Research (in alphabetical order)

Research of James C. Beidleman

My research interest is in Group Theory. Recently I have been concerned with products of groups, permutable subgroups, and finite groups in which the subnormal subgroups are normal or permute with certain classes of subgroups.
Let A and B be subgroups of a group G. A is said to permute with B if AB = { ab | a in A, b in B } is a subgroup of G. A is said to be permutable in G if A permutes with each subgroup of G. A and B is said to be a totally permutable (mutually permutable) pair provided that each subgroup of A permutes with each subgroup of B (A permutes with each subgroup of B and B permutes with each subgroup of A). A is said to be S-permutable if it permutes with all the Sylow subgroups of G.
Let the group G be a product of the subgroups A and B. A question that has been of interest for some time is: if A and B satisfy certain properties, what can be said about G? For example, if A and B are abelian (resp. nilpotent) and G is finite, then G is metabelian (resp. soluble). Another result is that if A and B form a totally permutable pair of supersoluble subgroups, then G is supersoluble.
A group G is called a T (resp. PT)-group provided that normality (resp. permutability) is transitive in G. That is, G is a T (resp. PT)-group provided that whenever H and K are subgroups of G such that H is normal (resp. permutable) in K and K is normal (resp. permutable) in G, then H is normal (resp. permutable) in G. A finite group is called a PST-group if S-permutability is transitive. A finite group is a PT (resp. PST)-group if all the subnormal subgroups of G are permutable (resp. S-permutable).

Recent publications on some of these topics include:

• J.C. Beidleman, A. Galoppo, H. Heineken and M. Manfredino, On certain products of soluble groups, Forum Mathematicum 13 (2001), 569-580.
• J.C. Beidleman and H. Heineken, Totally permutable torsion groups, J. Group Theory 2 (1999), 377-392.
• J.C. Beidleman, B. Brewster and D.J.S. Robinson, Criteria for permutability to be transitive in finite groups, J. Algebra 222 (1999), 400-412.
• J.C. Beidleman and H. Heineken, On the Hyperquasicenter of a Group, Journal of Group Theory 4 (2001), 199-206.
• J.C. Beidleman and H. Heineken, A Survey of Mutually and Totally Permutable Products in Infinite Groups, Quaderni di Matematica 8 (2002), 47-62.
• J.C. Beidleman and H. Heineken, Finite Soluble Groups whose Subnormal Subgroups Permute with Certain Classes of Subgroups, Journal of Groups Theory 6 (2003), 139-158.
• A. Ballester-Bolinches, J.C. Beidleman and H. Heineken, Groups in which Sylow Subgroups and Subnormal Subgroups Permute, Illinois Journal of Math. 47 (2003), 63-69.
• J.C. Beidleman and H. Heineken, Pronormal and Subnormal Subgroups and Permutability, Bollettino dell'Unione Mat. Italiana 6b (2003), 605-615.
• A. Ballester-Bolinches, J.C. Beidleman and H. Heineken, A Local Approach to Certain Classes of Finite Groups, Comm. in Algebra 31 (2003), 5931-5942.
• J.C. Beidleman, P. Hauck and H. Heineken, Totally Permutable Products of Certain Classes of Finite Groups, J. Algebra 276 (2004), 826-835.
• J.C. Beidleman and H. Heineken, Pairwise N-connected Products of Certain Classes of Finite Groups, Comm. in Algebra 32 (2004), 4741-4752.
• J.C. Beidleman and H. Heineken, Mutually Permutable Subgroups and Group Classes, Archiv der Mathematik 85 (2005), 18-30.

Research of Alberto Corso

Commutative Algebra and Algebraic Geometry find their roots in the study of algebraic equations in relation to the geometry of their solutions. Such a line of investigation goes back at least to Descartes and the idea of coordinatizing the plane.
Nowadays, though, commutative algebra and algebraic geometry study the solutions of those equations by forming an algebraic object, called a ring, which consists of the `generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations.
The methods in use are no longer from algebra alone, but also from analysis and topology. Conversely, they have been extensively used in those fields as well, and have proven useful in other fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.

Some of my research interests include: 

• Blow-up algebras: Rees algebras, associated graded rings, special fiber rings and Sally modules;
• Linkage and residual intersections theory; 
• Hilbert functions; 
• Integral closure of ideals; 
• Ideal theory of graphs; 
• Koszul homology.

Research of Edgar Enochs

My research interest concerns the coGalois theory of torsion free covers.
This is a joint program with a team including researchers in Almería, Spain (who frequently visit UK). We have now classified the compact absolute coGalois groups (they are products of finite dimensional p-adic Lie groups) and are now involved in the relative theory.

Some recent publications:

• Relative Homological Algebra (with Overtoun M.G. Jenda), de Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin 2000.
• Compact coGalois groups (with Juan Ramon García Rozas, Overtoun Jenda and Luis Oyonarte), Math. Proc. Camb. Phil. Soc. 128 (2000), 233-244.
• Envelopes and covers by modules of finite injective and projective dimensions (with Stephen Aldrich, Overtoun Jenda and Luis Oyonarte), J. Algebra 242 (2001), 447-459.

Research of Heide Gluesing-Luerssen

My primary area of research is in algebraic coding theory, that is, the theory of error-correcting codes. These codes play a crucial role for the reliability of data transmission via satellite, cell phones, internet communications, data storage on DVDs and where ever else data are transmitted and might undergo some transmission errors. Coding theory aims at preprocessing those data in such a way that the receiver is able to reconstruct the sent data from the erroneous ones if not too many errors happened during the transmission. My recent projects concern:
- duality for convolutional codes,
- isometries for convolutional codes,
- network coding,
- cyclic convolutional codes.

Research of David Leep

My primary area of research is in the algebraic theory of quadratic forms. I also study systems of quadratic forms and systems of homogeneous forms of higher degree. My work in these areas and other areas often involve finite fields, p-adic fields, number fields, and function fields. My other research interests involve Galois theory, especially those aspects of Galois theory that relate to quadratic form theory.

Research of Uwe Nagel

My research interests are mainly in Commutative Algebra and Algebraic Geometry and include combinatorial and computational aspects. Recent projects concern:
- Hilbert functions and Betti numbers of Gorenstein algebras,
- Liaison Theory,
- monomial ideals,
- complexity measures like the Castelnuovo-Mumford regularity,
- toric ideals arising in Phylogenetics and Algebraic Statistics.

Research of Avinash Sathaye

To be completed!