Pure Mathematics
Main areas of research activity are algebra, including group theory, semigroup theory, lattice theory, and computational group theory, and analysis, including fractal geometry, multifractal analysis, complex dynamical systems, Kleinian groups, and diophantine approximations.
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A note on the probability of generating alternating or symmetric groups
(201509)  Journal articleWe improve on recent estimates for the probability of generating the alternating and symmetric groups An and Sn. In particular, we find the sharp lower bound if the probability is given by a quadratic in n−1. This leads ... 
Lengths of words in transformation semigroups generated by digraphs
(20160808)  Journal articleGiven a simple digraph D on n vertices (with n≥2), there is a natural construction of a semigroup of transformations ⟨D⟩. For any edge (a, b) of D, let a→b be the idempotent of rank n−1 mapping a to b and fixing all vertices ... 
Idempotent rank in the endomorphism monoid of a nonuniform partition
(201602)  Journal articleWe calculate the rank and idempotent rank of the semigroup E(X,P) generated by the idempotents of the semigroup T(X,P), which consists of all transformations of the finite set X preserving a nonuniform partition P. We ... 
Ends of semigroups
(20160727)  Journal articleWe define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited ... 
Dimension conservation for selfsimilar sets and fractal percolation
(2015)  Journal articleWe introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic selfsimilar sets. For example, let K be a selfsimilar subset of ...