Turkish Journal of Mathematics, 44, 2062-2072 (2020).
deformfibred.pdf, also
http://arxiv.org/abs/2001.05953
An inversion formula for the primitive
idempotents of the trivial source algebra,
Journal of Pure and Applied Algebra 223, 5444-5454 (2019).
primitive.pdf, also
https://arxiv.org/abs/1809.10984
with Hatice Mutlu, A new canonical induction formula for
p-permutation modules,
Comptes Rendus Mathematique, 357, 327-332 (2019).
canonicalInduction.pdf, also
https://arxiv.org/abs/1811.02877
A general approach to Green functors using bisets,
Communications in Algebra 44 (12), 5351-5375 (2016).
generalgreenrev.pdf
with Merve Demirel, Simple functors of admissible linear categories,
Algebras and Representation Theory 19, 463-472 (2016).
admissibleRevised.pdf
with İpek Tuvay, A refinement of Alperin's conjecture for blocks
of the endomorphism algebra
of the Sylow permutation module,
Archiv der Mathematik 106, 15-20 (2016).
refinementAlperinFinal
Blocks of Mackey categories,
Journal of Algebra 446, 34-57 (2016).
blocksmackeyrev.pdf
Tornehave morphisms III: the reduced Tornehave morphism and the
Burnside unit functor,
Journal of Algebra 446, 19-33 (2016).
torne3rev.pdf
with İpek Tuvay, Real representation spheres and the real monomial
Burnside ring,
J. Algebra 353, 79-92 (2012). realrevised.pdf
Tornehave morphisms I: resurrecting the virtual permutation sets
annihilated by linearization,
Communications in Algebra 39, 355-395 (2011).
tornehave1rev.pdf
Tornehave morphisms II: the lifted Tornehave morphism and the dual
of the Burnside functor,
J. Pure and Applied Algebra 214, 1759-1777 (2010).
tornehave2rev.pdf
Rhetorical biset functors, rational p-biset functors and their semisimplicity
in characteristic zero,
J. Algebra 319, 3810-3853 (2008).
semisim_preprint.pdf
Genotypes of irreducible representations of finite p-groups,
J. Algebra, 306, 655-681 (2007).
genetic_preprint.pdf
Fibred permutation sets and the idempotents and units of monomial Burnside rings,
J. Algebra 281, 535-566 (2004).
with Ergün Yalçın, A new notion of rank for finite
supersolvable groups and free linear actions
on products of spheres,
J. Group Theory 6, 347-364 (2003).
Continuum quantum systems as limits of discrete quantum systems. IV.
Affine Canonical Transforms,
J. Math. Phys. 44, 1535-1553 (2003).
Continuum quantum systems as limits of discrete quantum systems. III. Operators,
J. Math. Phys. 42 (10), 4652-4668 (2001) .
Continuum quantum systems as limits of discrete quantum systems, II: state functions,
J. Phys. A: Math. Gen. 22, 4673-4682 (2001).
Continuum quantum systems as limits of discrete quantum systems. I: state vectors,
J. Functional Analysis 186, 153-166 (2001).
with Çağatay Candan, Tuğrul Hakioğlu,
M. Alper Kutay, Haldun Özaktaş,
The discrete harmonic
oscillator, Harper's equation, and the discrete fractional Fourier transform,
J. Phys. A: Math. Gen. 33, 2209-2222 (2000).
The discrete fractional Fourier transform and Harper's equation,
Mathematika (London) 47, 281-297 (2000).
Local representation theory and Mobius inversion,
Comm. Alg. 27 (7), 3377-3399 (1999).
On the contractibility of the orbit space of a G-poset of Brauer pairs,
J. Alg. 212, 460-465 (1999).
The dimension of a primitive interior G-algebra,
Glasgow Math. J. 41, 151-155 (1999).
Counting positive defect irreducible characters of a finite group,
New Zealand J. Math. 27, 167-176 (1998).
Alperin's fusion theorem and G-posets,
J. Group Theory 1, 357-369 (1998).
Defects of irreducible characters of p-solvable groups,
J. Alg. 202, 178-184 (1998).
The number of blocks with a given defect group,
Mathematika (London) 44, 368-373 (1997).
On p-soluble groups and the number of simple modules associated with a given Brauer pair,
Quart. J. Math. (Oxford) (Ser. 2) 48, 133-160 (1997).
Mobius inversion and the Lefschetz invariants of some p-subgroup complexes,
Comm. Alg. 24 (8), 2755-2769 (1996).
G-algebras, Clifford theory, and the Green correspondence,
J. Alg. 172, 335-353 (1995).
Modules with simple multiplicity modules,
J. Alg. 172, 152-158 (1995).
Induction, restriction and G-algebras,
Comm. Alg. 22 (15), 6349-6383 (1994).
Blocks of endomorphism algebras,
J. Alg. 168, 728-740 (1994).
Present MS students: Mahmut Esat Akın, Mert Akman.
Past PhD students:
Ergün Yaraneri, Inductions, restrictions, evaluations, and subfunctors ofPast MS students:
Ayşe Yaman, On the exponential map of the Burnside ring, 2002.This exam is part of the PhD Programme in Mathematics at Bilkent.
Exam outline and syllabus: qualsyl.pdf.
Past papers: qualSept22.pdf,
qualMay22.pdf,
qualMay21.pdf,
qualJan21.pdf,
qual18oct.pdf,
qual16june.pdf,
qual16jan.pdf,
qual15july.pdf,
qual14june.pdf,
qual14jan.pdf,
qual13sept.pdf,
qual12july.pdf,
qual11sept.pdf,
qual10june.pdf,
qual10jan.pdf,
qual09june.pdf,
qual09jan.pdf,
qual08may.pdf,
qual08jan.pdf,
qual06nov.pdf,
qual06aug.pdf.
Reproduction or adaptation without acknowledgement is plagiarism.
On why (-3)² = 9, OnMinusThreeSquared.pdf, 2023.
Lecture course on "Functorial Methods in Representation Theory",
Summer School, 7-18 August 2017 Nesin Mathematics Village, Serince.
Block Fusion Systems
BlockFusionSystems.pdf, 2017.
Some mathematics useful for understanding Plato, What a group is, according to Plato, groupplato.pdf, 2014.
A recreational introductory talk, Eudoxus: the origin of reasoning by creation and subtraction? creatsubtract.pdf, 2013.
A recreational colloquium paper, Why are Aristotle and Euclid so Modernistic? modern.pdf, 2007.
An appendix The duplication of the Square in Plato's Meno, commissioned for an ill-fated book proposal, menojul.pdf, 2006.
1983-87: Mathematical Tripos, Cambridge, UK;.
1987-92, D.Phil. in Mathematics, Oxford University, UK.
1992-93, Postdoc at Ecole Normale Superieure, Paris, France.
1993, Postdoc at Universitat Augsburg, Germany.
1994-96: Postdoc at University of Wales, Cardiff, UK.
1996-date: Department of Mathematics, Bilkent University.
1997-98: Fellowship at Mathematisches Institut, Friedrich Schiller Universitat, Jena, Germany.
2010-11: Sabbatical at Department of Mathematics, University of California, Santa Cruz.
2019-20: Sabbatical at Department of Mathematics, City, University of London.
Addresses and numbers.
Postal Address: Bilkent University, Department of Mathematics, Bilkent, Ankara, 06800 Turkey.
Office: 129 Fen A. (In the crescent-shaped building overlooking the circular fountain.)
Phone: +90 312 290 2120.
Fax: +90 312 290 5097.
Latest update: 16 April 2024