The encounter between "barbarian" conquerors with sedentary peoples possessing sophisticated cultural and political traditions is one of profound historical importance. The interaction has resulted in great cultural, religious, political, linguistic and demographic changes, in which inter alia whole previously distinct groups can disappear, not so much through physical destruction, but rather through assimilation and absorption. One such meeting of enormous dimensions was that of the Roman world with the various Germanic invaders. Another would be that of the Byzantine and Persian territories overrun by the Arab Muslim armies of the 7th century. While there is still much debate among historians about the exact nature of these encounters, there is no doubt that the resulting influence was not in one direction, but both sides were greatly affected by this experience. It is also clear that these meetings left an indelible impact on the further development of these two regions.

A different set of encounters is that of the nomadic peoples of the Eurasian steppe with their sedentary neighbours in the later Middle Ages, i.e. the Turkish and Mongol invasions of the Middle East in the 11th-14th centuries and the Khitan, Jurchen and Mongol invasions of China in the late 10th to mid-14th centuries. In the aftermath of all these instances, nomadic elites established long-term control over large swathes of the territory of sedentary society. Our research group seeks to examine the effects of this encounter in a comparative way, diachronically in the same territory and synchronically between the Islamic Middle East and China.

Infinite groups:
- Branch groups and automata groups, their subgroups, representations, presentations, and subgroup growth
- Zeta functions of nilpotent groups
- Rigid groups
- Redidual properties of the modular group

Finite groups:
- Asymptotic aspects of finite simple groups, and probabilistic aspects in particular
- Generation, and random generation, of finite simple groups
- Algorithms for matrix groups