Homepage for Laurence Barker

Group representation theorist, Associate Professor in the Department of Mathematics at Bilkent University.



Present lecture courses

Notes on inner product spaces, for both of the courses below: InnerProduct.pdf.

MATH 220: Linear Algebra, Section 1

Course specification: spec220spr24.pdf.

Quizzes and solutions: quiz220spr24.pdf.

Midterm with solutions: mid220spr24.pdf.

Exercises and solutions: Exercises220spr24.pdf.

Archives of some previous incarnations of the course,
including exam questions and solutions:

MATH 224: Linear Algebra 2

Course specification: spec224spr24.pdf.

Notes on vector spaces, with some exercises: LinAlgNotes1VectorSpaces.pdf.

Homeworks with some solutions: homework224spr24.pdf.

Quizzes and solutions: quiz224spr24.pdf.

Midterm with solutions: mid224spr24.pdf.

Archives of some previous incarnations of the course,
including exam questions and solutions:

Recent talks

The Puig category of a block and conjectural isomorphism invariance
of the multiplicities of the blocks
Yeditepe Mathematics Seminar, 9 December, 2022,

What is, and what should be, local structure of symmetry?,
Bilkent University Mathematics Club, 23 December, 2022,

Past lecture courses

The files arch[...].pdf, below, are archives of course materials, including past papers.

Undergraduate courses:

MATH 103: Introductory Mathematics

MATH 104: Thinking Mathematically 2
Course archive: arch104spr19.pdf.

MATH 110: Discrete Mathematics
Course archives: arch110fall14.pdfarch110fall13.pdfarch110fall12.pdf,

MATH 123: Abstract Mathematics 1
Course archive: arch123fall17.pdf.

MATH 124: Abstract Mathematics 2

MATH 132: Discrete and Combinatorial Mathematics
Course archives: arch132spr23.pdfarch132fall22.pdfarch132fall17.pdf,

MATH 210: Finite and Discrete Mathematics
Course archives: arch210spr16.pdfarch210spr15.pdf.

MATH 215: Mathematical Analysis
Course archive: arch215spr12.pdf.
Notes on countability: count215spr12.pdf
Notes on construction of the reals: conreal215spr12.pdf

MATH 220: Linear Algebra
Course archives: arch220fall23.pdfarch220fall11.pdf.

MATH 223: Linear Algebra 1

MATH 224: Linear Algebra 2
Course archives: arch224spr22.pdfarch224spr13.pdf.

MATH 227: Introduction to Linear Algebra
Course archives: arch227spr22.pdfarch227spr19.pdfarch227fall18.pdf,

MATH 313: Real Analysis 2

MATH 323: Algebra 1
Course archives: arch323fall20.pdf,   arch323fall16.pdf,   arch323fall14.pdf.

MATH 324: Algebra 2
Course archives: arch324spr23.pdf,

MATH 325: Representation Theory
Course archives: arch325fall23.pdfarch325spr21.pdfarch325spr18.pdf,

Postgraduate courses:

MATH 500: Mathematical Analysis

MATH 523: Algebra 1
Course archives: arch523fall20.pdf,  arch523fall16.pdf,  arch523fall12.pdf.

MATH 524: Algebra 2
Course archives: arch524spr23.pdfarch524spr15.pdf.

MATH 525: Group Representations
Course archives: arch525fall23.pdfarch525spr21.pdfarch525spr18.pdf,

MATH 527: Topics in Representation Theory
Course archive: arch527spr16.pdf.

MATH 616: Topics in Group Theory

Main research interests

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).

Research students

The theses below can be found at Bilkent PhD and MS theses in mathematics

Present MS students: Mahmut Esat Akın, Mert Akman.

Past PhD students:

Ergün Yaraneri, Inductions, restrictions, evaluations, and subfunctors of
Mackey functors
, 2008.

Olcay Coşkun, A correspondence of simple alcahestic group functors, 2008.

İpek Tuvay, Fusion systems in group representation theory, 2013.

Hatice Mutlu, Canonical induction, Green functors, Lefschetz invariant of
monomial G-posets
, 2019.

İsmail Alperen Öğüt, Deformations of some biset-theretic categories, 2020.

Past MS students:

Ayşe Yaman, On the exponential map of the Burnside ring, 2002.

Ergün Yaraneri, On monomial Burnside rings, 2003.

Olcay Coşkun, Modular representations and monomial Burnside rings, 2004.

Mehmet Uç, Green correspondence for Mackey functors, 2008.

Cihan Okay, The monomial Burnside functor, 2009.

İpek Tuvay, Real monomial Burnside rings and a decomposition of the tom Dieck map, 2009.

Volkan Dağhan Yaylioğlu, Mackey group categories and their simple functors, 2012.

Yasemin Büyükçolak, Canonical induction for trivial source rings, 2013.

Merve Demirel, Simple functors of admissible linear categories, 2013.

İsmail Alperen Öğüt, Biset functors and Brauer induction theorem, 2014.

Elif Doğan Dar, Blocks of quotients of Mackey functors, 2015.

Çisil Karagüzel, On some of the simple functors of the biset functor of p-permutation modules, 2016.

Andi Nika, The pandemic fusion system for the endomorphism algebras of p-permutation modules, 2018.

Utku Okur, Mackey decomposition for Brauer pairs, 2020.

Introduction to Group Theory

The following notes are incomplete and much in need of remedial treatment:

      Part 1, Introduction to sets algebranoteschap1.pdf.
      Part 2, Introductory number theory algebranoteschap2.pdf.
      Part 3, Abelian groups algebranoteschap3.pdf.
      Part 4, Lagrange's Theorem algebranoteschap4.pdf.
      Part 5, Normal subgroups and quotient groups algebranoteschap5.pdf.
      Part 6, The symmetric and alternating groups algebranoteschap6.pdf.
      Part 7, Permutation sets and Sylow's Theorem algebranoteschap7.pdf.

A Discrete Introduction to Conceptual Mathematics

Working draft versions of three chapters:

Chapter 1, Very clear deductive explanation.
Chapter 2, Graph theory. dincom2.pdf
Chapter 3, Mathematical induction and the Euclidian algorithm.
Chapter 4, Enumerative combinatorics and binomial coefficients. dincom4.pdf
Chapter 5, Correspondences and functions.
Chapter 6, Relations, equivalence relations, posets.
Chapter 7, Isomorphism.

Other Undergraduate Notes

Some Undergraduate Notes:

"Discrete" generally:
The courses MATH 110, 132, 210 all have similar
material. Some past exam papers and solutions are
collected in discretepastpapers.pdf.
For a larger collection, see the course archives, above, for MATH 110, 132, 210.

Some very incomplete introductory group theory notes: IntroNotesGroupTheory.pdf.

Note on permutation groups: permutation.pdf

Note on diagonalization and change of basis: diagonalisation2012.pdf

Note on determinants and the symmetric group: determinants.pdf

PhD Qualifying Exam

This exam is part of the PhD Programme in Mathematics at Bilkent.

Exam outline and syllabus: qualsyl.pdf.

Past papers: qualSept22.pdf,
qual08may.pdf qual08jan.pdfqual06nov.pdfqual06aug.pdf.

Notes and Essays

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.

Brief CV

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.

Email: my surname, then the at sign, then fen rabbit bilkent rabbit edu rabbit tr, where rabbit is to be replaced by a dot.

Latest update: 16 April 2024