**Homepage for Laurence Barker**

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

**Index:**

- Present lecture courses
- Recent talks
- Past lecture courses
- Main research interests
- Publications
- Research students
- An Introduction to Discrete Mathematics
- PhD Qualifying Exam
- Notes and essays
- Brief CV
- Addresses and numbers

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:

arch220fall23.pdf, arch220fall11.pdf.

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:

arch224spr22.pdf, arch224spr13.pdf.

of the multiplicities of the blocks

Yeditepe Mathematics Seminar, 9 December, 2022,

IsomInvarianceMultBeam22.pdf

Bilkent University Mathematics Club, 23 December, 2022,

ShouldLocalStructure22.pdf

Course archive: arch104spr19.pdf.

Course archives: arch110fall14.pdf, arch110fall13.pdf, arch110fall12.pdf,

arch110spr09.pdf, arch110spr08.pdf, arch110spr07.pdf.

Course archive: arch123fall17.pdf.

Course archives: arch132spr23.pdf, arch132fall22.pdf, arch132fall17.pdf,

arch132fall15.pdf, arch132spr14.pdf.

Course archives: arch210spr16.pdf, arch210spr15.pdf.

Course archive: arch215spr12.pdf.

Notes on countability: count215spr12.pdf

Notes on construction of the reals: conreal215spr12.pdf

Course archives: arch220fall23.pdf, arch220fall11.pdf.

Course archives: arch224spr22.pdf, arch224spr13.pdf.

Course archives: arch227spr22.pdf, arch227spr19.pdf, arch227fall18.pdf,

arch227spr17.pdf.

Course archives: arch323fall20.pdf, arch323fall16.pdf, arch323fall14.pdf.

Course archives: arch324spr23.pdf,

Course archives: arch325fall23.pdf, arch325spr21.pdf, arch325spr18.pdf,

arch325spr13.pdf.

**Postgraduate courses:**

**MATH 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:**

Mackey functors

Olcay Coşkun,

İpek Tuvay,

Hatice Mutlu,

monomial G-posets

İsmail Alperen Öğüt,

**Past MS students:**

Ergün Yaraneri,

Olcay Coşkun,

Mehmet Uç,

Cihan Okay,

İpek Tuvay,

Volkan Dağhan Yaylioğlu,

Yasemin Büyükçolak,

Merve Demirel,

İsmail Alperen Öğüt,

Elif Doğan Dar,

Çisil Karagüzel,

Andi Nika,

Utku Okur,

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.

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.

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

Note on

Note on

Note on

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.

**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