Applications of Grothendieck-Riemann-Roch theorem for stacks and stringy Chow ring of weighted blow-ups

In this thesis we prove three results. Firstly we apply the Grothendieck-Riemann-Roch theorem for stacks to root stacks to rederive the formula for parabolic bundles. Next we apply the same theorem to a quotient stack to derive a formula for equivariant Euler characteristic. When the quotient is obtained by an action on a smooth projective curve, we explicitly compute the Euler characteristic in terms of ramification data. This agrees with many previous results with different levels of generalities, thereby providing a unified way to prove the result in these settings. Lastly, we study the stringy Chow ring structure of weighted blow-ups with regular centres. We completely determine the ring structure and answering several questions regarding its finite-generation.