This involves the study of face numbers and flag numbers of polytopes and complexes. Connections with commutative algebra and with intersection homology of toric varieties, relations with polarity and symmetries and further applications in polytope theory are also studied. I am interested in using Algebraic shifting (generic initial ideals) both in the exterior algebra and commutative algebra contexts,

Research focuses on the study of bounds on the diameter of graphs of polyhedra and on the performance of randomized pivot rules.

A study is made of sharp transition results for monotone properties under symmetry, related discrete isoperimetric results, relations with random graph theory, percolation, first passage percolation, extremal combinatorics and harmonic analysis.