3.00 p.m. PUBLIC VIVAVOCE NOTIFICATION  Problems related to Invariant theory of Torus and finite groups Santosha Kumar Pattanayak Chennai Mathematical Institute. 240611 Abstract The thesis studies two major problems in Invariant theory. One direction of our work takes is the study of projective normality of the polarized variety $(P(V)/G, \mathcal L)$, where $V$ is a finite dimensional representation of a finite group $G$ over a field $K$ and $\mathcal L$ is the descent of the line bundle $\mathcal O(1)^{\otimes G}$ ($\mathcal O(1)$ denotes the ample generator of the Picard group of $\mathbb P(V)$). Another direction of our work is to study the semistable points for a maximal torus action on the homogeneous space $G/P$, where $G$ is semisimple simply connected algebraic group and $P$ is a parabolic subgroup of $G$. Both studies arose out of an attempt to understand the quotient ${S_n}\backslash({T}\backslash \backslash G_{2,n})^{ss}$.
