Probabilistic and Extremal Combinatorics

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The majority of this research is in the area of probabilistic combinatorics, which is comprised of the study of random combinatorial structures, randomized algorithms and probabilistic existence proofs for combinatorics (i.e. the probabilistic method). In recent years, this field has seen the development of a technique for proving that key statistics of a discrete stochastic process are likely to remain close to their expected trajectories as the process evolves. This method is known as the differential equations method for greedy algorithms and random graph processes. The investigator recently extended this technique to analyze the triangle-free process, a `controlled' random graph process that produces an interesting distribution on the space of triangle-free graphs. The investigator showed that a graph drawn from this distribution is likely to have independence number within a constant multiplicative factor of the smallest possible. In other words, the triangle-free process produces a Ramsey R(3,t) graph. The investigator shall continue the development of this method for the study of related `controlled' random processes, with an emphasis on processes that are interesting from the perspective of extremal combinatorics. Related questions are motivated by computer science. Here the investigator attempts to develop applications of the differential equations method to prove good performance of randomized algorithms without an explicit solution, or numerical approximation of the solution, of the associated differential equation (the need for such information often significantly complicates application of this method).This research project is an investigation of discrete mathematical objects, like networks or codes, whose evolution over time is determined by a sequence of random choices. We are particularly interested in processes where there is dependence between the choices made in different rounds. While such processes can be good models for the dynamics of real-world phenomenon, like the spread of a disease in a network or phase transitions in materials, the focus of this research is on processes that generate interesting mathematical objects. Randomness has long played an important role in the construction of sophisticated combinatorial objects, just as randomness plays an a central role in many algorithms in computer science. This research is expected to have an impact in multiple domains as the investigator shall develop general tools for understanding the evolution of these systems over time.