Title of Seminar: Algebraic Geometry Preprint Seminar 2023
Title of Talk: Full exceptional collections of vector bundles on rank 2 linear GIT quotients
Speaker: Jagadish Pine, TIFR, Mumbai
Date: October 10, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Beilinson proved that the derived category of coherent sheaves $D^b_{coh}(\mathbb{P}^N)$ admits a full exceptional collection of line bundles. Viewing $\mathbb{P}^N$ as a GIT quotient $\mathbb{C}^{N+1}//G_m$, one can pose the general question: Given a split reductive group $G$, and a linear representation $X$, when does a GIT quotient $X^{ss}//G$ admit a full exceptional collection of vector bundles? Since the variety $X^{ss}//G$ might have singularities, one must work with quotient stacks $[X^{ss}/G]$. When the rank of $G$ is $2$, under certain assumptions, $D^b_{coh}[X^{ss}/G]$ will provide a positive answer to this question. Using this result as building blocks, one can construct many examples that admit a full exceptional collection of vector bundles using quiver varieties. The talk is based on the arxiv preprint by Halpern-Leistner and Kimoi Kemboi titled "Full exceptional collections of vector bundles on rank $2$ linear GIT quotients".
Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Proper affine actions of hyperbolic groups
Speaker: Sourav Ghosh, Ashoka University
Date: October 10, 2023
Time: 16:30:00 Hours
Venue: A-369
Abstract: Classification of crystallographic groups by Bieberbach gave rise to the Auslander conjecture which states that any affine crystallographic group is virtually polycyclic. The conjecture is known to be true in lower dimensions but is still open in the general case. However, if one eases the assumption of cocompactness, then Margulis in a celebrated work showed that the new conjecture fails to hold. He showed that non-abelian free groups can act properly as affine transformations. In this talk, I will give an overview of the history and present some recent developments.
Title of Talk: Full exceptional collections of vector bundles on rank 2 linear GIT quotients
Speaker: Jagadish Pine, TIFR, Mumbai
Date: October 10, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Beilinson proved that the derived category of coherent sheaves $D^b_{coh}(\mathbb{P}^N)$ admits a full exceptional collection of line bundles. Viewing $\mathbb{P}^N$ as a GIT quotient $\mathbb{C}^{N+1}//G_m$, one can pose the general question: Given a split reductive group $G$, and a linear representation $X$, when does a GIT quotient $X^{ss}//G$ admit a full exceptional collection of vector bundles? Since the variety $X^{ss}//G$ might have singularities, one must work with quotient stacks $[X^{ss}/G]$. When the rank of $G$ is $2$, under certain assumptions, $D^b_{coh}[X^{ss}/G]$ will provide a positive answer to this question. Using this result as building blocks, one can construct many examples that admit a full exceptional collection of vector bundles using quiver varieties. The talk is based on the arxiv preprint by Halpern-Leistner and Kimoi Kemboi titled "Full exceptional collections of vector bundles on rank $2$ linear GIT quotients".
https://arxiv.org/abs/2202.12876
Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Proper affine actions of hyperbolic groups
Speaker: Sourav Ghosh, Ashoka University
Date: October 10, 2023
Time: 16:30:00 Hours
Venue: A-369
Abstract: Classification of crystallographic groups by Bieberbach gave rise to the Auslander conjecture which states that any affine crystallographic group is virtually polycyclic. The conjecture is known to be true in lower dimensions but is still open in the general case. However, if one eases the assumption of cocompactness, then Margulis in a celebrated work showed that the new conjecture fails to hold. He showed that non-abelian free groups can act properly as affine transformations. In this talk, I will give an overview of the history and present some recent developments.