IN ECONOMICS:
My main research interest is in microeconomics theory, which has five pillars:
Decision theory,
Game theory,
Social choice theory & Welfare economics,
General equilibrium theory,
Mechanism design theory
In principle, I am equally interested in these five, but thus far I mostly focused on 3 and in the near future, I am hoping to get some research done in the other four as well. For instance, the main theme in 1 is to model decision making situations, and the underlying subject is a human being or something similar to it, e.g. a computing machine. Three concepts play key role: (1) Utility, (2) Rationality and (3) Uncertainty and each has a philosophical and mathematical content.
IN MATHEMATICS:
At the moment, I am interested in discrete geometry, a branch of mathematics which covers the following topics: tiling, polygons and polyhedrons (i.e. Polyhedral combinatorics), dissections, packing problems, hyperplane arrangements, graph drawing and the Euclidean Ramsey theory. Here are examples of famous open problems in this field:
Voronoi conjecture
In polyhedral combinatorics the Hirsch conjecture has recently been disproved. Another topic that I am engaged to is graph theory. Some major open problem in this field are
In addition to discrete mathematics, I find geometry, mathematical logic, set theory, philosophy and foundations of mathematics, probability theory and optimization theory as fascinating areas!
IN APPLIED MATH:
Some of the other stuff that I am trying to do are Traffic Modeling and Geostatistics (as far as I know there is no immediate relation between these two fields). For the former, I need to learn basics of "Traffic (flow) Theory" which involves mathematical techniques such as PDE, and some simulation methods e.g. cellular automata. The latter is a part of statistical theory which addresses to spatial (or regional) variables. Actually, it might have some connections to Spatial Econometrics, a field which I know (perhaps a little) from a course I took several years ago. It is funny enough that Spatial Econometrics is a highly used tool in Urban Economics, and on other hand, traffic flow theory is also related to Urban Economics. Hence, there is an indirect connection between traffic theory and geostatistics.