Research Area: Mathematical Logic
Research Profile: Mathematical logic,universal algebra and computational complexity
Mathematical logic,universal algebra and computational complexity I am involved in the study and classification of general algebraic systems. This area of mathematics is often called Universal Algebra and got its start in the 1930s. In order to compare and classify algebras they are often grouped together according to the equations that they satisfy.
Borrowing and expanding on techniques and ideas from mathematical logic, classical abstract algebra, and also from newer branches of mathematics such as lattice theory and category theory, powerful tools have been developed to help organize and understand the structure of varieties (classes of algebras defined by equations) and the algebras they contain. Recent advances in the field have opened up a new area of study dealing with the local structure of finite algebras. This new local theory of finite algebras has not only been useful in solving several longstanding problems but it has also suggested a number of new and challenging research problems.
My current research program involves studying the computational complexity of subclasses of the Constraint Satisfaction Problem (CSP). Many well known complexity problems, such as graph coloring or Boolean satisfiability, can be naturally presented within the vast CSP framework. Recent work of Bulatov, Jeavons, Krokhin and others has established a strong connection between the CSP and universal algebra and some of the important open problems in the field can be expressed in purely algebraic terms.
Mathematical logic and universal algebra
On Research Leave