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Explore the emerging connections between semiclassical analysis and noncommutative geometry in this BIMSA Colloquium lecture. Learn how semiclassical Weyl's laws extend to a very large class of quantum spaces, building upon and simplifying recent results by McDonald-Sukochev-Zanin. Discover applications of these theoretical developments in contact geometry and sub-Riemannian geometry, including their relevance to Schrödinger operators arising from Grushin sub-Laplacians. Gain insights into how these two active fields within quantum theory are beginning to bridge together, opening new avenues for mathematical research and understanding of quantum spaces.
