A fundamental and widely used mathematical fact states that the arithmetic mean of a collection of non-negative real numbers is at least as large as its geometric mean. This is the most basic example of a large family of inequalities between symmetric functions that have attracted the interest of combinatorialists in recent years. This talk will present recent joint work with Jon Novak at UC San Diego, which unifies many such inequalities as corollaries of a fundamental monotonicity property of spherical functions on symmetric spaces. We will also discuss conjectural extensions of these results to even more general objects such as Heckman-Opdam hypergeometric functions and Macdonald polynomials.

The talk will be accessible to a broad mathematical audience and will not assume any knowledge of symmetric spaces or symmetric functions. However, the second half of the talk will assume familiarity with basic constructions of Lie theory, such as root systems and the Iwasawa decomposition. Details of the relevant work can be found in this pre-print.

The video of this talk is available on the IISc Math Department channel.

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Last updated: 03 Feb 2023