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We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

We provide a family of counterexamples to a first formulation of the dynamical Manin–Mumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of Abelian varieties under the action of group endomorphisms and for lines under the action of diagonal endomorphisms of $\mathbb{P}^1 \times \mathbb{ P}^1$.

It is a great honor for me to be invited by Tsinghua University to talk today. Tsinghua has, in its history, made many contributions to mathematics. The two most famous mathematicians in modern Chinese history have both been closely related to Tsinghua University. One is Professor Hua Lo-Keng, the other is Prof essor Chern Shiing-Sheng. Especially since Professor Chern is my teacher, I feel proud to be able to say some words in the opening ceremony for the Center for Advanced Study. I chose the title of my talk also for this reason. Due to my critical comments, I have decided to spend most of my time in giving a general talk in English. The remainder of the talk will be in Chinese.