Satya Deo

Research Summary:

My main interest has been the cohomological study of the group actions on finitistic spaces. These include various generalizations of Smith theorems on the cohomology structure of fixed point sets and Floyd's theorem on acyclicity of orbit spaces. Jointly with J.K.Maitra, we had earlier obtained results as to which groups can act freely on product of spheres.We are now looking at its possible generalizations.

We have also been studying the properties of spline modules on various polyhedral complexes embedded in the plane, or more generally, in any Euclidean space $R^n$. In this area, we have proved that under suitable conditions any basis for the spline module on a divided domain can always be extended to a basis when we take a subdivision of that domain. This result seems to be useful in approximation theory.

Jointly with R.Kulakarni, we have studied the following problem using methods of covering projections: Given a group, deteremine the lattice of all subgroups of that group; determine the set of conjugacy classes of subgroups of that group; determine the index of subgroups in their normalizers. We have solved the problem in a few interesting cases, e.g., for free groups on two or three generators, dihedral groups etc.

Publications:

Algebraic Topology, a Primer; Hindustan Book Agency, TRIM series Number 27, New Delhi, 2003.

Preprints:

1. Spline modules from a divided domain to a subdivided domain, (jointly with J.K.Maitra).
2. Branched coverings, Labelled coverings, and Transitive permutation groups (jointly with R.A.Kulkarni), in preparation.

1. Annual Conference on History of Mathematics, Raipur University, Raipur, Chhatisgarh, Nov 2003.
2. Annual conference of the International Academy of Physical Sciences, Gorakhpur University, Gorakhpur, Dec 2003.
3. AMS-India Joint meeting in Bangalore, Dec 2003.

1. Two lectures on ``Cohomological Dimension theory'' at the Indian Institute of Science, Bangalore.
2. Role of mathematics in computer science, popular talk, during the conference organised by SPM College, Allahabad.
3. Trends in the teaching and research in mathematics, Inaugural lecture, Refresher Course in Mathematics, University of Allahabad.

1. Appointed Chairman of several National Assessment and Accreditation Council (NAAC) Peer Teams for the Assessment of Degree Colleges of West Bengal, Maharashtra and Harayana states.
2. Member, National Board for Higher Mathematics (NBHM) serving on several subcommittees of NBHM.
3. Member, Visiting Committee of the Distance Education Council, New Delhi to assess UP Rajarshi Tandon Open University, Allahabad.

1. Gave a semester-course on Homotopy Theory for IInd Year students.
2. Gave a semester- course on Topology to First Year, IInd semester students.
3. Organised Colloquium Talks by Prof Norbert Schlomiuk and Dana Schlomiuk of the University of Montreal, Canada.