Algebraic and relational structures

点击翻译按钮获取中文摘要

Algebraic and relational structures arise in many different contexts, not only within mathematics, but in most of the sciences, and even in day to day life. For example, algebras known as groups are used to represent the symmetries of crystals and other physical systems and Boolean algebras are used to codify rules of logic. Many other common systems, in particular those arising from questions in computer science can be viewed in an algebraic way. Thus research on algebraic and relational structures can have an impact on several branches of the sciences and in particular on computer science.***  *Several important properties of algebraic systems can be expressed via equations. For example, the fact that addition is commutative is expressed by the equation x + y = y + x. It has proved useful to organize and classify algebraic systems according to the equations that they satisfy. These equationally defined classes of algebras, called varieties, are objects that I study in my research. I am interested in understanding the connection between various types of complexities that can arise in varieties and the structure of the algebras that belong to them.***  *A more recent direction that my research has taken relates to a problem from computational complexity. An important class of***problems, known as constraint satisfaction problems (CSPs), has a natural expression in terms of finite algebras and relational***structures. CSPs are ubiquitous in many areas of artificial intelligence, computer science, and discrete mathematics, such as***database theory, scheduling, and networking. A system of linear equations can be viewed as a special type of CSP. I am investigating a conjectured correlation between manageable complexity of subclasses of CSPs and the structure of the corresponding algebraic and relational systems.