Lennard's research interests include these topics:
- Banach space geometry and metric fixed point theory. He works mainly with Paddy Dowling and Barry Turett, trying to understand which Banach spaces support fixed-point-free non-expansive mappings on small sets (e.g., those that are weakly compact and convex).
- Convergence properties in Banach spaces. The uniform Kadec-Klee property is an analogue of uniform convexity that many classical nonreflexive spaces enjoy.
- Banach and Hilbert frames. Frames are non-linearly independent analogues of bases in Banach spaces that have many applications (e.g., in signal processing).
- Roundness and metric type. The notions of roundness, generalized roundness, and metric type are related to the isometric embedding of metric spaces into Hilbert and Banach spaces and to the classification of Banach spaces via uniform homeomorphisms.