The classical p-typical Witt vectors were contrived by Teichm?ller and Witt to build unramified extensions of the field of p-adic numbers from their residue fields in a functorial way. Dress and Siebeneicher introduced a fascinating generalization of them called "Witt-Burnside rings" in a group-theoretical way. In this talk, we will briefly review the basic theory of Witt vectors and Witt-Burnside rings. Recent developments in this area, in particular, some open problems concerned with Witt vector construction will be also dealt with.

Nonlinear wave equations have solutions with various types of behavior, such as dispersive waves, solitary waves (solitons), and blow-up in finite time. Heuristically, they can be distinguished by which is stronger on each solution, the dispersive effect or the nonlinear one. Rigorous analysis of the dynamics has been well developed in small neighborhoods around special solutions, typically the trivial one and some solitons, where all solutions exhibit the same behavior. However, rather little is known about the dynamics away from such neighborhoods: if and how different types of solutions can coexist or some solutions can change their behavior along time, etc. Numerical studies suggested that in some cases the two sets of solutions in stable regimes (dispersive waves and stable blow-up) are separated by a hypersurface of the third set of solutions which are unstable. Similar phenomena are well known for nonlinear diffusion equations, but they can be easily understood by the comparison principle, which does not apply to wave equations. In the joint work started with Wilhelm Schlag, we have rigorously obtained such a trichotomy in some simple settings such as the nonlinear Schrodinger and Klein-Gordon equations with unstable ground states, under some energy constraint. I will explain how we can construct the threshold hypersurface, describe the dynamics off and on the hypersurface, capture the stable transition between dispersion and blow-up, and thereby predict global behavior of solutions from the initial data. I will also discuss about open questions.

We aim to review the efforts for systematic and organized global collaborations in mathematics starting from late 19th century. We also summarize the activities in Korean math research community that are sometimes sporadic but are becoming increasingly organized and systematic.