Anupam Kumar Singh

Research Summary:

I have been working with Dr. Maneesh Thakur on problems related to the exceptional group of type G₂ over arbitrary fields. We completed our investigation of reality properties in G₂. An element is called real if and only if it is conjugate to its inverse. Given a group of type G₂ over a field k, one can realize it as the automorphism group of an octonion algebra over k. We were successful in determining the cases when reality holds and found counterexamples in other cases.

Motivated by some partial results in the direction of determining conjugacy classes and a question asked by Prof. R. S. Kulkarni to determine z-conjugacy classes, I (with Dr. M. Thakur) have been exploring anisotropic G₂ over arbitrary field k. Let G be an automorphism group of some octonion division algebra, which is an anisotropic group of type G₂. Two elements in G are called z-conjugate if and only if their centralizers are conjugate. We determine z-conjugacy classes of G and further, we have made computations of conjugacy classes in groups of type G₂.

I am investigating the "Reality" question in spin groups and other exceptional groups as well.

Preprints:

1. Reality properties of conjugacy classes in G₂ (submitted).
2. onjugacy classes in anisotropic G₂ (in preparation).

Conference/Workshops Attended:

1. International Conference on Algebra and Number Theory (11-16 December 2003) in University of Hyderabad.

2. Joint India-AMS mathematics meeting (17-20 December 2003) in IISc Bangalore.

3. Workshop on Linear Algebraic Groups, Quadratic Forms And Related Topics (1-5 February 2004) in Eilat organisd by European Research and Training Network "Algebraic K-theory, Linear Algebraic Groups and Related Structures" (HPRN--CT--2002--00287).

1. I helped in the tutorial sessions in VSSP-2003.
2. delivered two introductory lectures on Representation theory of finite groups in Basic Notions Seminar in January 2004.