Asymptotic Representation Theory, Enumeration, and Combinatorial Probability

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The project concerns problems from several areas of pure mathematics such as number theory, the study of the integers and their properties; representation theory, the study of symmetry; and probability theory, the mathematical study of randomness. These problems appear to have little in common with each other, nor do they seem to have any obvious applications, and yet a deeper look reveals many subtle connections between them and possible applications to other areas of applied and theoretical science. For example, the question "how many alternating sign matrices of order N exist?" which was asked by mathematicians in the 1980s in connection with an obscure algorithm proposed by the logician Charles Dodgson (aka Lewis Carroll) in the 19th century turned out to have connections to "square ice," an exotic form of water ice that was detected experimentally a few years ago by British researchers. Similarly, the Witten zeta function, which the investigator plans to study in connection with the enumeration of certain types of symmetry, was previously studied by other researchers (including the physicist Edward Witten) in a different context related to problems in quantum field theory. It is these sorts of connections that the investigator hopes to shed light on through his research, and which make pure mathematics such a fruitful area of study for the advancement of human knowledge.The specific problems the investigator proposes to study are questions at the interface of enumerative, algebraic and asymptotic combinatorics, with connections to other branches of mathematics, notably representation theory, probability theory, asymptotic analysis, and number theory. For example, one problem involves a new direction in asymptotic representation theory, namely that of finding asymptotic formulas for the number of n-dimensional representations of a Lie group as n grows large. The investigator's recent paper solved this problem for the group SU(3) using a difficult analysis of the so-called Witten zeta function associated with the group, which highlighted some intriguing connections between the problem and seemingly unrelated questions in analytic number theory and the theory of modular forms. In an additional part of the project, the investigator proposes to author a monograph on alternating sign matrices, which have been the subject of great interest since their discovery in the 1980s and are related to many contemporary research topics in combinatorics, probability theory and statistical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.