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We give a method to bound the entropy of measures on SL(d,R)/SL(d,Z) which are invariant under one parameter diagonal subgroups, in terms of entropy contributions from regions of the cusp corresponding to different parabolic groups. These bounds depend on an auxiliary linear functional on the Lie algebra of the Cartan group, which can then be optimized depending on the bound one is trying to obtain. We discuss how this method is utilized to obtain new sharp bounds for cusp entropies.

 

The Khovanov–Rozansky homology categorifies the classical Jones and HOMFLY-PT polynomials. In this talk, we will explore how the Khovanov-Rozansky homology of the (m, n)-torus knot can be derived from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. This result confirms a conjecture by Gorsky, Oblomkov, Rasmussen, and Shende. Our approach involves the geometry of Hilbert schemes of points and character D-modules. Numerous examples will be provided to introduce and clarify the main concepts.