Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Conformal mating by Schwarz reflection map of the deltoid
Speaker: Rashmita Hore, TIFR, Mumbai
Date: October 21, 2024
Time: 15:15:00 Hours
Venue: A-369
Abstract: Inspired by the Fatou-Sullivan dictionary between Kleinian groups and complex dynamics, it is natural to think of iterations of anti-holomorphic rational maps on the Riemann sphere as the complex dynamics counterpart of actions of Kleinian reflection groups. Schwarz reflections associated with (disjoint unions of) quadrature domains generate a class of anti-holomorphic dynamical systems. In certain cases, such systems provide a framework for conformally combining (mating) the dynamics of rational maps and that of reflection groups in the same dynamical plane. In this talk, we will demonstrate conformal matings between quadratic anti-holomorphic polynomials and reflection groups using Schwarz reflection maps, for the particular example of the deltoid reflection. The talk will be based on the following paper: https://smf.emath.fr/publications/dynamique-des-reflexions-de-schwarz-un-phenomene-daccouplement
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Period-index problem for hyperelliptic curves.
Speaker: Saurabh Gosavi, TIFR, Mumbai
Date: October 21, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Let $F$ be a field and $\alpha$ be in the Brauer group $Br(F)$. The period of $\alpha$ is its order in $Br(F)$, and the index of $\alpha$ is the square root of the dimension of the (unique) underlying division algebra of $\alpha$. It is well-known that the period divides the index and that they share the same prime factors, and so there exists $N(\alpha) \geq 1$ such that $ind(\alpha)$ divides $per(\alpha)^{N(\alpha)}$. When $F$ is the function field of a curve over a number-field, it is expected that there should exist a uniformly bound for the exponent $N(\alpha)$, i.e., there should exist an integer $N \geq 1$ such that for every $\alpha$ in $Br(F)$, one has $ind(\alpha)$ divides $per(\alpha)^{N}$. This is still an open problem. In this preprint, R. Parimala and J. Iyer show that when $C$ is a hyperelliptic curve over a totally imaginary number field and $\alpha$ is a period two Brauer class that is locally trivial, i.e., $\alpha$ becomes trivial when restricted to every completion of $F$, index of $\alpha$ also equals two. Following M. Lieblich, they prove this by showing that there is a rational point in the moduli space of stable $\alpha$-twisted locally free sheaves of rank two on $C$. Based on https://arxiv.org/pdf/2201.12780
Title of Talk: Conformal mating by Schwarz reflection map of the deltoid
Speaker: Rashmita Hore, TIFR, Mumbai
Date: October 21, 2024
Time: 15:15:00 Hours
Venue: A-369
Abstract: Inspired by the Fatou-Sullivan dictionary between Kleinian groups and complex dynamics, it is natural to think of iterations of anti-holomorphic rational maps on the Riemann sphere as the complex dynamics counterpart of actions of Kleinian reflection groups. Schwarz reflections associated with (disjoint unions of) quadrature domains generate a class of anti-holomorphic dynamical systems. In certain cases, such systems provide a framework for conformally combining (mating) the dynamics of rational maps and that of reflection groups in the same dynamical plane. In this talk, we will demonstrate conformal matings between quadratic anti-holomorphic polynomials and reflection groups using Schwarz reflection maps, for the particular example of the deltoid reflection. The talk will be based on the following paper: https://smf.emath.fr/publications/dynamique-des-reflexions-de-schwarz-un-phenomene-daccouplement
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Period-index problem for hyperelliptic curves.
Speaker: Saurabh Gosavi, TIFR, Mumbai
Date: October 21, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: Let $F$ be a field and $\alpha$ be in the Brauer group $Br(F)$. The period of $\alpha$ is its order in $Br(F)$, and the index of $\alpha$ is the square root of the dimension of the (unique) underlying division algebra of $\alpha$. It is well-known that the period divides the index and that they share the same prime factors, and so there exists $N(\alpha) \geq 1$ such that $ind(\alpha)$ divides $per(\alpha)^{N(\alpha)}$. When $F$ is the function field of a curve over a number-field, it is expected that there should exist a uniformly bound for the exponent $N(\alpha)$, i.e., there should exist an integer $N \geq 1$ such that for every $\alpha$ in $Br(F)$, one has $ind(\alpha)$ divides $per(\alpha)^{N}$. This is still an open problem. In this preprint, R. Parimala and J. Iyer show that when $C$ is a hyperelliptic curve over a totally imaginary number field and $\alpha$ is a period two Brauer class that is locally trivial, i.e., $\alpha$ becomes trivial when restricted to every completion of $F$, index of $\alpha$ also equals two. Following M. Lieblich, they prove this by showing that there is a rational point in the moduli space of stable $\alpha$-twisted locally free sheaves of rank two on $C$. Based on https://arxiv.org/pdf/2201.12780