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TITLE: [GS_M_APP] Thom's gradient conjecture for nonlinear evolution equations

ABSTRACT: The Łojasiewicz–Simon inequality is a powerful tool for proving the uniqueness of blowups in equations with gradient structure. It is therefore natural to seek a general theory describing the next-order behavior of solutions near these limits. In joint work with P.-K. Hung, we characterize both the convergence rate and the dominant mode of convergence for nonlinear evolution equations originally studied by Simon. As a consequence, we give an affirmative answer to Thom’s gradient conjecture, concerning the alignment of secant lines for convergent gradient flows, in the context of geometric problems.