PUBLIC VIVAVOCE NOTIFICATION 2.30 pm, http://meet.google.com/bujvznbcba Automorphism groups of Schubert varieties and rigidity of BottSamelsonDemazureHansen varieties Pinakinath Saha Chennai Mathematical Institute. 250620 Abstract This thesis consists of three problems. Throughout this thesis we consider $G$ to be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ to be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $G/B$ be the full flag variety consisting of all Borel subgroups of $G.$ For $w$ in $W,$ let $X(w)$ denote the Schubert variety in $G/B$ corresponding to $w.$ In the first problem, we show that given any parabolic subgroup $P$ of $G$ containing $B$ properly, there is an element $w$ in $W$ such that $P=Aut^0(X(w)).$ In the second problem, we consider $G =PSO(2n+1,\mathbb{C})(n \ge 3)$. Let $w$ be an element of Weyl group $W$ and $\underline{i}$ be a reduced expression of $w.$ Let $Z(w, \underline{i})$ be the BottSamelsonDemazureHansen variety (the desingularization of $X(w)$ corresponding to $\underline{i}$). Here we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i})$ vanish. In the third problem, we assume that the root system of $G$ is of type $F_{4}.$ In this problem, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that $Z(w_{0}, \underline{i})$ is rigid. Further, if $G$ is of type $G_{2},$ there is no reduced expression $\underline{i}$ of $w_{0}$ for which $Z(w_{0}, \underline{i})$ is rigid.
