BÌLKENT ALGEBRA DAY
March 24, 2001 (Saturday)

There will be a one day conference at Bilkent University on Saturday, March 24, 2001 on the
subject of Algebra covering various areas such as Algebraic Geometry, Group Theory,
Representation Theory and Module Theory. All talks will take place in the Math Department
seminar room located in the A-Blok of Science Building (SA-141.) Talks will start at 9:00 in
the morning. Tea, coffee and cookies will be available before the first talk.

PROGRAM

8:30  -   9:00   Tea and Coffee

9:00  -   9:30   Feza Arslan
Title: Open problems related to Hilbert functions
9:45 - 10:30   Selma Altinok
Title: Splines and their applications in R^3
10:45 - 11:30   Meral Tosun
Title: Curves on normal surfaces
11:45 -12:15    Bulent A. Ekin
Title: Some properties of partitions in terms of crank.

12:15 -  1:30    Lunch Break

1:30 -  2:00    Ali Osman Asar
Title:  Locally Nilpotent Minimal Non-P Groups
2:00 -  2:30    Mahmut Kuzucuoglu
Title:  Random walk around centralizers in locally finite simple groups
Title:  ES-modules
Title: Neat Submodules

3:45 -  4:15    Tea and Coffee Break

4:15 -  4:45    Li Huishi
Title:  Some Computational Problems in Quadric Solvable Polynomial Algebras
4:45 -  5:15     Laurance Barker
Title: Limits of group representations

ABSTRACTS

Speaker:   Feza Arslan
Title: Open Problems related to Hilbert functions.

Abstract:  Hilbert function is not only a very useful tool for investigating
geometric and algebraic properties, but also it has many interesting
applications. In this talk, we first give a short history of the
problems related with Hilbert functions and then discuss some open
problems.

Speaker:   Selma Altinok
Title:   Splines and their applications in R3.

Abstract:   In this seminar we mainly discuss splines on a three dimensional
polyhedral complex  D in R3 and the dimensions of the vector spaces of
splines on  D of degree less than or equal to k for specific examples either
by using Gröbner basis or by directly doing homology and dimension
calculations on a chain complex corresponding to D.

Speaker:    Meral Tosun
Title:    Curves on normal surfaces

Abstract:   A surface singularity is a pair (S,O) consisting of the spectrum
S=SpecR of a Noetherian two dimensional local ring R containing an algebraically
closed field isomorphic to the residue field  R/m of R with maximal
ideal m, and the closed point O of S. We will speak about the elements of
the ideals in passing through a resolution of S.

Speaker:   Ali Osman Asar
Title:   Locally Nilpotent Minimal Non-P Groups

Abstract:   Let G  be a group, and be a property of groups. If
every proper subgroup satisfies but G  itself does not satisfy it, then
G is called a  minimal non-p group.
In this talk, we will focus on locally nilpotent minimal non-P groups,
where P stands for minimal hypercentral or nilpotent-by-Chernikov.
The following are the main results that will be discussed during
the talk:

Theorem: Let  be a minimal non-hypercentral Fitting p-group.
If every proper subgroup of  is solvable, then G  is solvable.

Theorem: Let G  be a locally nilpotent  p-group in which
every proper subgroup is nilpotent-by-Chernikov.
Then is nilpotent-by-Chernikov.

Speaker:   Mahmut Kuzucuoglu
Title:   Random walk around centralizers in locally finite simple groups

Abstract:  Let be a group theoretical property. A group is called locally
X-group if every finite subset of  G  is contained  in a group which satisfies the
property X.

Centralizers provide a natural source to produce proper subgroups in non-abelian
groups. In this talk,  we give the structures of centralizers  for certain elements in
locally finite simple groups and mention some applications of these results.

Speaker:  Bülent A. Ekin
Title:   Some properties of partitions in terms of crank.

Abstract:   Let N(r, m, n)  (resp. M(r, m, n) ) denote the number of
partitions of whose ranks are congruent to modulo n. Atkin
and Swinnerton-Dyer gave the relations between the numbers N(r, m, n)
when  m = 5, 7  and  0  \leq  r,  k < m. Here, we show that the methods of
Atkin and Swinnerton-Dyer can be extended to prove the relations for
the crank.

Title:  ES-modules

Abstract:  Let R  be a ring and be a left exact preradical in the category
of right R-modules. Then we say that M  is an ES-module provided
that every exact submodule of M  is a direct summand of M. In this talk,
we give some properties of ES-modules and make it clear when the
finite direct sum of ES-modules is also an ES-module.

Speaker:  Laurance Barker
Title:  Limits of group representations

Abstract:  What should  it mean to say that a group representation
G --> GL(V)  is the limit of group representations G_n --> GL(V_n)
as n goes to infty?  The proposed answer, a reciprocity law'', is a rationale
for a standard usage of the term limit  in the phase space methods and
coherent state methods of quantum physics, optics, and signal processing.
The standard usage has been either in a weak rigorous form as limits of
complex numbers, or in a strong heuristic form that preserves algebraic structure.

Having proposed a mathematical definition of  limit (with properties
that conform to the standard usage) one can raise (if not answer) general
theoretical questions along the lines: what properties of representations
are preserved upon passing to the limit? Irreducibility? Isomorphism?

The limits have been used, in physics and engineering, as a means
whereby problems involving analysis and topology may be reduced to
problems involving only finite algebra and linear algebra. Very
speculatively, one might ask whether or not the same reduction method
could be of any use in connection with problems in fundamental mathematics.

Title:   Neat Submodules

Abstract:   In XIII. National Mathematics Symposium 6-9 September 2000,
Istanbul, under the heading of ''Genc Arastirmacilar''
I discussed our plan to solve the  following problems:

a. What is precisely the definition of Neat-submodules in terms of essential
submodules or high-submodules?

b. How we can define neat-injective envelope of any R-module?

c.  Does every R-module have a neat-injective envelope?

d.  If  neat-injective envelope exists for any R-module whether it is
unique up to isomorphism?

Recently, these problems have been solved and included in my Ph.D thesis
titled ''Neat-Injective Envelopes''. I would like to describe these results
and our future intentions in this direction.

D.K. Harrison, J.M. Irwin, C.L. Peercy and E.A. Walker.(1963),
High extension of abelian group, Acta Math. Acad. Sci. Hunger. 14, 319-330.

Stenström, B. (1967), Pure Submodules, Arkiv für Matematik 7, 159

Stenström, B. (1967), High Submodules and Purity, Arkiv für Matematik 7, 173-176.

Speaker:   Li Huishi
Title:  Some Computational Problems in Quadric Solvable Polynomial Algebras

Abstract:   Let be a field of characteristic 0, and let  A=k[a_1,...,a_n]  be a
gr, where solvable polynomial algebra is in the sense of A. Kandri-Rody and
V. Weispfenning (1990,  J. Symbolic Comput. 9, 1--26). After introducing
the  gr-filtration FA  on  A and exploring the compatibility of  FA with the
standard filtration FA on  A, it is shown, by using a double filtered-graded
transfer of data, that the Gelfand-Kirillov dimension of a finitely generated left
A-module can be algorithmically computed via using Gröbner bases.
As an application, the elimination lemma obtained from H. Li and
F. van Oystaeyen (2000, J. Algebra. 234, 101--127) for linear solvable
polynomial algebras is obtained for A. Moreover, the multiplicity of a
finitely generated left A-module can also be algorithmically computed
by means of Gröbner bases. Note that the class of quadric
solvable polynomial algebras properly contains all linear solvable
polynomial algebras.

Ergün Yalçin
Bilkent University
Bilkent, Ankara 06533 Turkey
Phone:  (312) 290 2106
Fax:  (312) 290 5097
Email: yalcine@fen.bilkent.edu.tr

Last update: March 14, 2001