Home page of Stephen Glasby
Stephen Glasby is Research Fellow at the University of Western Australia. He began his research in computational group theory developing algorithms for solvable groups (building on ideas by Kantor) but then turned to the matrix recognition project. Stephen first developed algorithms (with his coauthors Howlett, O’Brien, Leedham-Green) and did not analyze their complexity: for example writing representations over smaller fields (or writing projective representations modulo scalars over a smaller field). Now he develops algorithms and provides a rigorous complexity analysis for them. Recent such work involves the recognition of classical simple groups (joint with Praeger and Niemeyer). This required us to solve an interesting problem in “probabilistic geometry”: given a nondegenerate symplectic, unitary, or orthogonal space V and nondegenerate proper subspaces U and U’ with dim(U) + dim(U’) <= dim(V), what is the probability that the subspace U + U’ is nondegenerate and has dimension dim(U) + dim(U’)?
Stephen has published a series of papers that bound the number of composition factors of various groups in terms of various parameters. In a joint paper with Praeger, Rosa and Verret he proved that the number c(G) of composition factors of a primitive permutation group of degree n is at most (8/3)log_2(n) - 4/3 and that this bound is attained infinitely often. This paper and many others stem from problems he posed at the annual Research Retreat of the Center for the Mathematics of Symmetry and Computation (CMSC). Another result (with Giudici, Li, Verret) is remiscent of Legendre’s formula (a.k.a. Polignac’s formula): given a prime p, the largest k such that p^k divides n! is (n-s)/(p-1) where s is the sum of the “digits” of the p-adic expansion of n. Our result bounds the number c_p(G) of composition factors or order p of a completely reducible subgroup G of GL(n,p^f). We show that c_p(G) <= (c_p*n-s)/(p-1) where c_p is an explicit constant depending on p. Further, this result is best possible.
Stephen enjoys solving research problems, and working jointly with other mathematicians.