I have been too busy for about 8 years and the blog has been dormant, though I was often pondering agenda on various things I wanted to write about, plus finishing posts on what I promised in 2010 (and what I am still interested in).

For few years, one of focal points of my interest were certain completed Hopf algebroids over a noncommutative base which are sort of Heisenberg doubles of universal enveloping algebras of Hopf algebras. This Spring I have somewhat accidentally found one application in ordinary differential equations, which I suspect might also be of interest in study of certain formulas in the subject of renormalization of QFTs. am a bit puzzled with my own result, I have a feeling that one could get it without Hopf algebroids and that it must be somehow known in some form. That means I should discuss it with community, including in posts here. I hope this will be very soon.

Another of my more recent interests is related to strong shape theory. Jacob Lurie has written in his Higher Topos Theory book about the (infinity,1)-topos point of view on strong shape. In fact, we owe this perspective to much earlier work of Guenther from early 1990s, influenced by his PhD advisor, topologist Friedrich Bauer from Frankfurt. Šime Ungar wrote about a version of Blakers-Massey theorem in the setup of ordinary shape theory. For strong shape it seems there is no major result yet. Regarding that there is a Blakers-Massey result for infinity-topoi, it is wise to try interpreting it in the case of strong shape.

Most recently, and lead by some applied problems, I started being interested in the study of balance of incentives concerning agents in a generalized ecosystem. This is more general than what game theory teaches. Ecosystem is for me a system which involves units with some level of autonomous behaviour. In game theory, such units are (not called actors but) players and they have preferences, possibly multiple, which they want to satisfy. They also have some strategy. You want to find optimal strategy for some player when some model is given for the strategies of others. In modern society, organizations, on the internet and so on, the strategies of the rest of the world change continuously, somewhat stochastically and our information on them is only partial. What actions and interactions can we propose to learn more about such a world ? How to measure from inside the model of the game ? This is clearly more general than the game theory, and it is not only a theory, but an engineering concept. Second, managers of organizatons are interested in designing frameworks for ecosystems where some goals will be satisfied by locking the balance of behaviours of agents. For this the agents need to get certain incentives. This is very different from what is called business intelligence. BI is static, rule based and given by a company. Here we want to see the emerging logics, of course, one creates certain preconditions but they have very different character than BI. An interesting example of different kind is conceiving the protocols for a blockchain technology so that the ecosystem balance will work well. The very principles of blockchain in great part work because consensus and other protocols count on economical incentives of the participants. But there are secondary phenomena of the growth of the system, appearance of new intermediaries, pools, resource usage and so on, which require further development of algorithms and understanding of the wider context as well.