Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Naive A^1-connectedness of retract rational varieties
Speaker: Biman Roy, TIFR, Mumbai
Date: October 14, 2024
Time: 15:15:00 Hours
Venue: AG-77
Abstract: In $\mathbb{A}^1$-homotopy theory, introduced by Morel-Voevodsky, $\mathbb{A}^1$-connected component $\pi_0^{\mathbb{A}^1}(X)$ of a smooth variety $X$ plays the role of topological connected component of a topological space. A crude approximation of $\pi_0^{\mathbb{A}^1}(X)$ is the $\mathbb{A}^1$-chain connected component sheaf $\mathcal{S}(X)$ which involves the naive $\mathbb{A}^1$-homotopies. Asok-Morel, Balwe-Hogadi-Sawant showed that a smooth proper variety over a field $k$ is $\mathbb{A}^1$-connected if and only if it is $\mathbb{A}^1$-chain connected i.e. $\mathcal{S}(X)(Spec F)$ is trivial, for every finitely generated separable field extension $F/k$. Asok-Morel, Kahn-Sujatha proved that a smooth retract rational proper variety over a field $k$ is $\mathbb{A}^1$-chain connected. In a recent work Balwe-Rani improved this result by showing that a smooth retract rational proper variety over an infinite field $k$ is naively $\mathbb{A}^1$-connected i.e. $\mathcal{S}(X)$ is the trivial sheaf. This talk will be based on the article https://doi.org/10.48550/arXiv.2307.04371 by Chetan Balwe and Bandana Rani.
Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Combination Theorems in Complex Dynamics
Speaker: S. Viswanathan
Date: October 14, 2024
Time: 15:00:00 Hours
Venue: A-369
Abstract: In this talk we will see a few instances of combining (or mating) complex dynamical systems. Such theorems are used to (a) prove certain facts about the iterating functions, and (b) extract some geometric insights on the parameter spaces of these iterating functions. If time permits, we will introduce algebraic correspondences, and see why they provide the right framework for matings. The talk will be based on the following paper: https://ems.press/journals/emss/articles/13258390
Title of Talk: Naive A^1-connectedness of retract rational varieties
Speaker: Biman Roy, TIFR, Mumbai
Date: October 14, 2024
Time: 15:15:00 Hours
Venue: AG-77
Abstract: In $\mathbb{A}^1$-homotopy theory, introduced by Morel-Voevodsky, $\mathbb{A}^1$-connected component $\pi_0^{\mathbb{A}^1}(X)$ of a smooth variety $X$ plays the role of topological connected component of a topological space. A crude approximation of $\pi_0^{\mathbb{A}^1}(X)$ is the $\mathbb{A}^1$-chain connected component sheaf $\mathcal{S}(X)$ which involves the naive $\mathbb{A}^1$-homotopies. Asok-Morel, Balwe-Hogadi-Sawant showed that a smooth proper variety over a field $k$ is $\mathbb{A}^1$-connected if and only if it is $\mathbb{A}^1$-chain connected i.e. $\mathcal{S}(X)(Spec F)$ is trivial, for every finitely generated separable field extension $F/k$. Asok-Morel, Kahn-Sujatha proved that a smooth retract rational proper variety over a field $k$ is $\mathbb{A}^1$-chain connected. In a recent work Balwe-Rani improved this result by showing that a smooth retract rational proper variety over an infinite field $k$ is naively $\mathbb{A}^1$-connected i.e. $\mathcal{S}(X)$ is the trivial sheaf. This talk will be based on the article https://doi.org/10.48550/arXiv.2307.04371 by Chetan Balwe and Bandana Rani.
Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Combination Theorems in Complex Dynamics
Speaker: S. Viswanathan
Date: October 14, 2024
Time: 15:00:00 Hours
Venue: A-369
Abstract: In this talk we will see a few instances of combining (or mating) complex dynamical systems. Such theorems are used to (a) prove certain facts about the iterating functions, and (b) extract some geometric insights on the parameter spaces of these iterating functions. If time permits, we will introduce algebraic correspondences, and see why they provide the right framework for matings. The talk will be based on the following paper: https://ems.press/journals/emss/articles/13258390