My research 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 "The many faces of modern combinatorics"). 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 "A Combinatorial Miscellany", 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 it provides tools which, compared to classical ones, are often better suited for computations related to geometric and algebraic structures with a high degree of complexity. Combinatorial tools are also very well suited for computer experiments, which usually give a better insight into mathematical phenomena. Computer experiments play an important role throughout my research, particularly in discovering and testing formulas and algorithms.
A considerable part of my research is in modern Schubert calculus, which is concerned with enumerating geometric objects satisfying prescribed incidence constraints. The simplest example is: given 4 straight lines in general position in complex 3-space, how many lines meet all of them? (The answer is 2.) The basic questions in enumerative geometry were reformulated in the framework of intersection theory of flag varieties, and thus reduced to computations in the appropriate cohomology rings. Modern Schubert calculus is concerned with methods of performing these computations efficiently, in particular finding explicit combinatorial formulas for intersection numbers. Flag varieties have been the subject of considerable interest recently due to the subtle interplay between various areas connected to these spaces, such as: algebraic geometry, representation theory of Lie groups and their Weyl groups, algebraic topology, commutative algebra, and algebraic combinatorics.
Another part of my research, related to formal group laws, is underlied by the same idea of doing concrete computations, with applications to problems in topology. Formal group laws represent a powerful machinery used by topologists to classify topological spaces. Studying the combinatorics of formal group laws brings new insights to this area.
One of the main aims of my research in this area was to develop efficient clustering algorithms with fuzzy sets. Thus, I worked on several extensions of the Fuzzy c-Means clustering algorithm due to J. C. Bezdek, and I developed a package for clustering by implementing several classical algorithms and some of my own. I also worked on a number of theoretical problems related to clustering, such as: a generalized distance in graphs inspired by a clustering problem, defining separability of two fuzzy clusters etc. The most recent applications I worked on were to logical analysis of data LAD (joint with P. Hammer at Rutgers), and DNA sequencing (joint with B. Berger's group at MIT). For instance, in the latter project, I used the clustering package I developed to identify certain positions in a DNA sequence that turned out to be important for the detection of introns and exons.
My plans for the future in this area are to find further applications of clustering algorithms to computational biology and other areas of interest. I am also interested in applications of combinatorial algorithms and topology to computational biology.