Publications in Google Scholar
Curriculum Vitae in PDF
I am currently interested in interactions between theoretical computer science and, on the one hand, self-similar behaviour in group theory and algebra, and on the other hand its applications to complex and symbolic dynamical systems.
Self-similar, or fractal
, objects abound in mathematics; depending on context, they mean a space containing several almost disjoint copies of itself as subspaces; a group containing the direct product of copies of itself as a subgroup; or an algebra containing a matrix algebra over itself as a subalgebra. The fractalness
is algebraically encoded via the collection of inclusion maps of these subobjects in their common parent.
A self-similar group may be associated with any complex dynamical system, and yields an extremely potent algebraic invariant of that dynamical system up to isotopy and conjugation. I currently explore more deeply the connection between complex dynamics and fractal groups, and use it to extend the classification of degree-two polynomials (described by points in the Mandelbrot set) to arbitrary-degree rational functions.
All these questions immediately raise decidability and complexity questions: one the one hand, one would like an understanding of which problems are undecidable, usually in terms of their ability to simulate arbitrary Turing machines, and how high in the complexity hierarchy are the decidable problems; on the other hand, not only theoretical study, but also practical implementation of algorithms is fundamental in exploration and experimentation.
- The FR GAP package, for manipulation of groups and semigroups generated by automata
- The IMG GAP package, for manipulation of iterated monodromy groups and complex dynamics
- The Float GAP package, for manipulation of real, complex, interval etc. floating-point numbers
- The anq system, a C++ implementation of a quotient algorithm for groups, Lie algebras and associative algebras
How I Rank in Hell
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