Matrices are one of the most fundamental and useful objects in Mathematics. Set of matrices comes equipped with addition and multiplication laws and is called a matrix algebra. Central simple algebras are a generalisation of matrix algebras and are closely related to the theory of quadratic forms, hermitian forms and linear algebraic groups.
My research interests include the study of algebraic groups via central simple algebras. Using algebraic techniques, I have proved that classical algebraic groups over many fields, for example number fields, have trivial R-equivalence class only. This study has consequences in the study of rationality properties of varieties of algebraic groups and certain norm principles for quadratic forms.
Isotropy of involutions of central simple algebras is another difficult question to answer in general. I have contributed by giving weak isotropy criterion for a class of central simple algebras. This uses the newly developed notions of gauges.
I have also worked on problems concerning ramifications of central simple algebras over rational function fields and "subfields determining quaternion algebras" problem and have got some interesting partial results. The tools I use in my research are algebraic; e.g. Galois cohomology, valuations, classifications of quadratic forms etc.
After joing IISER I have also become interested in group theoretic computations using mathemtical softwares such as GAP, which I have used to find an interesting counterexample of an ortho-ambivalent group which is not strongly real.
- Alexey Galt, Amit Kulshrestha, Anupam Singh, Evgeny Vdovin, On Shalev's conjecture for type An and 2An, Journal of Group Theory 22 (2019), pp. 713-728.
- Dilpreet Kaur and Amit Kulshrestha, Quadratic forms in characteristic 2 and a Wedderburn decomposition over rationals, Journal of Algebra and Its Applications 18 (2019), 1950010.
- Dilpreet Kaur and Amit Kulshrestha, Strongly real special 2-groups, Communications in Algebra 43 (2015), pp. 1176-1193.
- Amit Kulshrestha, Strongly anisotropic involutions on central simple algebras, Communications in Algebra 39 (2011), pp. 1686-1704.
- Amit Kulshrestha and Raman Parimala, R-equivalence in adjoint classical groups over fields of virtual cohomological dimension 2, Transactions of the American Mathematical Society 360 (2008), pp. 1193-1221.