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
2:45 - 3:15 Adnan Tercan
Title: ES-modules
3:15 - 3:45 Arshad Imam
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
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 R^{3}.
Abstract:
In this seminar we mainly discuss splines on a three dimensional
polyhedral complex D in R^{3}
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 k 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 R passing through a resolution of S.
Speaker: Ali
Osman Asar
Title:
Locally Nilpotent Minimal Non-P Groups
Abstract:
Let G be a group, and P be a property of groups.
If
every proper subgroup satisfies P 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 G be a minimal non-hypercentral
Fitting
p-group.
If every proper subgroup of G 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 G is nilpotent-by-Chernikov.
Speaker:
Mahmut Kuzucuoglu
Title:
Random walk around centralizers in locally finite simple groups
Abstract: Let
X
be a group theoretical property. A group G 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 n whose ranks are congruent to r
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.
Speaker: Adnan
Tercan
Title:
ES-modules
Abstract: Let
R
be a ring and r 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.
Speaker: Arshad
Imam
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 k be a field of characteristic 0, and let A=k[a_1,...,a_n]
be a
quadric solvable polynomial algebra with respect to a graded monomial
ordering
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.
CORRESPONDENCE ADDRESS: