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Quantitative unique continuation is a fundamental property of partial differential equations, which asserts that the smallness of a solution at one scale quantitatively controls its behavior at larger scales. It plays an important role in the study of nodal sets and in problems from control theory. While Yau's conjecture on nodal set remains open in full generality, there has been significant progress in recent years. In this talk, I will present several new results on the nodal sets of elliptic and parabolic equations, which extend Yau’s conjecture to solutions with Gevrey regularity. Subsequently, I will explore applications of quantitative unique continuation to observability inequalities in control theory, which assert that global behavior can be determined from partial information.