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Explore advanced techniques in arithmetic dynamics through this mathematical seminar that examines the application of ultrafilters to study degenerating families of rational maps. Learn how to construct non-archimedean limits for one-parameter degenerating families of rational maps on the projective line, and discover Luo's innovative use of ultrafilters to build limits for arbitrary sequences of rational maps. Understand the formalization of this approach by Favre and Gong using Berkovich spaces, and see how their theory enables the derivation of uniformity results in arithmetic dynamics. Examine the proof that the number of common preperiodic points of two distinct monic degree d polynomials is uniformly bounded in terms of d, which establishes a conjecture of DeMarco-Krieger-Ye specifically for monic polynomials. Gain insights into the intersection of algebraic dynamics, non-archimedean geometry, and ultrafilter theory through this presentation delivered at the Joint IAS/Princeton University Groups and Dynamics Seminar by Jit Wu Yap from MIT.
