The investigator will study combinatorial models that arise in connection with certain structures on two-dimensional lattices, such as domino tilings, Young tableaux and alternating-sign matrices. One focus of the research will be enumeration questions, where the goal is to derive exact formulas for the number of combinatorial objects satisfying certain properties. Another emphasis will be on a probabilistic analysis of random combinatorial objects, and in particular their limit shapes, which are geometric shapes that arise in the asymptotic limit when the size of the model becomes large. More generally, the study of combinatorial models in a probabilistic setting raises many fascinating questions, for example questions about limiting fluctuations and about connections to random matrix theory. The technical toolbox of techniques that the investigator hopes to employ to attack such problems is very varied and involves both algebraic methods (e.g. generating functions, linear algebra), analytic methods (for example the calculus of variations) and general probabilistic techniques.
The scientific and educational value of the project is manifold. The combinatorial objects of interest have delighted pure mathematicians for many years for their inherent beauty and elegant structure. But also, amazingly, such seemingly "useless" mathematics has turned out to have deep connections to branches of physics and to some very applied mathematical disciplines like probability theory and random matrix theory. Thus, for example, alternating-sign matrices are related to "square ice", a simplified model for an ice crystal studied by statistical physicists, while at the same time appearing in the study of an algorithm for computing matrix determinants, which are a ubiquituous concept used in practically all the sciences and engineering disciplines. And Young tableaux, which originated with the mathematical study of symmetry, have been found to be related to random matrix theory, an important mathematical theory which has its origins in attempts by nuclear physicists to model the complex interactions in the nuclei of heavy elements. The combination of theoretical elegance and applied utility makes these research topics highly attractive from a scientific standpoint, and also very suitable as the focus for educational activities planned by the investigator, which would have as their goals to attract promising students to mathematics and the sciences and to promote public appreciation of the value of scientific research.
