Research Statement

Research Statement


I'm a pure mathematician, whose main research interests lie in categorical fuzzy topology, i.e., in application of methods of category theory to investigation of different fuzzy topological structures. Since the notion of fuzziness is based in various algebraic concepts such as, e.g., quantales, semi-quantales, frames, etc., I'm interested in the study of topological properties generated by them. More precisely, I'm trying to develop the theory of fuzzy topology, induced by an arbitrary variety of algebras, which has been recently called categorically-algebraic (catalg) topology. This new framework incorporates the most important approaches to fuzzy topology and provides convenient means of interaction between different theories. The main advantage of the new setting is the fact that the catalg framework ultimately erases the border between traditional and fuzzy developments, producing a theory, which underlines the algebraic essence of the whole (not only fuzzy) mathematics, thereby propagating algebra as the main driving force of modern exact sciences. The proposed machinery exploits the catalg version of the notion of topological system. This generalization provides a wide range of applications in different areas of computer science, mathematics and physics. For example, the concept is closely related to the so-called state property system, introduced to serve as the basic mathematical structure in the Geneva-Brussels approach to foundations of physics. On the other hand, contexts of formal concept analysis can also be put under the catalg roof.

Lately, I have turned my attention to exploring relationships between non-commutative topology (motivated by quantum structures and developed in the framework of C*-algebras or, more recently, quantales) and fuzzy topology, motivated by the challenge of finding a link between quantumness and fuzziness. The interest in that area arose from some results on quantale-like structures, e.g., quantale modules and algebras, obtained in my PhD thesis and further publications. The investigation gave rise to the concept of quantale algebroid, generalizing the notion of quantaloid, the latter having numerous applications in theoretical computer science, and, moreover, recently claimed to serve as a category-theoretic basis for many-valued mathematics.

My current research is related to a particular branch of categorical topology (called monoidal or lax-algebraic topology), which took its origin in the representation of the category of topological spaces and continuous maps as the category of lax algebras and lax homomorphisms for the canonical extension of the ultrafilter monad on the category of sets and maps to the category of sets and relations. Monoidal topology is based in the categorical concept of monad and the algebraic notion of quantale.