Modular Arithmetic: binary quadratic forms, Gauss composition, Modular forms, Poincaré series,
continued fractions, intersections numbers and linking numbers of modular geodesics.
Geometric Group Theory and Representation Theory: hyperbolic groups, Fuchsian groups, mapping class groups,
their moduli spaces, character varieties, compactification by actions on trees, bounded cohomology.
Arithmetic and Symbolic Dynamics: Diophantine approximation, continued fractions, interval exchanges,
translation surfaces, renormalisation, subshifts of low complexity and S-adic representations.
Topology of 3-manifolds: Surgery and cobordism distances between 3-manifolds, Gordian space of knots,
linking forms and Blanch field pairings, signatures and characteristic classes.
Singularity theory: topological classification of singular curves in surfaces (real or complex, algebraic or analytic),
using chord diagrams, linking forms, Witt-rings, cluster algebras.
Algebraic and Enumerative Combinatorics : graphs in surfaces, chord diagrams, their polynomial invariants and Hopf algebras.
Enumeration using symbolic methods (generating series) followed by analytic methods (singularity analysis).
Other areas of interests : all the rest, especially Dynamics, Game theory, Complexity theory, Persistence homology and Physics.