Key research themes
1. What are the combinatorial and incidence structures underlying Archimedean solids, specifically the rhombicosidodecahedron, and how can induced graphs reflect their properties?
This theme focuses on the combinatorial formalization of incidence relationships among vertices, edges, and faces in highly symmetric polyhedra. By axiomatizing local incidence structures inspired by the rhombicosidodecahedron, researchers can define nodal points with specific incidence counts and induce graphs reflecting shared region types. Understanding these structures provides insight into their geometric and graph-theoretic properties and offers frameworks to generalize to other uniform polyhedra.
2. How do the loci of triangle centers behave in Poncelet triangle families between conics, and under what conditions do these loci become conic (elliptic) curves?
Research in this area investigates the geometric properties of families of Poncelet triangles inscribed and circumscribed between two conics, especially the confocal pair of ellipses. A key question is which notable triangle centers sweep loci that are conics, typically ellipses, as the triangles vary, and which conditions or affine relationships characterize this behavior. Understanding these locus ellipticity phenomena sheds light on deep connections between classical geometry, complex analysis, and algebraic geometry.
3. Under what geometric and combinatorial conditions can polyhedra be inscribed or circumscribed to convex bodies in Euclidean spaces, and what are the existence results and counterexamples in various dimensions?
This theme examines the classical problem of inscribing or circumscribing polyhedra to convex bodies in Euclidean spaces of various dimensions. It addresses questions such as: Given a convex body in R^n, is there always a polyhedron with vertices on the boundary (inscribed) or one containing the body with each face touching the boundary (circumscribed)? The theme incorporates techniques from geometry, topology, and combinatorics to establish existence theorems, identify obstructions, and analyze special cases like squares inscribed in planar Jordan curves or cubes circumscribed around convex bodies. These fundamental results inform both geometric measure theory and combinatorial geometry.