The Principal Investigator (PI) will study a series of interrelated problems in enumerative combinatorics and probabilistic method. In recent years deep and unexpected connections have been found between algebraic enumeration and stochastic processes, in particular in the empirical process and in Brownian motion. The primary objective of this research is to explore this connection. The starting point is the probabilistic model of branching processes. On one hand, branching processes encode various combinatorial structures such as rooted trees, parking functions, and multi-colored structures. On the other hand, random graph evolution in the double jump" can be described by branching processes with the expected family size near one. From the algebraic standpoint, the PI plans to address the combi- natorial applications in branching processes and probabilistic results in combinatorics. She will investigate the behavior of branching processes with near critical offspring distributions by applying well-developed techniques in algebraic enumeration. She will also study the asymptotic properties of certain random structures from the probability theory of branching processes. From the stochastic standpoint, the PI plans to develop stochastic models for the theory of random graphs and random generated trees via branching processes, and to relate such models to the theory of Brownian motion. The emphasis is a combinatorial under- standing of the techniques and results of stochastic processes and calculus. The PI expects to apply the model to graph enumeration, and to investigate the distributive asymptotics of random graphs and other random structures.
In addition to the core research program outlined above, the PI intends to pursue three algebraic problems arising from the probabilistic method and extremal combinatorics. The first is a recurrence associated with Turan problems; the second is on discrepancy theory, and the third is on balancing vectors. The objective here is to extend our knowledge of algebraic structures on discrete systems and to develop new approaches in combinatorics.
