I.B.S. Passi

Research Summary:

The use of homological techniques in the study of Bass' conjecture on Hattori-Stallings rank of finitely generated projective modules over a group ring was pioneered by E. Eckmann. Working in this direction, a homological condition on a group that takes into account the arithmetic of the coefficient ring and implies the validity of the Bass' conjecture was studied (jointly with I. Emmanouil).

A study of the commutator subgroups of free nilpotent groups was carried out (jointly with N. Gupta); the results yield, in particular, a simple proof of the fundamental theorem of group rings.

The groups with the property that their integral group rings have hyperbolic unit groups were examined and some definitive results obtained (jointly with S. O. Juriaans and Dipendra Prasad).

Work on the book Algebra Vol.4, part of an ongoing project in collaboration with I. S. Luthar, continued throughout the year; this volume is now nearing completion.

Publications:

1. Integral Jodan Decomposition of Matrices, Linear Algebra and Its Applications, 355 (2002) 241-261 [jointly with K. W. Roggenkamp and Marcos Soriano].
2. Linear Groups, Professor Wazir Hasan Abdi Memorial Lectures Vol.1(2003), Department of Mathematics, Cochin University of Science and Technology, Cochin, pp. 53-64.

1. A contribution to Bass' conjecture [jointly with Ioannis Emmanouil].
2. Commutator subgroups of free nilpotent groups [jointly with Narain Gupta].
3. Hyperbolic unit groups [jointly with S. O. Juriaans and Dipendra Prasad].

1. Workshop on Topological Methods in Group Theory held at Institute of Mathematical Sciences, Chennai, July 2002; gave two lectures on Non-positive
   curvature and Group Theory.
2. Instructional Workshop and International Conference on Geometric Group Theory held at IIT-Guwahati, December, 2002; gave a course of lectures on Combinatorial Group Theory at the Workshop and a talk on Centralizers and Homology. Was Chairman of the Organising Committee of this programme.

Visits to other Institutes:

1. Institute of Mathematics & Statistics, University of Sao Paulo, Sao Paulo, Brazil, for one month during September - October 2002.
2. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, for one week during February-March 2003.

1. Prof Abdi Memorial Lecture entitled Linear Groups at the Annual Conference of the Ramanujan Mathematical Society held at Banaras Hindu University, Varanasi, June 2002.

2. Professor R. Vaidyanathswamy Endowment Lecture entitled "Geometric Group Theory" at the Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, July 2002.
3. Idempotents in Matrix Group Rings at Institute of Mathematics & Statistics, University of Sao Paulo, Sao Paulo, Brazil, October 2002.

Academic recognition/Awards:

1. Nominated as a Member of the Academic Council of Kurukshetra University, Kurukshetra for a period of two years 2002-04.

1. Gave a course of lectures on Combinatorial Group Theory in the Visiting Students Summer Programme of the Institute during May-June 2002.
2. Beginning January 2003, a weekly Seminar on Group Theory has been
   initiated which is planned to run for 2003; gave a series of lectures on Commutator Calculus in this programme.
3. Was Chairman of the following Committees of the National Board for Higher Mathematics:
   1. University Programmes Committee (until July 2002).
   2. Committee for the award of scholarships for postgraduate studies in Mathematics; HRI was one of the centres for the written test in September 2002 and the venue for the interviews in October 2002.
4. Was Convener (with D. Jatkar) of the HRI Colloquium during 2002-03; this is a joint programme of the Schools of Mathematics and Physics.