Please note that abstracts are not in alphabetical order.

On Summand Sum and Summand Intersection Property of Module
Mustafa Alkan
Hacettepe Üniversitesi
e-mail: alkan@hacettepe.edu.tr

R will be associative ring with identity and modules M unital left R-modules. In this work extending modules and lifting modules with the SSP (or SIP) are studied. A necessary and sufficient condition for a module M to have the SSP is that for every decomposition M=AÅB and f Î Hom(A,B) , Im(f) is a direct summand of B. Among others it is shown also that a (C₃) module with the SIP has the SSP, and a (D₃) module with SSP has the SIP. Also we show that for an extending module M, M is UC-module if and only if M has the SIP if and only if M has the SSIP. By using these results, we prove that for a quasi-continuous module M , M has the SIP if and only if E(M) has the SSIP.

Joint work with Abdullah Harmanc.

Splines and Fat Points
Selma Alt nok
Adnan Menderes Üniversitesi
e-mail: saltinok43@hotmail.com

In this talk we mainly discuss splines, fat points, especially in \mathbbR^d when d < 4. We give some examples to show the relationship of them and calculate the dimension of splines.

Model Companions of stable theories with automorphisms
John Baldwin
University of Illinois at Chicago
e-mail: jbaldwin@uic.edu

Let T be a stable theory and T_s the theory of models of T with an automorphism s. We introduce the new notion of T_s admitting obstructions. Main Theorem: For stable T, T_s has a model companion if and only if T_s does not admit obstructions. If T_s admits obstructions then T_s has the finite cover property.

This is joint work with Saharon Shelah.

Rotations of a 2-sphere, Euclidian motions of the plane, and limits of representations
Laurence Barker
Bilkent Üniversitesi
e-mail: barker@fen.bilkent.edu.tr

The textbook-writer's favorite example of so-called ``group contraction'' (no relation to topological contraction) is that where the rotation group SO(3) of a 2-sphere is ``contracted'' to Euclidian motion group E(2) on the plane. The ``contraction'' (a deformation of Lie algebras) is achieved by identifying the plane with the tangent at the north pole, and identifying plane translates with infinitesmal rotations about an axis through the equator. This scenario, called the phase space picture, is a simplification of a scenario called the configuration space picture. In the latter, SO(3) is replaced by a central extension U(1) ×U(2), which has an irreducible unitary action on \mathbb Cⁿ for each n Î \mathbb N. Meanwhile, E(2) is replaced by a central extension EHW(2), the group of Heisenberg-Weyl motions, which has an irreducible unitary action on L²(\mathbb R). We present the limit

(EHW(2) ® U(L2(\mathbb R))) = lim n  (U(1) ×U(2) ® U(n))

as an example of a limit of representations, and we shall use it to illustrate some general results and questions concerning limits of representations of Lie groups and Lie algebras. The initial motive for the material comes from certain discretization problems that arose, independently, in optics, signal analysis, and quantum physics.

The Si s Kebap Theorem for Groups of Finite Morley Rank
Ay se Berkman
ODTÜ
e-mail: berkman@arf.math.metu.edu.tr

This work concerns the classification of groups of finite Morley rank, a project which is centered around the following.

Cherlin - Zil'ber Conjecture: An infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field.

The partial answer we have so far is the following:

Generic Tame Theorem (GTT): A generic infinite simple tame K^*-group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field.

The proof is scattered in about a dozen articles and is based on the material of about twenty other articles. The main work is due to Altnel, Borovik and Cherlin.

The so-called Si s Kebap Theorem is used in the last steps of the proof of the GTT and is politically correct since it does not discriminate between even and odd types. It generalizes the classical involution theorem for groups of finite Morley rank (which I talked about in Antalya Algebra Days I) and it supplies an easier proof for a different version of the identification theorem for even type groups (joint work with A. Borovik and topic of my talk in Antalya Algebra Days II).

Si s Kebap Theorem: Assume that G is a simple K^*-group of finite Morley rank, is of p¢-type for some prime p, and contains a maximal p-torus D of Prüfer p-rank at least 3. Also assume that G is generated by the subgroups C_G^°(x) such that x is in D and has order p, and these subgroups are reductive. Then G is a Chevalley group over an algebraically closed field of characteristic ¹ p.

Dictionary: All definitions related to groups of finite Morley rank in general can be found in the book Groups of Finite Morley Rank by A. Borovik and A. Nesin, Oxford University Press, 1994.

Let G be a group of finite Morley rank, and let p be a prime number.

The connected component of G, denoted by G^°, is the intersection of all definable subgroups of finite index in G.

A group H is called quasi-simple if H¢=H and H/Z(H) is simple and non-abelian. A quasi-simple subnormal subgroup of G is called a component of G. The product of all components of G is called the layer of G and denoted by L(G), and E(G) stands for L(G)^°.

The group G is:

• connected, if G=G^°;
• a K^*-group, if every infinite simple proper definable and connected section of G is an algebraic group over an algebraically closed field;
• of p¢-type, if G does not contain any infinite abelian subgroups of exponent p;
• reductive, if G is connected and G=Z(G)E(G).

A p-torus of Prüfer rank n is isomorphic to the direct product of n copies of the quasicyclic p-group C_p^¥.

Joint work with Alexandre Borovik

Probabilistic and non-deterministic methods in algebra
Alexandre Borovik
UMIST
e-mail: borovik@umist.ac.uk

I will give an elementary introduction to recent results in the new emerging area of algebra concerned with the development and application of probabilistic and non-deterministic algorithms for algebraic problems. Examples of such algorithms are Monte Carlo algorithms for recognition of finite simple groups and genetic algorithms for solving equations in infinite groups. Some probabilistic algorithms (say, the Miller-Rabin primality test in computational number theory) have already became classical, while application of genetic algorithms to algebra is so recent that has been tested only at a limited range of problems. Time permitting, I will discuss some real life applications of the new methods.

On the Norms of the Interval Cauchy-Toeplitz and Cauchy-Hankel Matrices
Durmu s Bozkurt
Selçuk Üniversitesi
e-mail: dbozkurt@selcuk.edu.tr

Let T_n and H_n be Cauchy-Toeplitz and Cauchy-Hankel matrices, respectively. Then, we have obtained the following bounds for || T_n||₂ and ||H_n||₂:

æ  ú Ö æ ç è k + 1 k ö ÷ ø 2   + æ ç è k k + 1 ö ÷ ø 2   [ Y( 1,[(2k2 + 2k + 1)/(k2 + 2k + 1)] ) + Y( 1,[(2k + 1)/(k2 + 2k + 1)] ) ]   £ || Tn ||2 £ p 2sin( [(kp )/(k + 1)] )

where g=k/(k+1), h=(k+1)/k and n ³ 7 and

|| Hn ||2 £ p sin( [(kp )/(k + 1)] )

where || .||₂ is spectral norm of the matrix , g=(k+1)/k, h=k/(k+1), n ³ 7 and k ³ 7. Moreover, we have established the upper and lower bounds for the l_p norms of the interval Cauchy-Toeplitz and Cauchy-Hankel matrices, respectively.

Joint work with Necati Ta skara.

The Effciency on 2-generators of semi-direct product of groups and the efficiency of Standard Wreath Product
Ahmet Sinan Çevik
Bal kesir Üniversitesi
e-mail: scevik@balikesir.edu.tr

Let G be a semi-direct product of K by A where K and A are both cyclic groups of order n (n Î \Bbb N) and p (p is a prime), respectively. Then we prove that G has an efficient presentation on the minimal number (that is 2) of generators. Also let y be the set of all finite groups that have efficient presentations. We give sufficient conditions for the standard wreath product of two y-groups to be y-group.

Linear Growth of Primary Decompositions of Modules and Integral Closures
Nuri Çimen
Hacettepe Üniversitesi
e-mail: ncimen@hacettepe.edu.tr

Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. We say that I has linear growth of primary decompositions if there exists a positive integer l such that, for every n Î \Bbb N (\Bbb N denotes the set of positive integers), there exists a minimal primary decomposition of Iⁿ,

Iⁿ=q₁Ç...Çq_r with Ö{q_i}^ln Í q_i.

Irena Swanson proved that every ideal in every commutative Noetherian ring has linear growth of primary decompositions. Next, let M be a finitely generated R-module. We say that the ideal I has linear growth of primary decompositions with respect to M if there exists a positive integer l such that for every n Î \Bbb N, there exists a minimal primary decomposition of IⁿM in M,

IⁿM=Q₁Ç...ÇQ_s with Ö[(Q_i:M)]^nl Í (Q_i:M) for all i=1,...,s.

Later, Rodney Y. Sharp, by using injective modules, proved that every ideal of R has linear growth of primary decompositions with respect to every finitely generated R-module. In this work, we present another (maybe simpler) proof of this generalized result. Sharp also proved that every proper ideal in R has linear growth of primary decompositions for integral closures of ideals. We also extend this result to integral closures of ideals relative to finitely generated modules over R.

On t-complemented modules
Septimiu Crivei
Babes-Bolyai University, Romania
e-mail: crivei@math.ubbcluj.ro

Let t be a hereditary torsion theory on the category R-mod of left R-modules. A module A is called t-complemented if every submodule of A is t-dense in a direct summand of A. In [2], P.F. Smith, A.M. Viola-Prioli and J.E. Viola-Prioli considered and studied t-complemented modules, whose importance is due to their connection with extending modules. We establish some further properties on t-complemented modules, being especially interested in t-complemented t-injective modules and t-completely decomposable modules.

References

[1] S. Crivei, On t-complemented modules, Mathematica 46 (2001) (to appear).

[2] P.F. Smith, A.M. Viola-Prioli and J.E. Viola-Prioli, Modules complemented with respect to a torsion theory, Comm. Algebra 25 (1997), 1307-1326.

[3] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, 1994.

The Category of Smooth Representations of the Isometry Groups of Homogeneous Trees
Selçuk Demir
\.Istanbul Bilgi Üniversitesi
e-mail: sdemir@bilgi.edu.tr

We study the category of smooth representations of the isometry groups of homogeneous trees in a way similar to that of p-adic groups. In particular, we define the congruence Hecke algebras and prove that they are finitely generated. Then we prove the uniform admissibility theorem for these groups. Later it is proved that every smooth representation can be written as a direct sum of two subresentations such that one of these subresentations have only cuspidal irreducible subquotients while the other does not have any irreducible cuspidal subquotient. Using this it is proved that the category of admissible representations is noetherian. Later it is proved that if U is a congruence subgroup and a smooth representation is generated by its U-fixed vectors, then any of its subrepresentations also have the same property. These results are necessary to be able to develop a theory analogous to that of P. Schneider and U. Stuhler. Whenever it is possible, we give the same proofs which work also in the case of p-adic groups.

Annihilators of Ideals in Exterior Algebras
Songül Esin
Dogu s Üniversitesi
e-mail: sesin@dogus.edu.tr

It is well known that every Frobenius algebra is quasi-Frobenius, that is to say, if a finite dimensional algebra over a field F has a nondegenerate bilinear form B such that  B(xy,z)=B(x,yz) for all x,y,z Î A, then the maps  L® Ann_r(L)  and  R® Ann_l(R)  give inclusion preserving bijections between lattices of right and left ideals of A satisfying

(a) Ann_r(L₁+L₂)=Ann_r(L₁)ÇAnn_r(L₂)  ,   Ann_r(L₁ÇL₂)=Ann_r(L₁)+Ann_r(L₂)  

(b) Ann_l(R₁+R₂)=Ann_l(R₁)ÇAnn_l(R₂)   ,  Ann_l(R₁ÇR₂)=Ann_l(R₁)+Ann_l(R₂)

(for example see [2])

The most significant example of Frobenius algebras is the exterior algebra E(V) of a finite dimensional vector space V on which B is defined by

B(eI,eJ)=e{1,2,...,n}*(eIeJ)

where  {e₁,e₂,...,e_n}  is a basis for V  and e_I^*  is the dual of  e_I  in E^*; here

eI=ei1ei2...eik  ,  I={i1,i2,...,ik} Ì {1,2,...n} eJ=ej1ej2...ejk  ,  J={j1,j2,...,jk} Ì {1,2,...n}

stand for basis elements for E(e_Ie_J). In [1], by making use of this classical result, I.Dibag described annihilators of 2-vectors in E. The aim of this note is to extend this description to arbitrary m-vectors and to the case where the base field is of nonzero characteristic. The failure in some cases is also discussed.             

Joint work with Cemal Koç

References

[1] I.Dibag, Duality for Ideals in the Grassmann Algebra, Journal of Algebra 183, 24-37 (1996)                    

[2] Gregory Karpilovsky, Symetric and G-algebras, 1990

Groups of automorphisms of Chevalley algebras
Alev F rat
Ege Üniversitesi, \.Izmir
e-mail: firat@sci.ege.edu.tr

We present an extension of Segal theorem on Lie algebra automorphisms of upper triangular nilpotent matrices to the whole class of nilpotent subalgebras generated by positive parts of indecomposible Chevalley algebras.

Poincare duality in P.A. Smith theory
Bernhard Hanke
Universitaet Muenchen
e-mail: hanke@rz.mathematik.uni-muenchen.de

Let G=S¹, G=Z/p or more generally G be a finite p-group, where p is an odd prime. If G acts on a space whose cohomology ring fulfills Poincare duality (with appropriate coefficients k), we prove a mod 4 congruence between the total Betti number of X^G and a number which depends only on the k[G]-module structure of H^*(X;k). This improves the well known mod 2 congruences that hold for actions on general spaces.

How tall is the automorphism tower of a group?
Joel David Hamkins
The City University of New York and Carnegie Mellon
e-mail: hamkins@andrew.cmu.edu

The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group and so on, iterating transfinitely. Each group maps into the next using inner automorphisms and one takes a direct limit at limit stages. The question is whether the process ever terminates in a fixed point, a group which is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. Rae and Roseblade (1970) proved that the automorphism tower of any centerless Cernikov group terminates in finitely many steps. Hulse (1970) proved that the automorphism tower of any centerless polycyclic group terminates in countably many steps. Solving the problem for centerless groups, Thomas (1985) proved that the automorphism tower of any centerless group eventually terminates. In this talk, I will prove that every group has a terminating automorphism tower.

After this, I will discuss the set-theoretic aspects of the height of the automorphism tower of a group, and sketch my recent proof with Simon Thomas that it is consistent to have a group whose automorphism tower depends wildly on the set-theoretic background; indeed, in various models of set theory the automorphism tower of this very same group can be almost arbitrarily specified.

Cyclic Presentations and Admissible Words
\.Inci Gültekin
Atatürk Üniversitesi
e-mail: inciakarg@yahoo.com

In this study, let  d > 3 and d=2a+b+c be odd. In this case it is shown that 6-tuples (1,0,c,n,r,s) is admissible and obtained a word w. Hence we find out that the polynomial associated with the defining w is of form

f ( t )= n-1 å i=0  (-1 )i+1a2it2i,

where all indices are taken mod n.

Conformal-field-theoretic analogues of codes and lattices
Yi-Zhi Huang
Rutgers University
e-mail: yzhuang@math.rutgers.edu

We introduce and study completely-extendable conformal intertwining algebras. Based on results obtained in other papers, several examples are given. Duals of these algebras are constructed and nondegenerate such algebras are defined. We prove that the double dual of such a nondegenrate algebra is equal to itself. We explain that these nondegenerate algebras are the correct conformal-field-theoretic analogues of linear binary codes and nondegenerate rational lattices.

On the minimal neat injective modules containing a given module as a neat submodule
Archad \.Imam
Dokuz Eylül Üniversitesi
e-mail: iarchad@hotmail.com

The idea of neat-subgroups is well known in Abelian Group Theory (see e.g., [ 1] ,[ 5] ). In [ 3] we have generalized this idea in R-module and introduce the definition of neat-submodules in terms of essential submodules and consistently of high submodules. It is also well-known that over any ring R any module has an injective envelope (see e.g.,[ 2] ). In [ 4] we build up the structure of neat-injective envelope of an abelian group A in terms of its torsion part T(A) and torsion-free part A/T(A). In [ 5] we have developed a theory of neat-essential extensions of abelian groups. Existence of neat-injective envelopes for abelian groups is proved by Onishi [ 3] . In this paper we have proved the existence of neat-injective envelopes for any R-module. Firstly we define neat-monomorphism, neat-essential, maximal neat-essential extension and neat-injective envelope for any R-module. Our aim is to prove the following two theorems to show the existence of neat-injective envelope for any R-modules.

Theorem For any R-module M there is a maximal neat-essential extension a:M® E.

Theorem If a:M® E is a maximal neat-essential extension, then E is neat-injective envelope.

Corollary For every R-module there is a minimal neat-injective module containing a given module as a neat-submodule.

If time permits, I would also like to describe the possibility of giving structure of neat-injective envelopes for finitely generated modules over a principal ideal domain in terms of torsion submodules T(M) and the quotient module M/T(M).

   REFERENCES

[ 1] Fuchs.L. (1958). Abelian groups. Publishing House of the Academy of Sciences, Budapest.

[ 2] Faith, C., (1967). Lectures on Injectives Modules and Quotient Rings (Lecture Notes in Mathematics 49,             Springer-Verlag,1967).

[ 3] Imam.A. Neat-Submodles. (preprint-presented in Bilkent Algebra Day, March 24, 2001)

[ 4] Imam.A, Alizade.R and Ak nc .K. On the strucure of neat-injective envelopes of abelian groups.(preprint- presented in Antalya Algebra days II May 17-21, 2000).

[ 5] Imam.A.(2000) Neat-injective envelopes Ph.D thesis D.E.Ü, Izmir.

[ 6] Onishi, M. (1984). On minimal neat-injective groups containing a given group as a neat subgroup.  33.,Comment Math . Univ. St.Paul. No.2, 203-207.

[ 7] Stenström, B. (1967)., High Submodules and Purity, Arkiv för Matematik7, 173-176.

[ 8] Stenström, B. (1967)., Pure Submodules, Arkiv för Matematik7, 159

Joint work with: Refail Alizade., Yüksek Teknoloji Enstitüsü, Izmir, Turkey.

A Characterization of C₂(q) where q > 5
Ali Iranmanesh
Tarbiat Modarres Üniversitesi
e-mail: iranman@vax.ipm.ac.ir

The order of every finite group G can be expressed as a product of coprime positive integers m₁,...,m_t such that p(m_i) is a connected component of the prime graph of G. The integers m₁,...,m_t are called the order components of G. In several works it is proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups C₂(q) where q > 5 are also uniquely determined by their order components. As corollaries of this result, the validity for C₂(q) where q > 5 of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Be is obtained.

A joint work with B. Khosravi.

Generalizations of Lifting Modules
Derya Keskin
Hacettepe Üniversitesi
e-mail: keskin@hacettepe.edu.tr

In this note all rings are associative with identity and all modules are unital right modules.

Let M be a module. M is called a lifting module if for every submodule A of M there exists a decomposition M=M₁ÅM₂ such that M₁ £ A and AÇM₂ is small in M₂, equivalently, for every submodule A of M there exists a direct summand K of M such that K £ A and A/K is small in M/K. Dually, M is called extending if for every submodule A of M there exists a direct summand K of M such that A is essential in K. We call the module M Å-supplemented if every submodule of M has a supplement that is a direct summand of M. Every lifting module is Å-supplemented, but the converse is not true in general, see Example 2.1 in [].

In [], P.F. Smith and A. Tercan introduced (C₁₂)-modules to generalize extending modules. In this note, we shall be concerned with the following proper generalization of Å-supplemented modules which is dual to the condition (C₁₂): A module M is (D₁₂) if, for every submodule N of M, there exists a direct summand K of M and an epimorphism a:M/K® M/N such that Kera is small in M/K.

In this note, we obtain some properties and characterizations of (D₁₂)-modules. Also, the following fact is proved:

THEOREM. Let M be a non-zero module with finite corank. Then the following are equivalent.
(1) Every direct summand of M is a finite direct sum of hollow modules.
(2) Every direct summand of M is (D₁₂).

Joint work with W. Xue, Fujian Normal University.

References

[HKS] A. Harmanci, D. Keskin and P.F. Smith, On Å-supplemented Modules, Acta Math. Hungar. 83(1-2) (1999), 470-487.

[KX] D. Keskin and W. Xue, Generalizations of Lifting Modules, Acta Math. Hungar. 91(3) (2001), to appear.

[ST] P.F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra 21(6) (1993), 1809-1847.

Thompson's conjecture and random walks in classical symmetric domains
Alexander Klyachko
Bilkent Üniversitesi
e-mail: klyachko@fen.Bilkent.EDU.TR

Recall that singular spectrum of complex matrix A, is square root of spectrum of Hermitian matrix AA^*. Thompson conjecture claims that for existence of matrices A₁, A₂, and A₃=A₁A₂ with singular spectra s₁, s₂, s₃, it is necessary and sufficient existence of Hermitian matrices H₁, H₂, and H₃=H₁+H₂, with spectra log(s₁), log(s₂), and log(s₃). We will prove this conjecture using harmonic analysis on symmetric spaces.

On definable sets in the whole of p-adic numbers.
Ponomarev Konstantin
Ege Üniversitesi
e-mail: ponom@sci.ege.edu.tr

Subset of valued field is called formula set if it can be defined by some formula of predicate calculus in two-sorted language of valued fields. Let M and N be two formula subsets in valued field K. They are called equivalent sets if there is a bijection between these sets definable by some formula of restricted predicate calculus in the language of theory of valued fields without constant symbols. For ultra-product of fields of p-adic fields for different prime numbers p we define some characteristic numbers of formula sets by this equivalence.

X-Lifting Modules
Tamer Ko san
Hacettepe Üniversitesi, Ankara
e-mail: tkosan@hacettepe.edu.tr

In this work we study lifting property of a given module with respect to a class X of modules. Throughout R will denotean associative ring with identity and modules M will be unital right R-modules. We define [`Z]<sub>X</sub>(M), the X-cosingular submodule of M as follows: [`Z]_X(M) = rej_M(X) = Ç{Ker(f): f Î Hom(M,X),X Î X} the reject of X in M and [`Z]<sup>2</sup><sub>X</sub>(M) = [`Z]_X([`Z]_X(M)). Then

Theorem 1 Let X be a class of R-modules which is s-closed. Then the following hold for any R-module M.
1. If M is X-module then [`Z]_X(M) = 0
2. If N is a non-X-cosingular submodule of M and X is q-closed then every X-submodule of N is small in N.
3. If M is X-lifting and [`Z]<sub>X</sub>(M) is in X then [`Z]²_X(M) is X-coclosed in M.

Theorem 2 Let S_t denote the class of R-small modules. Let M be a module. Then [`Z]<sup>2</sup><sub>St</sub>(M) = Ç{A : A is coessential submodule of [`Z]_St(M) in M}.

Theorem 3 Let S_t denote the class of R-small modules.
i. Every R-module is non-S_t-cosingular if and only R is cosemisimple ring.
ii. If every S_t-module is M-injective then [`Z]<sub>St</sub>(M)ÇN = [`Z]_St(N) for every submodule N of M.

The main aim of this work is to prove the following Theorem:

Theorem 4 Let S_t denote the class of R-small modules and M a module. Let t_St = (T^{^}_St,F^{^}_St) be the torsion theory cogerated by S_t. Then the following are equivalent (1) M is T^{^}_St-lifting module.

(2) M = [`Z]<sup>2</sup><sub>St</sub>(M) ÅY, for some Y £ M such that Y and [`Z]²_St(M) are T^{^}_St-lifting and Y is [`Z]²_St(M)-projective and M is S_t-amply supplemented.

Joint work with Abdullah harmanc.

Fixed Points of Automorphisms in Finite and Locally Finite Groups
Mahmut Kuzucuoglu
ODTÜ
e-mail: matmah@metu.edu.tr

Let A be a subgroup of an automorphism group of G. We say that A acts fixed point freely on G if C_G(A)={  g Î G  |  g^a = g for all  a Î A  }={1}.

It is a consequence of the classification of finite simple groups that if an automorphism a acts fixed point freely on a finite simple group G, then G is soluble . We mention some results on fixed point free actions and some applications of these results to periodic residually finite groups.

Real regulators on Milnor Complexes
James D. Lewis
University of Alberta
e-mail: lewisjd@gpu.srv.ualberta.ca

Milnor K-theory, and algebraic K-theory play a role in the subject of algebraic cycles and motives. In this self-contained talk, I will explain the basics of Milnor K-theory, and introduce regulators on a certain Gersten-Milnor complex. This will be compared to a Beilinson type regulator. This talk is aimed at a general audience.

Geometric Goppa codes on fibre-product of Artin-Schreier and Kummer coverings
Mahmoud Shalalfeh
Bilkent Üniversitesi
e-mail: mahmoud@fen.bilkent.edu.tr

In this talk I will present a new family of smooth projective curves defined over a finite field F_q with many F_q-rational points using fibre product of Artin-Schreier curves. The ratio of the genus of these curves to the number of rational points is small enough to get geometric Goppa codes with good relative parameters.

Modular Vector Invariants of Finite Groups
Ugur Madran
Bilkent Üniversitesi
e-mail: madran@fen.bilkent.edu.tr

In this talk I will give an overview of modular vector invariants and some lower and upper degree bounds for the generators of the invariant algebra. More precisely, let G be any finite group such that p divides the order of G. Then for any representation of G as G Ì GL(n,\BbbF_p), we consider the action of G on A_n,m=\BbbF_p[x_1,1,¼,x_1,n;¼;x_m,1,¼,x_m,n] just by the equation: [g(x_i,1),¼,g(x_i,n)]^tr=[g_i,j]·[x_i,1,¼, x_i,n]^tr, where the exponent tr means transpose of a vector, and g_i,j's are the entries of g Î G.

The Z^* Functor for Rings whose Primitive Images are Artinian
A. Çigdem Özcan
Hacettepe Üniversitesi
e-mail: ozcan@hacettepe.edu.tr

Given a ring R, a subfunctor Z^* of the identity functor on the category of all right R-modules is defined by Z^*(M)={ m Î M : mR is a small module } for any R-module M. We prove that if the ring R satisfies the descending chain condition for right annihilators and R/P is an Artinian ring for every primitive ideal P then Z^*(M)={m Î M: mS=0 } for every right R-module M, where S is the left socle of R.

Joint work with: Patrick F. Smith, University of Glasgow.

On Commuting and Non-Commuting Complexes
Jonathan Pakianathan
University of Rochester
e-mail: jonpak@math.rochester.edu

In this talk, I will discuss joint work with Ergün Yalç n on various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise noncommuting (resp. commuting) sets of nontrivial elements of G.

We observe that NC(G) has only one positive dimensional component, BNC(G) and that this is always simply-connected. I will then discuss a wedge decomposition formula for BNC(G) derived from a recent result of Björner, Wachs and Welker and get some consequences from this structure theorem.

Finally I will discuss a sort of Ramsey duality between these complexes and complexes defined by Quillen and mention some results one gets from this duality.

Differential forms in model theory
David Pierce
ODTÜ
e-mail: dpierce@arf.math.metu.edu.tr

The operation of differentiation in calculus gives rise to the purely algebraic notion of a derivation. The existentially closed fields of characteristic zero with a given number of commuting derivations can be given a ``coordinate-free" characterization in terms of differential forms. The main tool is an algebraic generalization of the Frobenius Theorem of differential geometry. It becomes evident that the class of such existentially closed differential fields is elementary; that is, the theory of fields of characteristic zero with several commuting derivations has a model-companion. (This fact was proved earlier, but by complicated, ``non-coordinate-free" means.)

Hereditary categories with tilting objects
Idun Reiten
Norwegian University of Science and Technology (NTNU)
e-mail: idunr@math.ntnu.no

Let C be a hereditary abelian category over a field k, with finite dimensional homomorphism and extension spaces, and assume that C has a tilting object. Such categories have played an important role in the representation theory of finite dimensional algebras, in particular in connection with the quasitilted algebras introduced in joint work with Happel and Smaloe. For some time it was an open problem to classify all such categories. This was solved last year by Happel in case the field k is algebraically closed. In joint work with Happel we have extended the classification to hold for k being an arbitrary field.

Twin Buildings and Kac-Moody groups
Mark Ronan
University of Illinois at Chicago
e-mail: ronan@uic.edu

Twin buildings are finite dimensional objects that arise from (infinite dimensional) Kac-Moody groups. They are a generalization of spherical buildings for Chevalley groups. This talk will survey the main ideas, results and conjectures.

A Sharpening of Noether-Fleischmann Bound in the invariant theory of finite groups
Müfit Sezer
Purdue University
e-mail: msezer@math.purdue.edu

We consider linear representations of a finite group G on a finite dimensional vector space over a field of characteristic p ³ 0, relatively prime to |G|. A theorem to Noether for p=0 and to Fleischmann for p > 0 says that the ring of invariants is generated by homogeneous elements of degree at most |G|. Generalizing a sharpening of Noether bound due to Schmid, Domokos and Hegedüs we prove that if G is not cyclic, then the ring of invariants is generated by elements of degree at most [3/4]|G| if |G| is even, and at most [5/8]|G| if |G| is odd.

A mini-course on the Hodge conjecture
James D. Lewis
University of Alberta
e-mail: lewisjd@gpu.srv.ualberta.ca

The Hodge conjecture is the central conjecture in complex analytic and algebraic geometry. Despite the intractible nature of this conjecture, there are some interesting geometrical arguments that merit explanation, as well as a new angle from the point of view of regulators. My goal is to explain the Hodge conjecture, and suggest some ways that a new generation of geometers can approach it. Some familiarity with chapter 0 of the Griffiths-Harris text will be helpful.

On nilpotent ideals in the cohomology ring of a finite group
Ergün Yalç n
Bilkent Üniversitesi
e-mail: yalcine@fen.bilkent.edu.tr

In this talk, I will present some recent results of our joint work with Jonathan Pakianathan on the nilpotency degree of ideals in the cohomology ring of a finite group. These results are obtained by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. In particular, we consider Mui's essential cohomology conjecture (which states that the essential cohomology ideal has nilpotence degree 2), and show that it is equivalent to a conjecture about group actions on connected graphs.

On the Bounds for Condition Number of Interval Matrices
Necati Ta skara
Selçuk Üniversitesi
e-mail: ntaskara@selcuk.edu.tr

Let A^I be a diagonalizable interval matrix. Then, we have obtained k(A^I) £ (1+a_c /d_c )^{2n- 2} where k(A^I) is the condition number of the matrix A^I interval matrix.

Joint work with Durmu s Bozkurt

A Note on Weak (C11) Modules
Adnan Tercan
Hacettepe Üniversitesi
e-mail: tercan@hacettepe.edu.tr

All rings are associative and have identity elements and all modules are unital right modules. Let R be any ring. A right R-module M is called weak (C11)-module provided that every semisimple submodule of M has a complement which is a direct summand of M. Clearly every (C11)-module and in particular weak CS-modules are weak (C11)-modules but the converses are not true in general. We enjoy with the following Theorem which generalizes some well-known results due to Camillo and Yousef, Smith and Tercan and Armendariz. To this end we obtain that : Let M be a module such that every direct summand of M is a weak (C11)-module and M/Soc M has finite uniform dimension then there exist a semisimple submodule N of M and a submodule K of M with finite uniform dimension such that M is the direct sum of N and K. Since we could not settle so far the following remained open :
Whether every direct summand of a weak (C11)-module is also weak (C11) or not?

The classification problem for torsion-free abelian groups of finite rank
Simon Thomas
Rutgers University
e-mail: sthomas@math.rutgers.edu

In this talk, I will present some recent work which seeks to explain why no satisfactory system of complete invariants has been found for the torsion-free abelian groups of rank n > 1.

An Algebraic and Automata Theoretical Approach to Posets
Andreas Tiefenbach
ODTÜ e-mail: tiefenb@metu.edu.tr

We assign to each linear extension of a finite poset \mathbbP=(X,P) a homomorphism from the free semigroup X⁺ onto the semilattice ({ 1,... ,|X|},min). The related congruences are semilattice-congruences and thus their intersection g_P is a semilattice congruence too. The factor X⁺/g_P turns out to be isomorphic with the semilattice A(\mathbbP) of all nonempty antichains of \mathbbP.

We rediscover the well-known correspondence between linear extensions and maximal chains in A(\mathbbP). In addition we show that a set of linear extensions realizes the given poset if and only if their corresponding chains generate A(\mathbbP)¹ considered as lattice.

We also assign a language L(\mathbbP) to a given poset \mathbbP. We show how to find this language and define the language dimension of the poset \mathbbP. Here we show that a poset is an interval order if and only if it has language dimension 1.

Joint work with Stk Irk

On the problem of completeness of the outer automorphism group of an infinitely generated free group
Vladimir Tolstykh
Kemerovo State University
e-mail: vlad@offer.kuzb-fin.ru

Over ten years ago, Khramtsov [] proved that the outer automorphism group Out(F_n) of a free group F_n of finite rank n ³ 3 is complete, that it is centreless and all its automorphisms are inner. Recently Bridson and Vogtmann [] gave a new proof of completeness of Out(F_n). It seems very likely that the similar result holds for free groups of arbitrary rank at least three (note that the automorphism group of an arbitrary free group of rank at least two is complete, [,]). The following result provide some evidence for the latter conjecture.

Proposition. Let F be a free group of infinite rank. Then any automorphism of the group Out(F) preserves all conjugacy classes of soft involutions from Out(F).

In terminology introduced in [], we call an involution j of the group Aut(F) soft, if there is a basis of F such that j fixes or inverts the elements of this basis modulo the commutator subgroup [F,F]. An involution from Out(F) is also said to be soft, if its preimage in Aut(F) has a soft involution. It should be pointed out that the group Aut(F) (and hence the group Out(F)) is, speaking informally, full of soft involutions (see [] for details).

The proof of the Proposition uses the description of elements of prime order in Out(F) obtained by Culler [].

[BV] M. Bridson and K. Vogtmann. Automorphisms of automorphism groups of free groups. J. Algebra, 229 (2000), no. 2, 785-792.

[Cu] M. Culler. Finite groups of outer automorphisms of a free group. Contributions to group theory, 197-207, Contemp. Math., 33, Amer. Math. Soc., Providence, RI, 1984.

[DFo] J. Dyer, E. Formanek. The automorphism group of a free group is complete. J. London Math. Soc., 11 (1975), 181-190.

[Khr] D. G. Khramtsov, Completeness of groups of outer automorphisms of free groups, Group-theoretic investigations (Russian), Akad. Nauk SSSR Ural. Otdel., Sverdlovsk, (1990) 128-143.

[To] V. Tolstykh. The automorphism tower of a free group. J. London Math. Soc., 2 61 (2000), no. 2, 423-440.

Remarks on trees corresponding to surface singularities
Meral Tosun
Feza Gürsey Enstitüsü ve Y ld z Teknik Üniversitesi
e-mail: tosun@gursey.gov.tr

We give geometric features of the dual graph of a desingularization of a rational surface singularity.

Automorphic forms, Siegel-Weil Formula, L-functions
Çetin Urti s
University of Minnesota
e-mail: urtis@math.umn.edu

Automorphic forms are very special functions which have internal symmetries that arise in many technical phenomena, from number theory to theoretical physics. The theory of automorphic forms deals with very symmetrical or periodic functions on groups, and with their analytical and arithmetical properties. In the beginning there will be a short introduction to automorphic forms. About 70 years ago C.L. Siegel found an ingenious re-expression of theta series in terms of sum of powers of divisors as Eisenstein series. Eisenstein series and theta series are special kinds of automorphic forms and play an important role. In the 60's A. Weil translated Siegel's idea into modern language (the Weil representation). This was the origin of ``the theta correspondence'' which is a key technical device in automorphic forms. In effect, a theta correspondence builds a bridge between different species of automorphic forms (having different symmetries being defined on different groups). The Riemann's zeta function, which is the simplest L-function, will be an example.

Monoids and direct sum decompositions
Roger Wiegand
University of Nebraska - Lincoln
e-mail: rwiegand@math.unl.edu

Let R be a local (Noetherian, commutative) ring and M a finitely generated R-module. We let +(M) be the monoid of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of M. One can show that +(M) is a positive normal monoid, that is, it is isomorphic to the set of non-negative integer solutions of some finite system of homogeneous linear equations with integer coefficients, in a finite number of variables. The main theorem states the converse: Let L be a positive normal monoid containing a ``strictly positive'' element m (that is, for each element l Î L there is a positive integer n such that nm ³ l). Then there exist a one-dimensional local domain R and a finitely generated R-module M such that +(M) @ L, via an isomorphism taking [M] to m.

One can use this theorem to demonstrate spectacular failure of the Krull-Schmidt uniqueness theorem for direct-sum decompositions of modules. We will also discuss various possible directions in which the main theorem might be extended.

Building Noetherian and non-Noetherian integral domains
Sylvia Wiegand
University of Nebraska-Lincoln
e-mail: swiegand@math.unl.edu

We present examples of Noetherian and non-Noetherian integral domains which can be built using power series rings, homomorphic images and intersections.

Modules and Comodules
Robert Wisbauer
University of Duesseldorf, Germany
e-mail: wisbauer@math.uni-duesseldorf.de

In recent years the theory of coalgebras and Hopf algebras attracts considerable interest in particular because of the applications in various fields. In my talk I will outline the basic definitions of coalgebras and comodules and show the close connection to the theory of algebras and modules.

Quasithin groups and the classification of the finite simple groups
Stephen D. Smith
University of Illinois at Chicago
e-mail: smiths@math.uic.edu

Although the classification of the finite simple groups (CFSG) was announced around 1981, one part of that work-namely the classification of the ``quasithin" groups by G. Mason-was announced but never published. Since 1996, Michael Aschbacher and I have been preparing a new and more general treatment of that problem, and the result is now essentially ready for publication. The talk will include some history and background of the whole CFSG project, with an indication of the place of the quasithin groups within that effort; and some exposition of more recent work on revising and simplifying the CFSG, including some ideas used in the new theorem on quasithin groups.