In the analysis of singularities, uniqueness of limits often arises as an important question: that is, whether the geometry depends on the scales one takes to approach the singularity. In his seminal work, Simon demonstrated that Lojasiewicz inequalities, originally known in real algebraic geometry in finite dimensions, can be applied to show uniqueness of limits in geometric analysis in infinite dimensional settings. We will discuss some instances of this very successful technique and its applications.

In this talk, we address a question whether a mean-field approach for a large particle system is always a good approximation for a large particle system or not. For definiteness, we consider an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University).

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KAI-X Distinguished lecture