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Dept. of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI) has organised an colloquium on “Distinguished varieties in the polydisc and dilation of commuting contractions” by Dr. Sourav Pal, Associate Professor of Mathematics, IIT Bombay on 04 March 2024.

Title: Distinguished varieties in the polydisc and dilation of commuting contractions

Abstract

A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex algebraic variety that intersects $\mathbb D^n$ and exits the domain through the $n$-torus $\mathbb T^n$ without intersecting any other part of the topological boundary of $\mathbb D^n$. We show two different characterizations for a distinguished variety in the polydisc in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy on distinguished varieties in $\mathbb D^2$. We show that a distinguished variety in $\mathbb D^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb D^n$.