目：Order of Oscillating Sequences, MMA-MMLS, and Sarnak's Conjecture

间：2018.07.17（星期二）, 16:00-17:00

点：数学院南楼N913

要：In this talk, I will explain several concepts, a log-uniformly oscillation sequence, an oscillation sequence, an oscillation sequence of higher order, a minimal mean attractable (MMA) dynamical system, a minimal mean-L-stable (MMLS) dynamical system. Equicontinuous dynamical systems are clearly MLS. Feigenbaum dynamical systems are not equicontinuous globally but when they are restricted on minimal sets still equicontinuous. Furthermore, in this talk I will give two non-trivial examples of dynamical systems which are not equicontinuous even when they are restricted on minimal sets but MMLS. We will prove that any oscillating sequence is linearly disjoint with all MMA and MMLS dynamical systems. One of the consequences is that Sarnak’s conjecture holds for all MMA and MMLS dynamical systems. There are dynamical systems which are not MMLS. Therefore, we need to use the concept of an oscillation sequence of higher order. The Mobius sequence is an example of an oscillation sequence of higher order due to a result of Hua. In this talk, I will give another interesting example of an oscillation sequence of higher order. Furthermore, I will prove that any oscillation sequence of order $d\geq 2$ is linearly disjoint with all affine distal maps of the $d$-torus.

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

目：Expendability of Holomorphic Motions

间：2018.07.06（星期五）, 16:00-17:00

点：数学院南楼N913

要：In this talk, I will give a review of our work in the study of holomorphic motions of subsets in the Riemann sphere over hyperbolic Riemann surfaces.  I will discuss several conditions in the extension problem like the zero-winding number condition, the trivial trace-monodromy condition, the trivial monodromy condition, and the guiding isotopy condition. I will show that all these conditions are necessary in the study of the extension problem. I will use several counter-examples to show that the zero-winding number condition and the trivial trace-monodromy condition are not sufficient. Finally, I will prove that the trivial monodromy condition and the guiding isotopy condition are, indeed, sufficient.  In this talk, I will also explain how the study of the lifting problem, which used to be called as one of the most important problems in Teichmueller theory by Bers and Royden, plays an important role in the study of the extension problem in holomorphic motions.