Research Overview for a General Audience

My research is concerned with the development of new computational models, based on combinatorial structures, for various areas in algebra and geometry. This type of research is at the forefront of current mathematical research, being related to developments within mathematics itself (the current emphasis is more on computation than on generality and abstraction), to the advent of computers, as well as to recent applications to theoretical physics.

At the very heart of mathematics lies the study of certain geometric objects, such as algebraic varieties and topological spaces. Since this study is usually very hard, mathematicians associate certain algebraic structures to the geometric ones, such as symmetry groups and homology groups; the latter capture the essence of the geometric information. Various tools have been developed for studying these algebraic structures, such as representation theory, which is a fundamental tool for studying group symmetry by means of linear algebra (i.e., matrices). Although the passage to algebra is an important simplification, even calculations in this setup are usually very hard. It is here that combinatorics comes into the picture. Combinatorics could be described as the study of arrangements of objects according to specified rules; usually, simple rules can give rise to complex discrete structures, such as various types of puzzles. It turns out that combinatorial objects are particularly well suited for encoding complex algebraic or geometrical objects, while combinatorial methods are well suited for related computations. Combinatorial structures are also well suited for experiments using computer algebra systems; such experiments play an important part in my research, particularly during the stage of discovering certain formulas, properties etc.

My work is concerned with all the aspects mentioned above. More precisely, I am using combinatorial structures/methods to study certain algebraic varieties, certain topological spaces, and the representation theory of certain Lie algebras. I am particularly interested in flag varieties and Schubert varieties, which have many applications to representation theory and enumerative problems in geometry, being also a very useful testbed for a much larger class of varieties. My work involving representation theory is related to several recent developments, such as those involving quantum groups and their applications to theoretical physics.

A recent breakthrough, which started as a joint project with a colleague at MIT, consists of developing a new combinatorial model with many applications to algebra and geometry.

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany