Grant: Discovery Grants Program - Individual

This research weaves together two strands, one theoretical and one applied. On the theoretical side, this programme explores identities of symmetric functions from algebraic combinatorics. These identities, as well as a host of related identities in similar algebras, play a significant role in theoretical physics, algebraic geometry, and representation theory as well as combinatorics. Thus finding such identities is crucial for all these subjects. On the applied side this programme explores the security of anonymity networks. Anonymity networks--for example, Tor--conceal information about the relationship between senders and receivers. They are already a vital means of communication for political dissidents, journalists, and law enforcement officials, and as the common person's concern with electronic privacy grows, they will become more widespread. But how secure are they? This research uses techniques from combinatorics--jeu de taquin, lattice paths, and other bijective constructions--to derive new identities involving Schur symmetric functions. It also extends these results to other types of symmetric functions (e.g. Q-functions, Jack functions, Macdonald polynomials). These results will be particularly prized in physics in random matrix theory. Furthermore, solving these problems will require constructing new combinatorial techniques that will be applicable to a range of other combinatorial problems. This research also uses symmetric functions to derive practical metrics for anonymity networks, telling Alice how likely it is that an adversary can discover she is communicating with Bob. The applied goal is a spinoff of the theoretical goals. Simpler versions of the problem are solvable by classical enumerative techniques, but the full nature of the problem demands symmetric function theory from algebraic combinatorics, which goes beyond the normal tool chest of a computer scientist. At the same time, the problems are outside the domain of most combinatorialists, so being at the intersection of these two areas is an advantage for me and will lead to novel and significant results.