Overview Research Statement

Most of my research is situated at the interface of algebraic combinatorics, algebraic geometry, representation theory, and algebraic topology. A unifying theme of this research is the emphasis on combinatorics and computation. My work is part of an ongoing effort to perform concrete computations, to understand concrete spaces with combinatorial precision, and to build on the growing number of interactions between combinatorics and other areas of mathematics (surveyed in my paper [43]). These current trends in mathematics are due to the development of computer science, on the one hand (which stimulated research on the structures and problems relevant to it, and gave rise to experimental mathematics), and to developments within mathematics itself, on the other hand. Indeed, as A. Björner and R. Stanley point out in [6], “after an era where the fashion was to seek generality and abstraction, there is now much appreciation for concrete calculations, which are the hard problems”. The increasing role of combinatorics is due to the fact that combinatorial structures are particularly well suited for encoding complex mathematical objects, while combinatorial methods are well suited for related computations. Combinatorial structures are also very well suited for computer experiments, which usually give a better insight into mathematical phenomena. Computation based on combinatorial structures is a major part of my research, from the stage of computer experimentation (which I am actively pursuin